Cayley Diagrams of Small Groups

This page gives the Cayley diagrams, also known as Cayley graphs, of all groups of order less than 32. Their presentations are also given. The letters in the presentations correspond to the colours in the Cayley diagrams: black red green blue mauve grey.

The first and third columns, from order 4 onwards, contain links to pages with more and better Cayley diagrams.

Notation

N ⋊ H indicates a semidirect product of N by H. N is the normal subgroup.

QN, DN and DicN denote groups of order N (the quaternion, dihedral and dicyclic groups respectively).

Cyclic groups are denoted by C.

Contents

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31

Order   Name Presentation
generators as permutations		 Cayley diagram Orders of elements
Centre
Derived subgroup

Automorphisms
GAP no., name
(Sylow subgroup)
Schur mulitplier
1 Abelian 1 <> 1
1
1

1/1=1
1, 1

1
2 Abelian C2 < k | k2 >
k=(ab)		 1.2
C2
1

1 / 1 = 1
1, C2

1
3 Abelian C3
≅ A3 		< k | k3 >
k=(abc) 		1.32
C3
1

C2 / 1 = C2
1, C3

1
4 Abelian C4
< k | k4 >
k=(abcd) 		1.2.42
C4
1

C2 / 1 = C2
1, C4

1
C2 × C2 < r, g | r2, g2, rgrg >
r=(ab)(cd) g=(ac)(bd) b=(ad)(bc) 		1.23
C2×C2
1

D6 / 1 = D6
2, C2 x C2

2
5 Abelian C5 < k | k5 >
k=(abcde) 		1.54
C5
1

C4 / 1 = C4
1, C5

1
6 Abelian C6
C3 × C2 		< k | k6 >
k=(abcdef)

< k,r | k3, r2, krk-1r >
k=(abc) r=(de)

1.2.32.62
C6
1

C2 / 1 = C2
2, C6

1
Other D6
≅S3
≅ C3 ⋊ C2 		< k,r | k3, r2, krkr >
k=(abc) r=(bc) 		1.23.32
1
C3

D6 / D6 = 1
1, S3

1
7 Abelian C7 < k | k7 >
k=(abcdefg) 		1.76
C7
1

C6 / 1 = 1
1, C7

1
8 Abelian C8 < k | k8 >
k=(abcdefgh) 		1.2.42.84
C8
1

C22 / 1 = C22
1, C8

1
C4 × C2 < k,r | k4, r2, krk-1r >
k=(abcd) r=(ef) 		1.21+2.44
C4×C2
1

D8 / 1 = D8
2, C4 x C2

2
C2 × C2 × C2 < r,g,b | r2, g2, b2, rg2, gb2, rb2 >
r=(ab) g=(cd) b=(ef) 		1.27
C2×C2×C2
1

PSL(3,2) / 1 = PSL(3,2)
5, C2 x C2 x C2

2x2x2
Other D8
= C4 ⋊ C2 		< k,r | r4, r2, krkr >
k=(abcd) r=(ac)

< r,g,b | b2, g2, r2, bgbg, rbrg >
b=(ab)(cd) g=(ac)(bd) r=(bc)

1.21+4.42
C2
C2

D8 / C22 = C2
3, D8

2
Q8
a.k.a. Dic8 		< r,b | r4, b4, rbrrrb, rrbb >
b=(abcd)(ehgf) r=(afch)(bgde)

< r,g | r4, g4, rgrrrg,rrgg >
r=(abde)(fhcg) g=(acdf)(egbh) b=(bcef)(agdh)

1.2.46
C2
C2

S4 / C22 = D6
4, Q8

1
9 Abelian C9 < k | k9 >
k=(abcdefghi) 		1.32.66
C9
1

C6 / 1 = C6
1, C9

1
C3 × C3 < k, r | k3, r3, krkkrr >
k=(abc) r=(def) 		1.38
C3×C3
1

GL(2,3) / 1 = GL(2,3)
2, C3 x C3

3
10 Abelian C10
≅ C5 × C2 		< k | k10 >
k=(abcdefghij)

< k,r | k5, r2, krk-1r  >
k=(abcde) r=(fg)

