metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.6Q8, C42.5S3, C4.6Dic6, C6.3(C2×Q8), (C4×C12).3C2, (C2×C4).74D6, C6.2(C4○D4), C4⋊Dic3.5C2, C2.5(C2×Dic6), C3⋊1(C42.C2), C2.6(C4○D12), Dic3⋊C4.1C2, (C2×C6).11C23, (C2×C12).72C22, C22.35(C22×S3), (C2×Dic3).2C22, SmallGroup(96,77)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.6Q8
G = < a,b,c | a12=b4=1, c2=a6b2, ab=ba, cac-1=a-1, cbc-1=a6b-1 >
Subgroups: 106 in 56 conjugacy classes, 33 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C2×C4, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C2×Dic3, C2×C12, C2×C12, C42.C2, Dic3⋊C4, C4⋊Dic3, C4×C12, C12.6Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, Dic6, C22×S3, C42.C2, C2×Dic6, C4○D12, C12.6Q8
Character table of C12.6Q8
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 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 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | -2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ15 | 2 | -2 | -2 | 2 | -1 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -√3 | √3 | -1 | -√3 | -√3 | 1 | 1 | √3 | -√3 | √3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ16 | 2 | -2 | -2 | 2 | -1 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | √3 | -√3 | -1 | √3 | √3 | 1 | 1 | -√3 | √3 | -√3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ17 | 2 | -2 | -2 | 2 | -1 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | -√3 | -√3 | 1 | -√3 | √3 | -1 | -1 | √3 | √3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ18 | 2 | -2 | -2 | 2 | -1 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | √3 | √3 | 1 | √3 | -√3 | -1 | -1 | -√3 | -√3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ19 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | -2i | 0 | 0 | 2i | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | 0 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 2i | 0 | 0 | 2i | 0 | 0 | 0 | -2i | -2i | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | -2 | 2 | -2 | 2 | 0 | 2i | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | -2i | 0 | 0 | -2i | 0 | 0 | 0 | 2i | 2i | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 2i | 0 | 0 | -2i | complex lifted from C4○D4 |
| ρ23 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√3 | i | -√-3 | √3 | -i | √-3 | √3 | -√3 | -i | -√-3 | √-3 | i | complex lifted from C4○D12 |
| ρ24 | 2 | -2 | 2 | -2 | -1 | 0 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √3 | -√-3 | -i | -√3 | √-3 | -i | √3 | -√3 | -√-3 | i | i | √-3 | complex lifted from C4○D12 |
| ρ25 | 2 | -2 | 2 | -2 | -1 | 0 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√3 | √-3 | -i | √3 | -√-3 | -i | -√3 | √3 | √-3 | i | i | -√-3 | complex lifted from C4○D12 |
| ρ26 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√3 | -i | √-3 | √3 | i | -√-3 | √3 | -√3 | i | √-3 | -√-3 | -i | complex lifted from C4○D12 |
| ρ27 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √3 | i | √-3 | -√3 | -i | -√-3 | -√3 | √3 | -i | √-3 | -√-3 | i | complex lifted from C4○D12 |
| ρ28 | 2 | -2 | 2 | -2 | -1 | 0 | 2i | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √3 | √-3 | i | -√3 | -√-3 | i | √3 | -√3 | √-3 | -i | -i | -√-3 | complex lifted from C4○D12 |
| ρ29 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √3 | -i | -√-3 | -√3 | i | √-3 | -√3 | √3 | i | -√-3 | √-3 | -i | complex lifted from C4○D12 |
| ρ30 | 2 | -2 | 2 | -2 | -1 | 0 | 2i | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√3 | -√-3 | i | √3 | √-3 | i | -√3 | √3 | -√-3 | -i | -i | √-3 | complex lifted from C4○D12 |
►Regular action on 96 points
(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)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 85 29 84)(2 86 30 73)(3 87 31 74)(4 88 32 75)(5 89 33 76)(6 90 34 77)(7 91 35 78)(8 92 36 79)(9 93 25 80)(10 94 26 81)(11 95 27 82)(12 96 28 83)(13 40 54 63)(14 41 55 64)(15 42 56 65)(16 43 57 66)(17 44 58 67)(18 45 59 68)(19 46 60 69)(20 47 49 70)(21 48 50 71)(22 37 51 72)(23 38 52 61)(24 39 53 62)
