metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12:2Q8, C4:2Dic6, C4.4D12, C12.27D4, C42.4S3, C3:1(C4:Q8), C6.1(C2xD4), C6.2(C2xQ8), (C4xC12).2C2, C2.4(C2xD12), (C2xC4).73D6, C4:Dic3.4C2, C2.4(C2xDic6), (C2xC6).10C23, (C2xDic6).2C2, (C2xC12).85C22, C22.34(C22xS3), (C2xDic3).1C22, SmallGroup(96,76)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12:2Q8
G = < a,b,c | a12=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 138 in 68 conjugacy classes, 41 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2xC4, C2xC4, C2xC4, Q8, Dic3, C12, C2xC6, C42, C4:C4, C2xQ8, Dic6, C2xDic3, C2xC12, C2xC12, C4:Q8, C4:Dic3, C4xC12, C2xDic6, C12:2Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2xD4, C2xQ8, Dic6, D12, C22xS3, C4:Q8, C2xDic6, C2xD12, C12:2Q8
Character table of C12:2Q8
| 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 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | orthogonal lifted from D4 |
| 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 D6 |
| 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 S3 |
| ρ12 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √3 | 1 | -√3 | -√3 | -1 | √3 | -√3 | √3 | -1 | -√3 | √3 | 1 | orthogonal lifted from D12 |
| ρ14 | 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 |
| ρ15 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √3 | -1 | √3 | -√3 | 1 | -√3 | -√3 | √3 | 1 | √3 | -√3 | -1 | orthogonal lifted from D12 |
| ρ16 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√3 | -1 | -√3 | √3 | 1 | √3 | √3 | -√3 | 1 | -√3 | √3 | -1 | orthogonal lifted from D12 |
| ρ17 | 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 |
| ρ18 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√3 | 1 | √3 | √3 | -1 | -√3 | √3 | -√3 | -1 | √3 | -√3 | 1 | orthogonal lifted from D12 |
| ρ19 | 2 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ20 | 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 |
| ρ21 | 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 |
| ρ22 | 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 |
| ρ23 | 2 | -2 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ24 | 2 | -2 | 2 | -2 | -1 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √3 | -√3 | 1 | -√3 | √3 | 1 | √3 | -√3 | -√3 | -1 | -1 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ25 | 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 |
| ρ26 | 2 | -2 | 2 | -2 | -1 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√3 | √3 | 1 | √3 | -√3 | 1 | -√3 | √3 | √3 | -1 | -1 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ27 | 2 | -2 | 2 | -2 | -1 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √3 | √3 | -1 | -√3 | -√3 | -1 | √3 | -√3 | √3 | 1 | 1 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ28 | 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 |
| ρ29 | 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 |
| ρ30 | 2 | -2 | 2 | -2 | -1 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√3 | -√3 | -1 | √3 | √3 | -1 | -√3 | √3 | -√3 | 1 | 1 | √3 | symplectic lifted from Dic6, Schur index 2 |
►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 26 66 16)(2 27 67 17)(3 28 68 18)(4 29 69 19)(5 30 70 20)(6 31 71 21)(7 32 72 22)(8 33 61 23)(9 34 62 24)(10 35 63 13)(11 36 64 14)(12 25 65 15)(37 50 80 94)(38 51 81 95)(39 52 82 96)(40 53 83 85)(41 54 84 86)(42 55 73 87)(43 56 74 88)(44 57 75 89)(45 58 76 90)(46 59 77 91)(47 60 78 92)(48 49 79 93)
(1 80 66 37)(2 79 67 48)(3 78 68 47)(4 77 69 46)(5 76 70 45)(6 75 71 44)(7 74 72 43)(8 73 61 42)(9 84 62 41)(10 83 63 40)(11 82 64 39)(12 81 65 38)(13 85 35 53)(14 96 36 52)(15 95 25 51)(16 94 26 50)(17 93 27 49)(18 92 28 60)(19 91 29 59)(20 90 30 58)(21 89 31 57)(22 88 32 56)(23 87 33 55)(24 86 34 54)