1.2.54.104
C10
1

C4 / 1 = C4
2, C10
1
Other D10
= C5⋊C2 		< k,r | k5, r2, krkr >
k=(abcde) r=(be)(cd) 		1.25.54
1
C5

C5⋊C4 / D10 = C2
1, D10
1
11 Abelian C11 < k | k11 >
k=(abcdefghijk) 		1.1110
C11
1

C10 / 1 = C10
1, C11

1
12 Abelian C12
≅ C4 × C3 		< k | k12 >
k=(abcdefghijkl)

< k, r | k3, r4, krkkrrr >
k=(abc) r=(defg)

1.2.32.42.62.124
C12
1

C22 / 1 = C22
2, C12
C4
1
C6 × C2
≅ C3 × C2 × C2 		< k,r | k6, r2, krkkkkkrr >
k=(abcdef) r=(gh) 		1.23.32.66
C6×C2
1

D12 / 1 = D12
5, C6 x C2
C22
2
Other direct products D12
= D6 ⋊ C2
≅ D6 × C2 		< k,r | k6, r2, krkr >
k=(abcdef) r=(bf)(ce)

< k,r | k3, r2, g2, krkr, kgkkg, rgrg >
k=(abc) r=(bc) g=(de)

1.21+6.32.62
C2
C3

D12 / D6 = C2
4, D12
C22
2
Other Dic12
≅ C3 ⋊ C4 		< b,r | b6, r4, b3r2, brbrrr >
b=(abc)(pr)(qs) r=(bc)(pqrs) 		1.2.32.46.62
C2
C3

D12 / D6 = C2
1, C3 : C4
C4
1
A4
= (C2×C2) ⋊ C3 		< k,r | k3, r2, (kr)3 >
k=(abc) r=(ab)(cd)
The first diagram resembles a truncated tetrahedron, whose rotational symmetry group is A4.

< r,b,e | r2, g2, e3, rgrg, geree, rgegee >
r=(ab)(cd) g=(ac)(bd) e=(bcd)
The second diagram shows more clearly how the C3 (grey) acts on the C2×C2 (colour).

1.23.38
1
C22

S4 / A4 = C2
3, A4
C22
2
13 Abelian C13 < k | k13 >
k=(abcdefghijklm) 		1.1312
C13
1

C12 / 1 = C12
1, C13

1
14 Abelian C14
≅ C7 × C2 		< k | k14 >
k=(abcdefghijklmn)

< k,r | k7, r2, krk6r2 >
k=(abcdefg) r=(pq)

1.2.76.146
C14
1

C6 / 1 = C6
2, C14

1
Other D14
= C7 ⋊ C2 		Here are three ways of drawing a Cayley diagram for D14. The first one I regard as usefully reflecting the structure of the group. The other two are given to show that it is possible to draw them like this, and omitted for other dihedral groups.

< k,r | k7, r2, krkr >
k=(abcdefg) r=(bg)(cf)(de)

< r,g | r2, g2, (rg)7 >
r=(bg)(cf)(de) g=(af)(be)(cd)

< k,r,g | k7, r2, g2, kgr >
k=(abcdefg) r=(bg)(cf)(de) g=(ag)(bf)(ce)

1.27.76
1
C7

C7⋊C6 / D14 = C3
1, D14

1
15 Abelian C15
≅ C5 × C3 		< k | k15 >
k=(abcdefghijklmno)

< k,r | k3, r5, krkkrrr >
k=(abcde) r=(mno)

1.32.54.158
C15
1

C4×C2 / 1 = C4×C2
1, C15

1
16 Abelian C16
 < k | k16 >
k=(abcdefghijklmnop) 		1.2.42.84.168
C16
1

C4×C2 / 1 = C4×C2
1, C16

1
C4 × C4 < k,r | k4, r4, krk-1r-1 >
k=(abcd) r=(efgh) 		1.23.412
C4×C4
1

(C22×A4)⋊C2 / 1 = (C22×A4)⋊C2
2, C4 x C4

2
C4 × C2 × C2 < r,g,e | r2, g2, e4, rgrg, rere-1, gege-1 >
r=(ab) g=(cd) e=(efgh) 		1.21+6.48
C4×C2×C2
1

"(((D8×C2)⋊C2)⋊C3)⋊C2" / 1 = ?
10, C4 x C2 x C2

2x2x2
C2 × C2 × C2 × C2 < r,g,b,m | r2, g2, b2, m2, rgrg, rbrb, rmrm, gbgb, gmgm, bmbm >
r=(ab) g=(cd) b=(ef) m=(gh) 		1.215
C2×C2×C2×C2
1

