direct product, metabelian, soluble, monomial
Aliases: A4×D12, C4⋊(S3×A4), C3⋊1(D4×A4), (C4×A4)⋊3S3, C12⋊1(C2×A4), D6⋊1(C2×A4), (C3×A4)⋊6D4, (C12×A4)⋊3C2, (C22×D12)⋊C3, (S3×C23)⋊1C6, (C22×C12)⋊1C6, (C2×A4).15D6, C22⋊2(C3×D12), C6.3(C22×A4), C23.18(S3×C6), (C6×A4).20C22, (C2×S3×A4)⋊3C2, C2.4(C2×S3×A4), (C2×C6)⋊1(C3×D4), (C22×C4)⋊3(C3×S3), (C22×C6).3(C2×C6), SmallGroup(288,920)
Series: Derived ►Chief ►Lower central ►Upper central
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
►On 36 points
(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