(1 68 35 39)(2 67 36 38)(3 66 25 37)(4 65 26 48)(5 64 27 47)(6 63 28 46)(7 62 29 45)(8 61 30 44)(9 72 31 43)(10 71 32 42)(11 70 33 41)(12 69 34 40)(13 90 60 83)(14 89 49 82)(15 88 50 81)(16 87 51 80)(17 86 52 79)(18 85 53 78)(19 96 54 77)(20 95 55 76)(21 94 56 75)(22 93 57 74)(23 92 58 73)(24 91 59 84)
G:=sub<Sym(96)| (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,85,29,84)(2,86,30,73)(3,87,31,74)(4,88,32,75)(5,89,33,76)(6,90,34,77)(7,91,35,78)(8,92,36,79)(9,93,25,80)(10,94,26,81)(11,95,27,82)(12,96,28,83)(13,40,54,63)(14,41,55,64)(15,42,56,65)(16,43,57,66)(17,44,58,67)(18,45,59,68)(19,46,60,69)(20,47,49,70)(21,48,50,71)(22,37,51,72)(23,38,52,61)(24,39,53,62), (1,68,35,39)(2,67,36,38)(3,66,25,37)(4,65,26,48)(5,64,27,47)(6,63,28,46)(7,62,29,45)(8,61,30,44)(9,72,31,43)(10,71,32,42)(11,70,33,41)(12,69,34,40)(13,90,60,83)(14,89,49,82)(15,88,50,81)(16,87,51,80)(17,86,52,79)(18,85,53,78)(19,96,54,77)(20,95,55,76)(21,94,56,75)(22,93,57,74)(23,92,58,73)(24,91,59,84)>;
G:=Group( (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,85,29,84)(2,86,30,73)(3,87,31,74)(4,88,32,75)(5,89,33,76)(6,90,34,77)(7,91,35,78)(8,92,36,79)(9,93,25,80)(10,94,26,81)(11,95,27,82)(12,96,28,83)(13,40,54,63)(14,41,55,64)(15,42,56,65)(16,43,57,66)(17,44,58,67)(18,45,59,68)(19,46,60,69)(20,47,49,70)(21,48,50,71)(22,37,51,72)(23,38,52,61)(24,39,53,62), (1,68,35,39)(2,67,36,38)(3,66,25,37)(4,65,26,48)(5,64,27,47)(6,63,28,46)(7,62,29,45)(8,61,30,44)(9,72,31,43)(10,71,32,42)(11,70,33,41)(12,69,34,40)(13,90,60,83)(14,89,49,82)(15,88,50,81)(16,87,51,80)(17,86,52,79)(18,85,53,78)(19,96,54,77)(20,95,55,76)(21,94,56,75)(22,93,57,74)(23,92,58,73)(24,91,59,84) );
G=PermutationGroup([[(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),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,85,29,84),(2,86,30,73),(3,87,31,74),(4,88,32,75),(5,89,33,76),(6,90,34,77),(7,91,35,78),(8,92,36,79),(9,93,25,80),(10,94,26,81),(11,95,27,82),(12,96,28,83),(13,40,54,63),(14,41,55,64),(15,42,56,65),(16,43,57,66),(17,44,58,67),(18,45,59,68),(19,46,60,69),(20,47,49,70),(21,48,50,71),(22,37,51,72),(23,38,52,61),(24,39,53,62)], [(1,68,35,39),(2,67,36,38),(3,66,25,37),(4,65,26,48),(5,64,27,47),(6,63,28,46),(7,62,29,45),(8,61,30,44),(9,72,31,43),(10,71,32,42),(11,70,33,41),(12,69,34,40),(13,90,60,83),(14,89,49,82),(15,88,50,81),(16,87,51,80),(17,86,52,79),(18,85,53,78),(19,96,54,77),(20,95,55,76),(21,94,56,75),(22,93,57,74),(23,92,58,73),(24,91,59,84)]])
C12.6Q8 is a maximal subgroup of
C42.2D6 C24.13Q8 C42.264D6 C8⋊Dic6 C42.14D6 C42.19D6 Dic6.3Q8 D12.3Q8 D4.3Dic6 Q8.5Dic6 C42.213D6 C42.215D6 C42.72D6 C42.76D6 C42.77D6 C42.274D6 C42.277D6 C42.89D6 C42.90D6 C42.94D6 C42.96D6 C42.100D6 D4⋊5Dic6 C42.105D6 C42.113D6 C42.118D6 C42.119D6 Dic6⋊10Q8 Q8⋊6Dic6 D12⋊10Q8 C42.132D6 C42.134D6 C42.140D6 C42.234D6 C42.145D6 C42.147D6 S3×C42.C2 C42.236D6 C42.157D6 C42.159D6 C42.161D6 C42.168D6 C42.174D6 C42.176D6 C36.6Q8 C62.37C23 C62.39C23 C12.25Dic6 Dic5.7Dic6 C20.Dic6 C60.24Q8
C12.6Q8 is a maximal quotient of
C6.(C4⋊Q8) (C2×C4).Dic6 C12⋊4(C4⋊C4) (C2×C42).6S3 C42⋊11Dic3 C36.6Q8 C62.37C23 C62.39C23 C12.25Dic6 Dic5.7Dic6 C20.Dic6 C60.24Q8
Matrix representation of C12.6Q8 ►in GL4(𝔽13) generated by
| 0 | 1 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 4 | 11 |
| 3 | 6 | 0 | 0 |
| 7 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 12 | 1 |
| 8 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 0 | 0 | 8 | 10 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [0,12,0,0,1,12,0,0,0,0,6,4,0,0,0,11],[3,7,0,0,6,10,0,0,0,0,12,12,0,0,0,1],[8,5,0,0,0,5,0,0,0,0,8,0,0,0,10,5] >;
C12.6Q8 in GAP, Magma, Sage, TeX
C_{12}._6Q_8 % in TeX
G:=Group("C12.6Q8"); // GroupNames label
G:=SmallGroup(96,77);
// by ID
G=gap.SmallGroup(96,77);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,217,55,218,86,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=1,c^2=a^6*b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^-1>;
// generators/relations
Export