A8 / 1 = A8
14, C2 x C2 x C2 x C2

2x2x2x2x2x2
C8 × C2 < k,r | k8, r2, krk-1r >
k=(abcdefgh) r=(ij)

Compare this diagram with those for D16, "Modular", and "Quasidihedral" below. These four diagrams show C2 acting in all four possible ways on C8, whose automorphism group is C2×C2.

1.23.44.88
C8×C2
1

D8×C2 / 1 = D8×C2
5, C8 x C2
2
Other direct products D8 × C2 < k,r,g | k4, r2, g2, krkr, kgkkkg, rgrg >
k=(abcd) r=(ac) g=(pq) 		1.21+2+8.44
C2×C2
C2

"(((C4×C2)⋊C2)⋊C2)⋊C2" / C22 = ?
11, C2 x D8

2x2x2
Q8 × C2 < r,b,g | r4, b4, gr, rrbb, rgr-1g, bgb-1g >
r=(abcd)(efgh) b=(aecg)(fbhd) g=(pq) 		1.21+2.412
C2×C2
C2

"(((C24)⋊C3)⋊C2)⋊C2" / C22 = ?
12, C2 x Q8

2
2x2
Other D16
= C8 ⋊ C2
 < k,r | k8, r2, krkr >
k=(abcdefgh) r=(bh)(cg)(df) 		1.21+8.42.84
C2
C4

"(D8×C2)⋊C2" / D8 = C22
7, D16

2
Modular
= C8 ⋊ C2 		< k,r | k8, r2, krkkkr >
k=(abcdefgh) r=(bf)(dh)

< k,r | k8, r8, krkr >

1.21+2.44.88
C4
C2

D8×C2 / C22 = C22
6, (C4 x C2) : C2

1
Quasidihedral, a.k.a. semidihedral
= C8 ⋊ C2 		< k,r | k8, r2, krkkkkkr >
k=(abcdefgh) r=(bd)(cg)(fh) 		1.21+4.46.84
C2
C4

D8×C2 / D8 = C2
8, QD16

1
Dic16

a.k.a. Q16 		< b,r | b8, r4, b4r2, (br)4, rbr3b >
b=(abcdefgh)(pqrstuvw) r=(apet)(bwfs)(cvgr)(duhq) 		1.2.410.84
C2
C4

(D8×C2)⋊C2 / D8 = C22
9, Q16

1
C4 ⋊ C4 < k,r | k4, r4, krkr3 >
k=(abcd) r=(bd)(efgh) 		1.23.412
C2×C2
C2

C24⋊C2 / C22 = ?
4, C4 : C4

2
(C2 × C2) ⋊ C4 < r,g,e | r2, g2, e4, rgrg, ereeeg, egeeer >
r=(ab)(cd) g=(ac)(bd) e=(bc)(pqrs) 		1.23+4.48
C2×C2
C2

C24⋊C2 / C22 = ?
3, (C4 x C2) : C2

2x2
Pauli
= D8 ⋊ C2
= Q8 ⋊ C2
= (C4×C2) ⋊ C2 		< k,r,b | k4, r2, b2, krkr, kbkkkb, kkrbrb >
k=(abcd)(efgh) r=(bd)(eg) b=(ae)(bf)(cg)(dh)

The C2 acts on the D8 by the permutation (b,aab),(ab,aaab).

The upper diagram may make it clearer what happens: there are two black-and-red D8s, with the inner square of one rotated through π relative to the other. The lower diagram is prettier, having some of the symmetry of a cube.

