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## G = A4×D12order 288 = 25·32

### Direct product of A4 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — A4×D12
 Chief series C1 — C3 — C2×C6 — C22×C6 — C6×A4 — C2×S3×A4 — A4×D12
 Lower central C2×C6 — C22×C6 — A4×D12
 Upper central C1 — C2 — C4

Generators and relations for A4×D12
G = < a,b,c,d,e | a2=b2=c3=d12=e2=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 778 in 138 conjugacy classes, 27 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, C12, C12, A4, A4, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C24, C3×S3, C3×C6, D12, D12, C2×C12, C3×D4, C2×A4, C2×A4, C22×S3, C22×C6, C22×D4, C3×C12, C3×A4, S3×C6, C4×A4, C4×A4, C2×D12, C22×C12, C22×A4, S3×C23, C3×D12, S3×A4, C6×A4, D4×A4, C22×D12, C12×A4, C2×S3×A4, A4×D12
Quotients: C1, C2, C3, C22, S3, C6, D4, A4, D6, C2×C6, C3×S3, D12, C3×D4, C2×A4, S3×C6, C22×A4, C3×D12, S3×A4, D4×A4, C2×S3×A4, A4×D12

Smallest permutation representation of A4×D12
On 36 points
Generators in S36
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 17 34)(2 18 35)(3 19 36)(4 20 25)(5 21 26)(6 22 27)(7 23 28)(8 24 29)(9 13 30)(10 14 31)(11 15 32)(12 16 33)
(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 32 33 34 35 36)
(1 3)(4 12)(5 11)(6 10)(7 9)(13 23)(14 22)(15 21)(16 20)(17 19)(25 33)(26 32)(27 31)(28 30)(34 36)

G:=sub<Sym(36)| (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,17,34)(2,18,35)(3,19,36)(4,20,25)(5,21,26)(6,22,27)(7,23,28)(8,24,29)(9,13,30)(10,14,31)(11,15,32)(12,16,33), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,33)(26,32)(27,31)(28,30)(34,36)>;

G:=Group( (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,17,34)(2,18,35)(3,19,36)(4,20,25)(5,21,26)(6,22,27)(7,23,28)(8,24,29)(9,13,30)(10,14,31)(11,15,32)(12,16,33), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,33)(26,32)(27,31)(28,30)(34,36) );

G=PermutationGroup([[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(1,17,34),(2,18,35),(3,19,36),(4,20,25),(5,21,26),(6,22,27),(7,23,28),(8,24,29),(9,13,30),(10,14,31),(11,15,32),(12,16,33)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,23),(14,22),(15,21),(16,20),(17,19),(25,33),(26,32),(27,31),(28,30),(34,36)]])

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 size 1 1 3 3 6 6 18 18 2 4 4 8 8 2 6 2 4 4 6 6 8 8 24 24 24 24 2 2 6 6 8 ··· 8

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 6 6 6 6 type + + + + + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D4 D6 C3×S3 D12 C3×D4 S3×C6 C3×D12 A4 C2×A4 C2×A4 S3×A4 D4×A4 C2×S3×A4 A4×D12 kernel A4×D12 C12×A4 C2×S3×A4 C22×D12 C22×C12 S3×C23 C4×A4 C3×A4 C2×A4 C22×C4 A4 C2×C6 C23 C22 D12 C12 D6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 1 1 1 2 2 2 2 4 1 1 2 1 1 1 2

Matrix representation of A4×D12 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 12 0 0 0 1 12 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 12 0 1 0 0 12 1 0
,
 3 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 2 6 0 0 0 7 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 11 7 0 0 0 7 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,12,12,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[3,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[2,7,0,0,0,6,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[11,7,0,0,0,7,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4×D12 in GAP, Magma, Sage, TeX

A_4\times D_{12}
% in TeX

G:=Group("A4xD12");
// GroupNames label

G:=SmallGroup(288,920);
// by ID

G=gap.SmallGroup(288,920);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-3,197,92,648,271,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^12=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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