Cayley Diagrams of Small Groups

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

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.
Commutator subgroup.
1 Abelian 1 <> 1

1

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

C2

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

C3

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

C4

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

C2×C2

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

C5

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

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

1.2.32.62

C6

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

1

C3
7 Abelian C7 < k | k7 >
k=(abcdefg) 		1.76

C7

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

C8

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

C4×C2

1
C2 × C2 × C2 < r,g,b | r2, g2, b2, rgrg, gbgb, rbrb >
r=(ab) g=(cd) b=(ef) 		1.27

C2×C2×C2

1
Other D8
= C4 ⋊ C2 		< k,r | r4, r2, krkr >
k=(abcd) r=(ac) 		1.21+4.42

C2

C2
Q8
a.k.a. Dic8 		< r,b | r4, b4, rbrrrb, rrbb >
r=(abcd)(efgh) g=(ahcf)(bgde)

< r,g | r4, g4, rgrrrg,rrgg >
r=(abcd)(efgh) b=(aecg)(fbhd)

1.2.46

C2

C2
9 Abelian C9 < k | k9 >
k=(abcdefghi) 		1.32.66

C9

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

C3×C3

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

1

C5
11 Abelian C11 < k | k11 >
k=(abcdefghijk) 		1.1110

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
C6 × C2
≅ C3 × C2 × C2 		< k,r | k6, r2, krkkkkkrb >
k=(abcdef) r=(gh) 		1.23.32.66

C6×C2

1
Other direct products D12
= C6 ⋊ C2
≅ D6 × C2 		< k,r | k6, rrsup>2, 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
Other Dic12
≅ C3 ⋊ C4 		< b,r | b6, r4, brbbbr >
b=(abc)(pr)(qs) r=(bc)(pqrs) 		1.2.32.46.62

C2

C3
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

C2×C2
13 Abelian C13 < k | k13 >
k=(abcdefghijklm) 		1.1312

C13

1
14 Abelian C14
≅ C7 × C2 		< k | k14 >
k=(abcdefghijklmn) 		1.2.76.146

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)2 >
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
15 Abelian C15
≅ C5 × C3 		< k | k15 >
k=(abcdefghijklmno)

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

1.32.54.158

C15

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

C16

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

Compare this diagram with those for D16, "Modular", and "Quasdiihedral" below. These four diagrams correspond to C2 acting in the four possible ways on C8. The automorphism group of C8 is C2×C2.

1.23.44.88

C8×C2

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

C4×C4

1
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
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
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
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
Other D16
= C8 ⋊ C2
 < k,r | k8, r2, krkr >
k=(abcdefgh) r=(bh)(cg)(df) 		1.21+8.42.84

C2

C4
Modular
= C8 ⋊ C2 		< k,r | k8, r2, krkkkr >
k=(abcdefgh) r=(bf)(dh) 		1.21+2.44.88

C4

C2
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
Dic16

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

C2

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

C2×C2

C2
(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
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
17 Abelian C17 < k | k17 >
k=(abcdefghijklmnopq) 		1.1716

C17

C1
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

C1
C6 × C3
= C3 × C3 × C2 		< k,r | k6, r3, krk-1r-1 >
k=(abcdef) r=(jkl) 		1.2.38.68

C6×C3

C1
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)

1.2.32.62.96.186

C3

C3
Other D18 < k,r | k9, r2, krkr >
k=(abcdefghi) r=(bi)(ch)(dg)(ef) 		1.29.32.96

1

C9
(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
19 Abelian C19 < k | k19 >
k=(abcdefghijklmnopqrs) 		1.1918

C19

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

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

1.2.42.54.104.204

C20

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
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
Other Dic20
≅ C5 ⋊C2 C4 		< b,r | b5, r4, brbrrr >
b=(abcde)(pr)(qs) r=(be)(cd)(pqrs)

< k,r | k5, r4, krkrrr >

1.2.410.54.104

C2

C5
Frob20
≅ C5 ⋊C4 C4 		< k,g | k5, g4, kgkkggg >
k=(abcde) g=(bced) 		1.25.410.54

1

C5
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
Other C7 ⋊ C3 < k,r | k7, r3, krkrrrrr >
k=(abcdefg) r=(bce)(dgf) 		1.314.76

1

C7
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
Other D22
= C11 ⋊ C2 		< k,r | k11, r2, krkr >
k=(abcdefghijk) r=(bk)(cj()di)(eh)(fg) 		1.211.1110

1

C11
23 Abelian C23 < k | k23 >
k=(abcdefghijklmnopqrstuvw) 		1.2322

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

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

1.23.32.44.66.128

C12×C2

1
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
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
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
D6 × C4 < k,r,g | k4, r2, g3, krkr, kgk-1g-1, rgr-1g-1 >
k=(abc) r=(bc) g=(pqrs) 		1.27.32.48.62.124

C4

C3
Dic12 × C2
≅ C6 ⋊C2 C4 		< b,r,g | b6, r4, 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+1+1.32.412.66

C2×C2

C3
Q8 × C3 < r,b,g | r4, b4, g2, rrbb, rgr-1g-1, bgb-1g-1 >
r=(abcd)(efgh) b=(aecg)(fbhd) g=(pq)

These Cayley diagrams of direct products are tedious and uninformative.

1.2.32.48.62.1212

C3×C2

C2
A4 × C2 ≅ (C2×C2×C2) ⋊ C3 < 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
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
Q24

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

1.2.32.42+12.62.124

C2

C6
C3 ⋊ C8 < k,r | k3, r8, krkr7 >
k=(abc) r=(bc)(defghijk) 		1.2.32.42.62.812.124

C4

C3
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
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
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
25 Abelian C25 < k | k25 >
k=(abcdefghijklmnopqrstuvwxy) 		1.54.2520

C25

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

C5×C5

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

1

C13
27 Abelian C27 < k | k27 >
k=(abcdefghijklmnopqrstuvwxyzæ)

1.32.96.2718

C27

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

1.38.918

C9×C3

1
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

C3×C3×C3

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

C3

C3
(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
28 Abelian C28
= C7 × C4 		< k | k28 >
k=(abcdefghijklmnopqrstuvwxyzæð)

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

1.2.42.76.2118

C28

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

C14×C2

1
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
Other Dic28
≅ C7 ⋊C2 C4 		< b,r | b7, r4, brbrrr >
b=(abcdefg)(pr)(qs) r=(bg)(cf)(de)(pqrs) 		1.2.414.76.146

C2

C7
29 Abelian C29 < k | k29 >
k=(abcdefghijklmnopqrstuvwxyzæðñ) 		1.2928

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)

1.2.32.54.62.104.1510.306

C30

1
Other direct products D10 × C3 < k,r,g | k5, r2, g2, krkr, kgk-1g-1, rgr-1g-1 >
k=(abcde) r=(be)(cd) g=(pq) 		1.25.32.54.610.158

C3

C5
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
Other D30
C15 ⋊ C2 		< k,r | k15, r2, abab >
k=(abcdefghijklmno) r=(bo)(cn)(dm)(el)(fk)(gj)(hi) 		1.27.32.54.158

1

C15
31 Abelian C31 < k | k31 >
k=(abcdefghijklmnopqrstuvwxyzæðñçþ) 		1.3130

C31

1

Another page giving Cayley diagrams of groups with fewer than 32 elements, but using orange triangles to denote central extensions.
Some more Cayley diagrams
Some more pages on groups

Copyright N.S.Wedd 2007