 1.27.48
C4
C2

S4×C2 / C22 = ?
13, C4 : C4

2x2
17 Abelian C17 < k | k17 >
k=(abcdefghijklmnopq) 		1.1716
C17
1

C16 / 1 = C16
1, C17

1
18 Abelian C18
= C9 × C2 		< k | k18 >
k=(abcdefghijklmnopqr)

< k,r | k9, r2, krk-1r-1 >
k=(abcdefghi) r=(mn)

1.2.32.62.96.186
C18
1

C6 / 1 = C6
2, C18
C9
1
C6 × C3
= C3 × C3 × C2 		< k,r | k6, r3, krk-1r-1 >
k=(abcdef) r=(jkl) 		1.2.38.68
C6×C3
1

GL(2,3) / 1 = GL(2,3)
5, C6 x C3
C32
3
Other direct products D6 × C3

≅ (C3 × C3) ⋊ C2
with the C2 interchanging the generators of the two C3s

< k,r,g | k3, r2, g3, krkr, kgkkgg, rgrgg >
k=(abc) r=(bc) g=(pqr)

< k,r,g | k3, r3, g2, krkkrr, grgkk, gkgrr >
k=(abc) r=(def) g=(ad)(be)(cf)

1.23.38.66
C3
C3

D12 / D6 = C2
3, C3 x S3
C32
3
Other D18 < k,r | k9, r2, krkr >
k=(abcdefghi) r=(bi)(ch)(dg)(ef) 		1.29.32.96
1
C9

"(C9⋊)⋊C2" / D18 = C3
1, D18
C9
1
(C3 × C3) ⋊ C2
with the C2 acting separately on the two C3s 		< k,r,g | k3, r3, c2, krkkrr, kgkg, rgrgr >
k=(abc) r=(def) g=(bc)(ef) 		1.29.38
1
C3×C3

"((C32⋊Q8)⋊C3)⋊C2" / C32⋊C2 = ?
4, (C3 x C3) : C2
C32
3
19 Abelian C19 < k | k19 >
k=(abcdefghijklmnopqrs) 		1.1918
C19
1

C18 / 1 = C18
1, C19
20 Abelian C20
 = C5 × C4 		< k | k20 >
k=(abcdefghijklmnopqrst)

< k,r | k5, r4, krk-1r-1 >
k=(abcde) r=(mnop)

1.23.54.1012
C10×C2
1

C4×C2 / 1 = C4×C2
2, C20
C4
1
C10 × C2
= C5 × C2 × C2 		< k,r | k10, r2, krk-1r-1 >
k=(abcdefghij) r=(mn)

There is little interest in Cayley diagrams of direct products. This one is like a man who walks round in circles (black), and takes his hat on and off (red). The hat does not affect the walking, and the walking does not affect the hat.

1.2.42.54.1012
C10×C2
1

D6×C4 / 1 = D6×C4
5, C10 x C2
C22
2
Other direct products D20
= C10 ⋊ C2
≅ D10 × C2 		< k,r | k10, r2, krkr >
k=(abcdefghij) r=(bj)(ci)(dh)(eg)

< k,b,g | k5, r2, g2, krkr, kgk-1g-1, rgr-1g-1 >
k=(abcde) r=(be)(cd) g=(pq)

1.21+10.54.104
C2
C5

"C2 x (C5 : C4)" / D10 = ?
4, D20
C22
2
Other Dic20
≅ C5 ⋊C2 C4 		< b,r | b5, r4, b5r2, brbrrr >
b=(abcde)(pr)(qs) r=(be)(cd)(pqrs)

< k,r | k5, r4, krkrrr >

1.2.410.54.104
C2
C5

C2×(C5⋊C2C4) / D10 = C2
1, C5 : C4
C4
1
Frob20
≅ C5 ⋊C4 C4 		< k,r | k5, r2, kgkkggg >
k=(abcde) r=(bd)(ce)

< k,g | k5, g4, (kr)5 >
k=(abcde) g=(bced)

1.25.410.54
1
C5

C5⋊C4C4 / C5⋊C4C4 = 1
3, C5 : C4
C4
1
21 Abelian C21
= C7 × C3 		< k | k21 >
k=(abcdefghijklmnopqrstu)

< k,r | k7, r3, krk-1r-1 >
k=(abcdefg) r=(pqr)

1.32.76.2112
C21
1

C6×C2 / 1 = C6×C2
2, C21

1
Other Frob21
≅ C7 ⋊ C3 		< k,r | k7, r3, krk5r2 >
k=(abcdefg) r=(bce)(dgf) 		1.314.76
1
C7

Frob42 / Frob21 / C2

1
22 Abelian C22
= C11 × C2 		< k | k22 >
k=(abcdefghijklmnopqrstuv)

