non-abelian, soluble, monomial
Aliases: C12.5S4, A4⋊2Dic6, (C3×A4)⋊3Q8, C4.1(C3⋊S4), C6.28(C2×S4), (C4×A4).1S3, C3⋊2(A4⋊Q8), (C2×A4).8D6, (C2×C6)⋊3Dic6, (C12×A4).1C2, C6.7S4.2C2, (C22×C12).4S3, (C22×C6).19D6, C22⋊(C32⋊4Q8), (C6×A4).13C22, C2.3(C2×C3⋊S4), C23.1(C2×C3⋊S3), (C22×C4).2(C3⋊S3), SmallGroup(288,907)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for A4⋊Dic6
G = < a,b,c,d,e | a2=b2=c3=d12=1, e2=d6, cac-1=eae-1=ab=ba, ad=da, cbc-1=a, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 540 in 108 conjugacy classes, 27 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C2×C4, Q8, C23, C32, Dic3, C12, C12, A4, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C3×C6, Dic6, C2×Dic3, C2×C12, C2×A4, C22×C6, C22⋊Q8, C3⋊Dic3, C3×C12, C3×A4, Dic3⋊C4, C4⋊Dic3, C6.D4, A4⋊C4, C4×A4, C2×Dic6, C22×C12, C32⋊4Q8, C6×A4, C12.48D4, A4⋊Q8, C6.7S4, C12×A4, A4⋊Dic6
Quotients: C1, C2, C22, S3, Q8, D6, C3⋊S3, Dic6, S4, C2×C3⋊S3, C2×S4, C32⋊4Q8, C3⋊S4, A4⋊Q8, C2×C3⋊S4, A4⋊Dic6
Character table of A4⋊Dic6
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | |
| size | 1 | 1 | 3 | 3 | 2 | 8 | 8 | 8 | 2 | 6 | 36 | 36 | 36 | 36 | 2 | 6 | 6 | 8 | 8 | 8 | 2 | 2 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 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 | -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 | -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 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | 1 | 1 | 1 | 1 | -2 | 1 | 1 | 1 | -2 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | 1 | 1 | -2 | orthogonal lifted from D6 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 1 | 1 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -2 | -2 | -2 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | -2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | -2 | -2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -2 | 1 | 1 | -√3 | √3 | √3 | -√3 | 0 | √3 | √3 | -√3 | 0 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ15 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | √3 | √3 | -√3 | -√3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ16 | 2 | -2 | -2 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -2 | -√3 | √3 | √3 | -√3 | -√3 | 0 | -√3 | 0 | √3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ17 | 2 | -2 | -2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -2 | 1 | 1 | √3 | -√3 | -√3 | √3 | 0 | -√3 | -√3 | √3 | 0 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ18 | 2 | -2 | -2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -2 | 1 | √3 | -√3 | -√3 | √3 | -√3 | √3 | 0 | -√3 | √3 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ19 | 2 | -2 | -2 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -2 | √3 | -√3 | -√3 | √3 | √3 | 0 | √3 | 0 | -√3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ20 | 2 | -2 | -2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -2 | 1 | -√3 | √3 | √3 | -√3 | √3 | -√3 | 0 | √3 | -√3 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ21 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -√3 | -√3 | √3 | √3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ22 | 3 | 3 | -1 | -1 | 3 | 0 | 0 | 0 | -3 | 1 | -1 | -1 | 1 | 1 | 3 | -1 | -1 | 0 | 0 | 0 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ23 | 3 | 3 | -1 | -1 | 3 | 0 | 0 | 0 | 3 | -1 | 1 | -1 | 1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ24 | 3 | 3 | -1 | -1 | 3 | 0 | 0 | 0 | 3 | -1 | -1 | 1 | -1 | 1 | 3 | -1 | -1 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ25 | 3 | 3 | -1 | -1 | 3 | 0 | 0 | 0 | -3 | 1 | 1 | 1 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ26 | 6 | 6 | -2 | -2 | -3 | 0 | 0 | 0 | -6 | 2 | 0 | 0 | 0 | 0 | -3 | 1 | 1 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C3⋊S4 |
| ρ27 | 6 | 6 | -2 | -2 | -3 | 0 | 0 | 0 | 6 | -2 | 0 | 0 | 0 | 0 | -3 | 1 | 1 | 0 | 0 | 0 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3⋊S4 |
| ρ28 | 6 | -6 | 2 | -2 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from A4⋊Q8, Schur index 2 |
| ρ29 | 6 | -6 | 2 | -2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | -1 | 1 | 0 | 0 | 0 | -3√3 | 3√3 | -√3 | √3 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ30 | 6 | -6 | 2 | -2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | -1 | 1 | 0 | 0 | 0 | 3√3 | -3√3 | √3 | -√3 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 72 points
Generators in S72