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), (1,26,66,16)(2,27,67,17)(3,28,68,18)(4,29,69,19)(5,30,70,20)(6,31,71,21)(7,32,72,22)(8,33,61,23)(9,34,62,24)(10,35,63,13)(11,36,64,14)(12,25,65,15)(37,50,80,94)(38,51,81,95)(39,52,82,96)(40,53,83,85)(41,54,84,86)(42,55,73,87)(43,56,74,88)(44,57,75,89)(45,58,76,90)(46,59,77,91)(47,60,78,92)(48,49,79,93), (1,80,66,37)(2,79,67,48)(3,78,68,47)(4,77,69,46)(5,76,70,45)(6,75,71,44)(7,74,72,43)(8,73,61,42)(9,84,62,41)(10,83,63,40)(11,82,64,39)(12,81,65,38)(13,85,35,53)(14,96,36,52)(15,95,25,51)(16,94,26,50)(17,93,27,49)(18,92,28,60)(19,91,29,59)(20,90,30,58)(21,89,31,57)(22,88,32,56)(23,87,33,55)(24,86,34,54)>;
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), (1,26,66,16)(2,27,67,17)(3,28,68,18)(4,29,69,19)(5,30,70,20)(6,31,71,21)(7,32,72,22)(8,33,61,23)(9,34,62,24)(10,35,63,13)(11,36,64,14)(12,25,65,15)(37,50,80,94)(38,51,81,95)(39,52,82,96)(40,53,83,85)(41,54,84,86)(42,55,73,87)(43,56,74,88)(44,57,75,89)(45,58,76,90)(46,59,77,91)(47,60,78,92)(48,49,79,93), (1,80,66,37)(2,79,67,48)(3,78,68,47)(4,77,69,46)(5,76,70,45)(6,75,71,44)(7,74,72,43)(8,73,61,42)(9,84,62,41)(10,83,63,40)(11,82,64,39)(12,81,65,38)(13,85,35,53)(14,96,36,52)(15,95,25,51)(16,94,26,50)(17,93,27,49)(18,92,28,60)(19,91,29,59)(20,90,30,58)(21,89,31,57)(22,88,32,56)(23,87,33,55)(24,86,34,54) );
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)], [(1,26,66,16),(2,27,67,17),(3,28,68,18),(4,29,69,19),(5,30,70,20),(6,31,71,21),(7,32,72,22),(8,33,61,23),(9,34,62,24),(10,35,63,13),(11,36,64,14),(12,25,65,15),(37,50,80,94),(38,51,81,95),(39,52,82,96),(40,53,83,85),(41,54,84,86),(42,55,73,87),(43,56,74,88),(44,57,75,89),(45,58,76,90),(46,59,77,91),(47,60,78,92),(48,49,79,93)], [(1,80,66,37),(2,79,67,48),(3,78,68,47),(4,77,69,46),(5,76,70,45),(6,75,71,44),(7,74,72,43),(8,73,61,42),(9,84,62,41),(10,83,63,40),(11,82,64,39),(12,81,65,38),(13,85,35,53),(14,96,36,52),(15,95,25,51),(16,94,26,50),(17,93,27,49),(18,92,28,60),(19,91,29,59),(20,90,30,58),(21,89,31,57),(22,88,32,56),(23,87,33,55),(24,86,34,54)]])
C12:2Q8 is a maximal subgroup of
C4.Dic12 C12.47D8 C12.2D8 C24:9Q8 C12.14Q16 C24:8Q8 C8:5D12 C4.5D24 C12:4Q16 C8:Dic6 C42.14D6 C42.20D6 C8.D12 Q8.14D12 C12:SD16 D12:3Q8 D12:4Q8 C4:Dic12 Dic6:3Q8 Dic6:4Q8 C12.50D8 C12.38SD16 D4.2D12 Q8:4Dic6 Q8:5Dic6 C12:7Q16 C42.62D6 C42.65D6 C42.68D6 C42.71D6 C12.16D8 C12:4SD16 C12.17D8 C12.9Q16 C12.SD16 C12:3Q16 C12.Q16 C42.274D6 C42.276D6 C42.89D6 C42.90D6 C42.92D6 C42.99D6 D4xDic6 C42.106D6 D4:6Dic6 D12:24D4 D4:6D12 C42.117D6 Q8xDic6 Dic6:10Q8 Q8:7Dic6 Q8xD12 D12:10Q8 C42.135D6 C42.141D6 C42.144D6 C42.148D6 C42.156D6 C42.165D6 C42.238D6 S3xC4:Q8 C42.241D6 C36:2Q8 Dic3:Dic6 C12:3Dic6 C12:6Dic6 Dic5:Dic6 C60:Q8 C60:8Q8
C12:2Q8 is a maximal quotient of
(C2xC4):Dic6 (C22xC4).85D6 C24:9Q8 C24:8Q8 C24.13Q8 C8:Dic6 C12:4(C4:C4) (C2xDic6):7C4 C42:10Dic3 C36:2Q8 Dic3:Dic6 C12:3Dic6 C12:6Dic6 Dic5:Dic6 C60:Q8 C60:8Q8
Matrix representation of C12:2Q8 ►in GL4(F13) generated by
| 0 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 6 | 2 |
| 0 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 10 | 4 | 0 | 0 |
| 4 | 3 | 0 | 0 |
| 0 | 0 | 6 | 8 |
| 0 | 0 | 7 | 7 |
G:=sub<GL(4,GF(13))| [0,1,0,0,12,0,0,0,0,0,7,6,0,0,0,2],[0,1,0,0,12,0,0,0,0,0,12,0,0,0,0,12],[10,4,0,0,4,3,0,0,0,0,6,7,0,0,8,7] >;
C12:2Q8 in GAP, Magma, Sage, TeX
C_{12}\rtimes_2Q_8 % in TeX
G:=Group("C12:2Q8"); // GroupNames label
G:=SmallGroup(96,76);
// by ID
G=gap.SmallGroup(96,76);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,217,103,218,50,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
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