< k,r | k11, r2, krk-1r-1 >
k=(abcdefghijk) r=(pq)

1.2.1110.2210
C22
1

C10 / 1 = C10
2, C22

1
Other D22
= C11 ⋊ C2 		< k,r | k11, r2, krkr >
k=(abcdefghijk) r=(bk)(cj()di)(eh)(fg) 		1.211.1110
1
C11

"(C11 : C5) : C2" / D22 = C5
1, D22

1
23 Abelian C23 < k | k23 >
k=(abcdefghijklmnopqrstuvw) 		1.2322
C23
1

C22 / 1 = C22
1, C23

1
24 Abelian C24
= C8 × C3 		< k | k24 >
k=(abcdefghijklmnopqrstuvwv)

< k,r | k8, r3, krk-1r-1 >
k=(abcdefgh) r=(mn)

1.2.32.42.62.84.126.246
C24
1

C23 / 1 = C23
2, C24
C8
1
C12 × C2
= C6 × C4
= C4 × C3 × C2 		< k,r | k12, r2, krk-1r-1 >
k=(abcdefghijkl) r=(pq)

< k,r | k6, r4, krk-1r-1 >
k=(abcdef) r=(pqrs)

1.23.32.44.66.128
C12×C2
1

D8×C2 / 1 = D8×C2
9, C12 x C2
C4×C2
2
C3 × C2 × C2 × C2
= C6 × C2 × C2
= C2 × C2 × C2 × C3 		< r,g,e | r2, g2, e6, rgrg rere-1, gege-1 >
r=(ab) g=(cd) e=(pqrstu)

< r,g,b,e | r2, g2, b2, e3, rgrg, gbgb, brbr, rere-1, gege-1, bebe-1, >
r=(ab) g=(cd) b=(ef) e=(ghi)

1.27.32.614
C3×C2×C2×C2
1

PSL(3,2)×C2 / 1 = PSL(3,2)×C2
15, C6 x C2 x C2
C23
2x2x2
Other direct products D12 × C2
= D6 × C2 × C2 		< k,r,g | k6, r2, g2, krkr, kgk-1g-1, rgr-1g-1 >
k=(abcdef) r=(bf)(ce) g=(pq) 		1.215.32.66
C2×C2
C3

S4×D6 / D6 = S4
14, C2 x C2 x S3
C23
2x2x2
D8 × C3 < k,r,g | k4, r2, g3, krkr, kgk-1g-1, rgr-1g-1 >
k=(abcd) r=(bd) g=(pqr) 		1.25.32.42.610.124
C3×C2
C2

D8×C2 / C22 = ?
10, C3 x D8
D8
2
D6 × C4 < k,r,g | k3, r2, g4, krkr, kgk-1g-1, rgr-1g-1 >
k=(abc) r=(bc) g=(pqrs) 		1.27.32.48.62.124
C4
C3

S3×C22 / S3 = ?
5, C4 x S3
C4×C2
2
Dic12 × C2
≅ C6 ⋊C2 C4 		< b,r,g | b6, r4, b3r2, g2, brbbbr, gbgb-1, grgr-1 >
b=(abc)(mo)(np) r=(bc)(mnop) g=(st)

< k,r | k6, r4, krkr3 >
k=(abcdef) r=(bf)(ce)(pqrs)

1.21+2.32.412.66
C22
C3

D8×D6 / D6 = ?
7, C2 x (C3 : C4)
C4×C2
2
Q8 × C3 < r,b,g | r4, b4, g3, rrbb, rgr-1g-1, bgb-1g-1 >
r=(abcd)(efgh) b=(aecg)(fbhd) g=(pqr)

These Cayley diagrams of direct products are tedious and uninformative.