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 13)(10 14)(11 15)(12 16)(25 47)(26 48)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 37)(34 38)(35 39)(36 40)(49 69)(50 70)(51 71)(52 72)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)
(1 47 71)(2 48 72)(3 37 61)(4 38 62)(5 39 63)(6 40 64)(7 41 65)(8 42 66)(9 43 67)(10 44 68)(11 45 69)(12 46 70)(13 27 53)(14 28 54)(15 29 55)(16 30 56)(17 31 57)(18 32 58)(19 33 59)(20 34 60)(21 35 49)(22 36 50)(23 25 51)(24 26 52)
(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)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 14 7 20)(2 13 8 19)(3 24 9 18)(4 23 10 17)(5 22 11 16)(6 21 12 15)(25 68 31 62)(26 67 32 61)(27 66 33 72)(28 65 34 71)(29 64 35 70)(30 63 36 69)(37 52 43 58)(38 51 44 57)(39 50 45 56)(40 49 46 55)(41 60 47 54)(42 59 48 53)
G:=sub<Sym(72)| (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16)(25,47)(26,48)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40)(49,69)(50,70)(51,71)(52,72)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68), (1,47,71)(2,48,72)(3,37,61)(4,38,62)(5,39,63)(6,40,64)(7,41,65)(8,42,66)(9,43,67)(10,44,68)(11,45,69)(12,46,70)(13,27,53)(14,28,54)(15,29,55)(16,30,56)(17,31,57)(18,32,58)(19,33,59)(20,34,60)(21,35,49)(22,36,50)(23,25,51)(24,26,52), (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)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,14,7,20)(2,13,8,19)(3,24,9,18)(4,23,10,17)(5,22,11,16)(6,21,12,15)(25,68,31,62)(26,67,32,61)(27,66,33,72)(28,65,34,71)(29,64,35,70)(30,63,36,69)(37,52,43,58)(38,51,44,57)(39,50,45,56)(40,49,46,55)(41,60,47,54)(42,59,48,53)>;
G:=Group( (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16)(25,47)(26,48)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40)(49,69)(50,70)(51,71)(52,72)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68), (1,47,71)(2,48,72)(3,37,61)(4,38,62)(5,39,63)(6,40,64)(7,41,65)(8,42,66)(9,43,67)(10,44,68)(11,45,69)(12,46,70)(13,27,53)(14,28,54)(15,29,55)(16,30,56)(17,31,57)(18,32,58)(19,33,59)(20,34,60)(21,35,49)(22,36,50)(23,25,51)(24,26,52), (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)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,14,7,20)(2,13,8,19)(3,24,9,18)(4,23,10,17)(5,22,11,16)(6,21,12,15)(25,68,31,62)(26,67,32,61)(27,66,33,72)(28,65,34,71)(29,64,35,70)(30,63,36,69)(37,52,43,58)(38,51,44,57)(39,50,45,56)(40,49,46,55)(41,60,47,54)(42,59,48,53) );
G=PermutationGroup([[(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,13),(10,14),(11,15),(12,16),(25,47),(26,48),(27,37),(28,38),(29,39),(30,40),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(33,37),(34,38),(35,39),(36,40),(49,69),(50,70),(51,71),(52,72),(53,61),(54,62),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68)], [(1,47,71),(2,48,72),(3,37,61),(4,38,62),(5,39,63),(6,40,64),(7,41,65),(8,42,66),(9,43,67),(10,44,68),(11,45,69),(12,46,70),(13,27,53),(14,28,54),(15,29,55),(16,30,56),(17,31,57),(18,32,58),(19,33,59),(20,34,60),(21,35,49),(22,36,50),(23,25,51),(24,26,52)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,14,7,20),(2,13,8,19),(3,24,9,18),(4,23,10,17),(5,22,11,16),(6,21,12,15),(25,68,31,62),(26,67,32,61),(27,66,33,72),(28,65,34,71),(29,64,35,70),(30,63,36,69),(37,52,43,58),(38,51,44,57),(39,50,45,56),(40,49,46,55),(41,60,47,54),(42,59,48,53)]])
Matrix representation of A4⋊Dic6 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 12 | 0 |
| 0 | 0 | 1 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 11 | 11 | 0 |
| 0 | 0 | 8 | 2 | 1 |
| 0 | 0 | 5 | 12 | 0 |
| 2 | 9 | 0 | 0 | 0 |
| 9 | 2 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 3 | 6 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 | 1 |
| 0 | 0 | 8 | 1 | 0 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,12,2,0,0,0,0,1,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,11,1,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,11,8,5,0,0,11,2,12,0,0,0,1,0],[2,9,0,0,0,9,2,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[3,7,0,0,0,6,10,0,0,0,0,0,1,5,8,0,0,0,0,1,0,0,0,1,0] >;
A4⋊Dic6 in GAP, Magma, Sage, TeX
A_4\rtimes {\rm Dic}_6 % in TeX
G:=Group("A4:Dic6"); // GroupNames label
G:=SmallGroup(288,907);
// by ID
G=gap.SmallGroup(288,907);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,28,85,36,451,1684,6053,782,3534,1350]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^12=1,e^2=d^6,c*a*c^-1=e*a*e^-1=a*b=b*a,a*d=d*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations
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