non-abelian, soluble, monomial
Aliases: D6:2S4, A4:2D12, A4:C4:S3, (C3xA4):3D4, C23.7S32, C2.15(S3xS4), C6.15(C2xS4), (C2xA4).7D6, (S3xC23):2S3, C3:1(A4:D4), (C22xC6).7D6, (C6xA4).7C22, C22:2(C3:D12), (C2xS3xA4):2C2, (C2xC3:S4):2C2, (C3xA4:C4):1C2, (C2xC6):2(C3:D4), SmallGroup(288,858)
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=dad-1=eae=ab=ba, cbc-1=a, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 942 in 134 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C23, C32, Dic3, C12, A4, A4, D6, D6, C2xC6, C2xC6, C22:C4, C2xD4, C24, C3xS3, C3:S3, C3xC6, D12, C2xDic3, C3:D4, C2xC12, S4, C2xA4, C2xA4, C22xS3, C22xC6, C22wrC2, C3xDic3, C3xA4, S3xC6, C2xC3:S3, D6:C4, C3xC22:C4, A4:C4, C2xD12, C2xC3:D4, C2xS4, C22xA4, S3xC23, C3:D12, C3:S4, S3xA4, C6xA4, D6:D4, A4:D4, C3xA4:C4, C2xC3:S4, C2xS3xA4, A4:D12
Quotients: C1, C2, C22, S3, D4, D6, D12, C3:D4, S4, S32, C2xS4, C3:D12, A4:D4, S3xS4, A4:D12
Character table of A4:D12
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 3 | 3 | 6 | 18 | 36 | 2 | 8 | 16 | 12 | 12 | 36 | 2 | 6 | 6 | 8 | 16 | 24 | 24 | 12 | 12 | 12 | 12 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | -2 | -2 | 0 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ8 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | -1 | -1 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 2 | -1 | -1 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ10 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 1 | 1 | -1 | -2 | 1 | 0 | 0 | -√3 | -√3 | √3 | √3 | orthogonal lifted from D12 |
| ρ11 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 1 | 1 | -1 | -2 | 1 | 0 | 0 | √3 | √3 | -√3 | -√3 | orthogonal lifted from D12 |
| ρ12 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | 0 | -2 | -2 | 2 | 1 | 1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | complex lifted from C3:D4 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | 0 | -2 | -2 | 2 | 1 | 1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | complex lifted from C3:D4 |
| ρ14 | 3 | 3 | -1 | -1 | -3 | 1 | -1 | 3 | 0 | 0 | 1 | -1 | 1 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from C2xS4 |
| ρ15 | 3 | 3 | -1 | -1 | 3 | -1 | 1 | 3 | 0 | 0 | 1 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from S4 |
| ρ16 | 3 | 3 | -1 | -1 | -3 | 1 | 1 | 3 | 0 | 0 | -1 | 1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from C2xS4 |
| ρ17 | 3 | 3 | -1 | -1 | 3 | -1 | -1 | 3 | 0 | 0 | -1 | 1 | 1 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from S4 |
| ρ18 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ19 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 2 | 2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3:D12 |
| ρ20 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | -3 | 0 | 0 | 2 | -2 | 0 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from S3xS4 |
| ρ21 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | -3 | 0 | 0 | -2 | 2 | 0 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from S3xS4 |
| ρ22 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4:D4 |
| ρ23 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3 | -1 | 1 | 0 | 0 | 0 | 0 | -√3 | √3 | √3 | -√3 | orthogonal faithful |
| ρ24 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3 | -1 | 1 | 0 | 0 | 0 | 0 | √3 | -√3 | -√3 | √3 | orthogonal faithful |
►On 36 points
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(14 20)(16 22)(18 24)(26 32)(28 34)(30 36)
(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 30 15)(2 16 31)(3 32 17)(4 18 33)(5 34 19)(6 20 35)(7 36 21)(8 22 25)(9 26 23)(10 24 27)(11 28 13)(12 14 29)
(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 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 36)(24 35)
G:=sub<Sym(36)| (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(14,20)(16,22)(18,24)(26,32)(28,34)(30,36), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,30,15)(2,16,31)(3,32,17)(4,18,33)(5,34,19)(6,20,35)(7,36,21)(8,22,25)(9,26,23)(10,24,27)(11,28,13)(12,14,29), (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)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,36)(24,35)>;
G:=Group( (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(14,20)(16,22)(18,24)(26,32)(28,34)(30,36), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,30,15)(2,16,31)(3,32,17)(4,18,33)(5,34,19)(6,20,35)(7,36,21)(8,22,25)(9,26,23)(10,24,27)(11,28,13)(12,14,29), (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)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,36)(24,35) );
G=PermutationGroup([[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(14,20),(16,22),(18,24),(26,32),(28,34),(30,36)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(1,30,15),(2,16,31),(3,32,17),(4,18,33),(5,34,19),(6,20,35),(7,36,21),(8,22,25),(9,26,23),(10,24,27),(11,28,13),(12,14,29)], [(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),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,36),(24,35)]])
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 | 0 | 1 | 12 |
| 0 | 0 | 1 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 11 | 7 | 0 | 0 | 0 |
| 6 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 7 | 11 | 0 | 0 | 0 |
| 11 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
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],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,12,12,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[11,6,0,0,0,7,11,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1],[7,11,0,0,0,11,6,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,12] >;
A4:D12 in GAP, Magma, Sage, TeX
A_4\rtimes D_{12} % in TeX
G:=Group("A4:D12"); // GroupNames label
G:=SmallGroup(288,858);
// by ID
G=gap.SmallGroup(288,858);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,85,36,234,1684,3036,782,1777,1350]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^12=e^2=1,c*a*c^-1=d*a*d^-1=e*a*e=a*b=b*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations
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