1.2.32.46.62.1212
C3×C2
C2

S4×C2 / C22 = ?
11, C3 x Q8
Q8
1
A4 × C2 ≅ (C2×C2×C2) ⋊ C3 < k,r | k3, r2, (kr)3 central, (kr)6 >
k=(abc) r=(ab)(cd)(pq)

< k,r,g | k3, r2, g2, (kr)3, kgk-1g-1, rgr-1g-1 >
k=(abc) r=(ab)(cd) g=(pq)

< r,g,b,e | r2, g2, b2, e3, rgrg, gbgb, brbr, gere-1, bege-1, rebe-1 >
r=(ab) g=(cd) b=(rf) e=(ace)(bdf)

1.27.38.68
C2
C2×C2

S4 / A4 = C2
13, C2 x A4
C23
2
Other D24
= C12 ⋊ C2 		< k,r | k12, r2, krkr>
k=(abcdefghijkl) r=(bl)(ck)(dj)(ei)(fh) 		1.21+12.32.42.62.124
C2
C6

D8×C3 / D12 = C2
6, D24
D8
2
Dic24

< b,r | b12, r4, b6r2, brbrrr >
b=(abcdefghijkl)(mnopqrstuvwx) r=(asgm)(brhx)(cqiw)(dpjv)(eoku)(fnlt)

1.2.32.42+12.62.124
C2
C6

D8×C3 / D12 = C2
4, C3 : Q8
Q8
1
C3 ⋊ C8 < k,r | k3, r8, krkr7 >
k=(abc) r=(bc)(defghijk) 		1.2.32.42.62.812.124
C4
C3

D6×C22 / S3 = D6×C22
1, C3 : C8
C8
1
SL(2,3)
≅ Q8 ⋊ C3 		< k,r | k6, r4, krkrkr >
k=(abcdef)(gh) r=(gahd)(ecbf)

< r,b,g,e | r4, b4, g4, e3, rrbb, bbgg, ggrr, rbgrbg, rebege>

1.2.38.46.68
C2
Q8

S4 / A4 = C2
3, SL(2,3)
Q8
1
C3 ⋊ D8
 < k,r,g | k3, g4, r2, gkgggk, rgrg, rkr-1k-1 >
k=(abc) g=(ghij)(bc) r=(hj)		 1.21+2+6.32.46.66
C2
C6

D6×C22 / D12 = C2
8, (C6 x C2) : C2
D8
2
S4
≅ (C2×C2) ⋊ D6 		< k,r | k4, r2, (kr)3 >
k=(abcd) r=(ab)

< k,r | k3, r2, (kr)4 >
k=(abc) r=(cd)

< b,g,r,e | b2, g2, r2, e3, bgbg, rgrb, rbrg, ege2b, ebe2bg >
b=(hi)(jk) g=(hj)(ik) r=(ij) e=(hij)

1.23+6.38.46
1
A4

S4 / S4 = 1
12, S4
D8
2
25 Abelian C25 < k | k25 >
k=(abcdefghijklmnopqrstuvwxy) 		1.54.2520
C25
1

C20 / 1 = C20
1, C25

1
C5 × C5 < k,r | k5, r5, krk-1r-1>
k=(abcde) r=(fghij) 		1.524
C5×C5
1

GL(2,5) / 1 = GL(2,5)
2, C5 x C5
1
5
26 Abelian C26
= C13 × C2 		< k | k26 >
k=(abcdefghijklmnopqrstuvwxyz)

< k,r | k13, r2, krk-1r-1 >
k=(abcdefghijklm) r=(pq)

1.2.1312.2612
C26
1

C12 / 1 = C12
2, C26

1
Other D26
= C13 ⋊ C2 		< k,r | k11, r2, krkr >
k=(abcdefghijklm) r=(bm)(cl)(dk)(ej)(fi)(gh) 		1.213.1312
1
C13

"(C13 : C4) : C3" / D26 = C6
1, D26

1
27 Abelian C27 < k | k27 >
k=(abcdefghijklmnopqrstuvwxyzæ)

1.32.96.2718
C27
1

C18 / 1 = C18
1, C27

1
C9 × C3 < k,r | k9, r3, krk-1r-1 >
k=(abcdefghi) r=(pqr)

1.38.918
C9×C3
1

"C2 x (((C3 x C3) : C3) : C2)" / 1 = ?
2, C9 x C3

3
C3 × C3 × C3 < k,r,g | k3, r3, g3, krk-1r-1, rgr-1g-1, gkg-1g-1 >
k=(abc) r=(def) g=(ghi)

1.327
C33
1

Gl(3,3) / 1 = GL(3,3)
5, C3 x C3 x C3

3x3x3
Other C9 ⋊ C3 < k,r | k9, r3, krk5r-1 >
k=(abcdefghi) r=(beh)(cif) 		1.32+6.918
C3
C3

