# Groups of orders 12, 24 and 48 containing A4

This page gives information on groups of orders 12, 24 and 48 which "contain" A4, as a normal subgroup, as a quotient group, or as a combination of these.

The information is currently not quite complete. The green table cell indicates missing information, some of the other information may be wrong.

O Name components, GAP no. Presentation Orders of Elements Generating Permutations (and matrices) Cayley Diagram Centre Derived Group Sylow-2-subgroup
12 A4

PSL(2,3)

Tetrahedral group
C22⋊ C3 < a,b,s | a2=b2=s3=1, ba=ab, sa=bs, sb=abs > 1.23.38 (bc)(ad)

(bdc)

1 C22 C22
24 A4×C2 A4 × C2 1.21+6.38.68 ( direct product ) C2 C22 C23
S4

PGL(2,3)

Octahedral group
A4 ⋊ C2 < a,b,s,t | a2=b2=s3=t2=1, ba=ab, sa=bs, sb=abs, ta=bt, tb=at, ts=s2> 1.23+6.38.46 (abcd)

(bdc)

1 A4 D8
SL(2,3)

Binary tetrahedral group
C2 ↑ A4 < a,b,c,s | a2=b2=c, c2=1, s3=1, ba=abc, sa=bs, sb=abs > 1.2.38.46.68 (bcgf)(adhe)

(bef)(cgd)

C2 Q8 Q8
48 A4×C4 A4×C4, 31 1.21+6.38.42+6.68.1216 ( direct product ) C4 C22 C4×C22
A4×C2×C2 A4×C2×C2, 49 1.23+12.38.624 ( direct product ) C22 C22 C24
Full octahedral group S4×C2, 48 1.219.38.412.68 ( direct product ) C2 A4 D8×C2
SL(2,3)×C2 SL(2,3)×C2, 32 1.23.38.412.624 ( direct product ) C22 Q8 Q8×C2
A4⋊C4 A4⋊C4

30

< a,b,s,t | a2=b2=s3=t4=1, ba=ab, sa=bs, sb=abs, ta=bt, tb=at, ts=s2> 1.21+6.38.424.68 (abcd)(pqrs)

(bdc)

C2 A4 C22⋊C4
GL(2,3) (C2↑A4)⋊C2

29

< a,b,c,s,t | a2=b2=c, c2=1, ba=abc, s3=t2=(st)2=1, sa=bs, sb=abs, ta=bt, tb=at, ts=s2> 1.21+12.38.46.68.812 (abfdhgce)

(bef)(cgd)

C2 SL(2,3) quasidihedral
Binary octahedral group C2↑(A4⋊C2)

28

< s,t | s6=t8=1, ts=s5t3 > 1.2.38.418.68.812 (abij)(cekm)(dnlf)(gpoh), (bcd)(efg)(jkl)(mno) C2 SL(2,3) Q16
C2↑(A4×C2) C2↑(A4×C2)
C4↑A4

33

< a,b,c,d,s | a2=b2=d2=c, c2=1, s3=1, d central, ba=abc, sa=bs, sb=abs > 1.2.38.414.624 ? C4 Q8 Pauli
C42⋊C3 C42⋊C3

3

< a,b,s | a4=b4=s3=1, ba=ab, ca=bc, cb=a3b3> 1.23.332.412 (aecg)(bfdh)(imko)(jnlp), (bdc)(eni)(fok)(gml)(hpj) 1 C42 C42
(C22,C22)⋊C3

The pullback of A4×A4 by C3
(C22,C22)⋊C3
C22⋊A4

50

< a,b,p,q,s | a2=b2=p2=q2=s3=1, a,b,p,q all commute, sa=bs, sb=abs, sp=qs, sq=pqs > 1.215.332 (ab)(cd), (pq)(rs), (abc)(pqr) 1 C24 C24

The image to the left shows how how some of the groups listed above are related (direct products are omitted from this diagram). The names of the groups, as given above, are shown in black. Green lines join groups to their normal subgroups. Red lines join groups to their quotient groups. The blue above each group shows it broken down into components; the light blue arcs indicate what the central extensions by C2 act on.

We can expand the diagram above to include all the groups listed in the table. The resulting diagram is below. The meanings of the colours of the lines is:

 red enlarge the group by adding a centre dashed red enlarge the group by adding two "centres", such that neither is central green enlarge the group by building a semidirect product of it blue enlarge the group by forming a direct product pink enlarge the group using it to build a semidirect product of something else

## Notes

Orange triangles and up-arrows ↑ are used on this page to indicate central extensions.

The letters used for the permutations relate to the grids used used in the page GL(2,3), with the letters in these positions.

Copyright N.S.Wedd 2008,2009