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 [×2], C3, C4 [×2], C4 [×6], C22, C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×4], Dic3 [×4], C12 [×2], C12 [×2], C2×C6, C42, C4⋊C4 [×6], C2×Dic3 [×4], C2×C12, C2×C12 [×2], C42.C2, Dic3⋊C4 [×4], C4⋊Dic3 [×2], C4×C12, C12.6Q8
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D6 [×3], C2×Q8, C4○D4 [×2], Dic6 [×2], C22×S3, C42.C2, C2×Dic6, C4○D12 [×2], 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 32 67 13)(2 33 68 14)(3 34 69 15)(4 35 70 16)(5 36 71 17)(6 25 72 18)(7 26 61 19)(8 27 62 20)(9 28 63 21)(10 29 64 22)(11 30 65 23)(12 31 66 24)(37 50 78 94)(38 51 79 95)(39 52 80 96)(40 53 81 85)(41 54 82 86)(42 55 83 87)(43 56 84 88)(44 57 73 89)(45 58 74 90)(46 59 75 91)(47 60 76 92)(48 49 77 93)
(1 84 61 37)(2 83 62 48)(3 82 63 47)(4 81 64 46)(5 80 65 45)(6 79 66 44)(7 78 67 43)(8 77 68 42)(9 76 69 41)(10 75 70 40)(11 74 71 39)(12 73 72 38)(13 94 26 56)(14 93 27 55)(15 92 28 54)(16 91 29 53)(17 90 30 52)(18 89 31 51)(19 88 32 50)(20 87 33 49)(21 86 34 60)(22 85 35 59)(23 96 36 58)(24 95 25 57)
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), (1,32,67,13)(2,33,68,14)(3,34,69,15)(4,35,70,16)(5,36,71,17)(6,25,72,18)(7,26,61,19)(8,27,62,20)(9,28,63,21)(10,29,64,22)(11,30,65,23)(12,31,66,24)(37,50,78,94)(38,51,79,95)(39,52,80,96)(40,53,81,85)(41,54,82,86)(42,55,83,87)(43,56,84,88)(44,57,73,89)(45,58,74,90)(46,59,75,91)(47,60,76,92)(48,49,77,93), (1,84,61,37)(2,83,62,48)(3,82,63,47)(4,81,64,46)(5,80,65,45)(6,79,66,44)(7,78,67,43)(8,77,68,42)(9,76,69,41)(10,75,70,40)(11,74,71,39)(12,73,72,38)(13,94,26,56)(14,93,27,55)(15,92,28,54)(16,91,29,53)(17,90,30,52)(18,89,31,51)(19,88,32,50)(20,87,33,49)(21,86,34,60)(22,85,35,59)(23,96,36,58)(24,95,25,57)>;
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), (1,32,67,13)(2,33,68,14)(3,34,69,15)(4,35,70,16)(5,36,71,17)(6,25,72,18)(7,26,61,19)(8,27,62,20)(9,28,63,21)(10,29,64,22)(11,30,65,23)(12,31,66,24)(37,50,78,94)(38,51,79,95)(39,52,80,96)(40,53,81,85)(41,54,82,86)(42,55,83,87)(43,56,84,88)(44,57,73,89)(45,58,74,90)(46,59,75,91)(47,60,76,92)(48,49,77,93), (1,84,61,37)(2,83,62,48)(3,82,63,47)(4,81,64,46)(5,80,65,45)(6,79,66,44)(7,78,67,43)(8,77,68,42)(9,76,69,41)(10,75,70,40)(11,74,71,39)(12,73,72,38)(13,94,26,56)(14,93,27,55)(15,92,28,54)(16,91,29,53)(17,90,30,52)(18,89,31,51)(19,88,32,50)(20,87,33,49)(21,86,34,60)(22,85,35,59)(23,96,36,58)(24,95,25,57) );
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)], [(1,32,67,13),(2,33,68,14),(3,34,69,15),(4,35,70,16),(5,36,71,17),(6,25,72,18),(7,26,61,19),(8,27,62,20),(9,28,63,21),(10,29,64,22),(11,30,65,23),(12,31,66,24),(37,50,78,94),(38,51,79,95),(39,52,80,96),(40,53,81,85),(41,54,82,86),(42,55,83,87),(43,56,84,88),(44,57,73,89),(45,58,74,90),(46,59,75,91),(47,60,76,92),(48,49,77,93)], [(1,84,61,37),(2,83,62,48),(3,82,63,47),(4,81,64,46),(5,80,65,45),(6,79,66,44),(7,78,67,43),(8,77,68,42),(9,76,69,41),(10,75,70,40),(11,74,71,39),(12,73,72,38),(13,94,26,56),(14,93,27,55),(15,92,28,54),(16,91,29,53),(17,90,30,52),(18,89,31,51),(19,88,32,50),(20,87,33,49),(21,86,34,60),(22,85,35,59),(23,96,36,58),(24,95,25,57)])
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
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