"((C3 x C3) : C3) : C2" / C32 = ?
4, C9 : C3

1
(C3 × C3) ⋊ C3 < k,r,g | k3, r3, g3, krk-1r-1, rgkkgg, krgrrgg >
k=(abc)(def)(ghi) r=(adg)(beh)(cfi) g=(bdi)(cge)

1.32+24
C3
C3

"(((C3 x C3) : Q8) : C3) : C2" / C32 = ?
3, (C3 x C3) : C3

3x3
28 Abelian C28
= C7 × C4 		< k | k28 >
k=(abcdefghijklmnopqrstuvwxyzæð)

< k,r | k7, r4, krk-1r-1 >
k=(abcdefg) r=(mnop)

1.2.42.76.146.2812
C28
1

C6×C2 / 1 = C6×C2
2, C28
C4
1
C14 × C2
= C7 × C2 × C2 		< k,r | k14, r2, krk-1r-1 >
k=(abcdefg) r=(pq) 		1.23.76.1418
C14×C2
1

D6×C6 / 1 = D6×C6
4, C14 x C2
C22
2
Other direct products D28
= C14 ⋊ C2
≅ D14 × C2 		< k,r | k14, r2, krkr >
k=(abcdefghijklmn) r=(bn)(cm)(dl)(ek)(fj)(gj)

< k,r,g | k7, r2, g2, krkr, kgk-1g-1, rgr-1g-1 >
k=(abcdefg) r=(bg)(cf)(de) g=(pq)

1.21+2+12.76.146
C2
C7

(C7⋊C6)×C2 / D14 = C6
3, D28
C22
2
Other Dic28
≅ C7 ⋊C2 C4 		< b,r | b14, r4, b7r2, brbrrr >
b=(abcdefg)(pr)(qs) r=(bg)(cf)(de)(pqrs) 		1.2.414.76.146
C2
C7

(C7⋊C6)×C2 / D14 = C6
1, C7 : C4
C4
1
29 Abelian C29 < k | k29 >
k=(abcdefghijklmnopqrstuvwxyzæðñ) 		1.2928
C29
1

C28 / 1 = C28
1, C29

1
30 Abelian C30
= C15 × C2
= C10 × C3
= C6 × C5
= C5 × C3 × C2
 < k | k30 >
k=(abcdefghijklmnopqrstuvwxyzæðñç)

< k,r | k15, r2, krk-1r-1 >
k=(abcdefghijklmno) r=(pq)

< k,r | k10, r3, krk-1r-1 >
k=(abcdefghij) r=(klm)

< k,r | k6, r5, krk-1r-1 >
k=(abcdef) r=(ghijk)

1.2.32.54.62.104.158.308
C30
1

C4×C2 / 1 = C4×C2
4, C30

1
Other direct products D10 × C3 < k,r,g | k5, r2, g3, krkr, kgk-1g-1, rgr-1g-1 >
k=(abcde) r=(be)(cd) g=(pq) 		1.25.32.54.610.158
C3
C5

(C5⋊C4)×C2 / D10 = C22
2, C3 x D10

1
D6 × C5 < k,r,g | k3, r2, g5, krkr, kgk-1g-1, rgr-1g-1 >
k=(abc) r=(bc) g=(pqrst) 		1.23.32.54.1012.158
C5
C3

D6×C4 / D6 = C4
1, C5 x S3

1
Other D30
C15 ⋊ C2 		< k,r | k15, r2, abab >
k=(abcdefghijklmno) r=(bo)(cn)(dm)(el)(fk)(gj)(hi) 		1.215.32.54.158
1
C15

(C5⋊C4)×D6 / D6 = ?
3, D30

1
31 Abelian C31 < k | k31 >
k=(abcdefghijklmnopqrstuvwxyzæðñçþ) 		1.3130
C31
1

C30 / 1 = C30
1, C31

1

Other pages giving Cayley diagrams of groups with fewer than 32 elements:

Some more Cayley diagrams
Some more pages on groups

Copyright N.S.Wedd 2007