direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2xDic12, C6:1Q16, C8.16D6, C4.8D12, C12.31D4, C12.31C23, C24.18C22, C22.14D12, Dic6.7C22, C3:1(C2xQ16), (C2xC8).4S3, (C2xC24).6C2, (C2xC4).82D6, (C2xC6).19D4, C6.12(C2xD4), C2.14(C2xD12), C4.29(C22xS3), (C2xDic6).4C2, (C2xC12).90C22, SmallGroup(96,112)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xDic12
G = < a,b,c | a2=b24=1, c2=b12, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 130 in 60 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2xC4, C2xC4, Q8, Dic3, C12, C2xC6, C2xC8, Q16, C2xQ8, C24, Dic6, Dic6, C2xDic3, C2xC12, C2xQ16, Dic12, C2xC24, C2xDic6, C2xDic12
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2xD4, D12, C22xS3, C2xQ16, Dic12, C2xD12, C2xDic12
Character table of C2xDic12
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 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 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | -2 | -2 | 2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -2 | 2 | 2 | -2 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | -2 | -2 | 2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 2 | -2 | -2 | 2 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | -2 | -2 | 2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | √3 | -√3 | -√3 | √3 | -√3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ16 | 2 | -2 | -2 | 2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -√3 | √3 | √3 | -√3 | √3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ17 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -√3 | -√3 | √3 | -√3 | -√3 | √3 | √3 | √3 | orthogonal lifted from D12 |
| ρ18 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | √3 | √3 | -√3 | √3 | √3 | -√3 | -√3 | -√3 | orthogonal lifted from D12 |
| ρ19 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -√2 | √2 | -√2 | √2 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | -√2 | -√2 | √2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | √2 | √2 | -√2 | -√2 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | -√2 | √2 | √2 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ21 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -√2 | -√2 | √2 | √2 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | √2 | -√2 | -√2 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ22 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | √2 | -√2 | √2 | -√2 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | √2 | √2 | -√2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ23 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√2 | -√2 | √2 | √2 | √3 | -√3 | √3 | -√3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ3+ζ87+ζ85ζ3 | symplectic lifted from Dic12, Schur index 2 |
| ρ24 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √2 | √2 | -√2 | -√2 | √3 | -√3 | √3 | -√3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ32+ζ85ζ32+ζ85 | symplectic lifted from Dic12, Schur index 2 |
| ρ25 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√2 | √2 | -√2 | √2 | -√3 | √3 | √3 | -√3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ32+ζ85ζ32+ζ85 | symplectic lifted from Dic12, Schur index 2 |
| ρ26 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √2 | -√2 | √2 | -√2 | -√3 | √3 | √3 | -√3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ3+ζ87+ζ85ζ3 | symplectic lifted from Dic12, Schur index 2 |
| ρ27 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√2 | √2 | -√2 | √2 | √3 | -√3 | -√3 | √3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ32+ζ83+ζ8ζ32 | symplectic lifted from Dic12, Schur index 2 |
| ρ28 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√2 | -√2 | √2 | √2 | -√3 | √3 | -√3 | √3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ3+ζ8ζ3+ζ8 | symplectic lifted from Dic12, Schur index 2 |
| ρ29 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √2 | √2 | -√2 | -√2 | -√3 | √3 | -√3 | √3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ32+ζ83+ζ8ζ32 | symplectic lifted from Dic12, Schur index 2 |
| ρ30 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √2 | -√2 | √2 | -√2 | √3 | -√3 | -√3 | √3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ3+ζ8ζ3+ζ8 | symplectic lifted from Dic12, Schur index 2 |
►Regular action on 96 points
Generators in S96
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 49)(47 50)(48 51)
(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 40 13 28)(2 39 14 27)(3 38 15 26)(4 37 16 25)(5 36 17 48)(6 35 18 47)(7 34 19 46)(8 33 20 45)(9 32 21 44)(10 31 22 43)(11 30 23 42)(12 29 24 41)(49 91 61 79)(50 90 62 78)(51 89 63 77)(52 88 64 76)(53 87 65 75)(54 86 66 74)(55 85 67 73)(56 84 68 96)(57 83 69 95)(58 82 70 94)(59 81 71 93)(60 80 72 92)
G:=sub<Sym(96)| (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,49)(47,50)(48,51), (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), (1,40,13,28)(2,39,14,27)(3,38,15,26)(4,37,16,25)(5,36,17,48)(6,35,18,47)(7,34,19,46)(8,33,20,45)(9,32,21,44)(10,31,22,43)(11,30,23,42)(12,29,24,41)(49,91,61,79)(50,90,62,78)(51,89,63,77)(52,88,64,76)(53,87,65,75)(54,86,66,74)(55,85,67,73)(56,84,68,96)(57,83,69,95)(58,82,70,94)(59,81,71,93)(60,80,72,92)>;
G:=Group( (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,49)(47,50)(48,51), (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), (1,40,13,28)(2,39,14,27)(3,38,15,26)(4,37,16,25)(5,36,17,48)(6,35,18,47)(7,34,19,46)(8,33,20,45)(9,32,21,44)(10,31,22,43)(11,30,23,42)(12,29,24,41)(49,91,61,79)(50,90,62,78)(51,89,63,77)(52,88,64,76)(53,87,65,75)(54,86,66,74)(55,85,67,73)(56,84,68,96)(57,83,69,95)(58,82,70,94)(59,81,71,93)(60,80,72,92) );
G=PermutationGroup([[(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,61),(35,62),(36,63),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,49),(47,50),(48,51)], [(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)], [(1,40,13,28),(2,39,14,27),(3,38,15,26),(4,37,16,25),(5,36,17,48),(6,35,18,47),(7,34,19,46),(8,33,20,45),(9,32,21,44),(10,31,22,43),(11,30,23,42),(12,29,24,41),(49,91,61,79),(50,90,62,78),(51,89,63,77),(52,88,64,76),(53,87,65,75),(54,86,66,74),(55,85,67,73),(56,84,68,96),(57,83,69,95),(58,82,70,94),(59,81,71,93),(60,80,72,92)]])
C2xDic12 is a maximal subgroup of
C6.Q32 C24.8D4 C2.Dic24 C12.4D8 C8.8D12 C12:4Q16 C8.D12 Dic12:C4 D12.32D4 Dic6.32D4 Dic6.D4 D4.D12 Dic3:Q16 D6:Q16 C42.36D6 C4:Dic12 Dic12:9C4 C8.2D12 Dic3:5Q16 D6:2Q16 C24.18D4 C16.D6 C24.82D4 C24.4D4 Q8.10D12 C24.22D4 C24.31D4 C24.26D4 D8.9D6 D4.13D12 C2xS3xQ16 D8.10D6
C2xDic12 is a maximal quotient of
C12.14Q16 C24:8Q8 C12:4Q16 C23.40D12 Dic6.32D4 C4:Dic12 Dic6:3Q8 C24.82D4
Matrix representation of C2xDic12 ►in GL4(F73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 57 | 16 | 0 | 0 |
| 57 | 57 | 0 | 0 |
| 0 | 0 | 7 | 7 |
| 0 | 0 | 66 | 14 |
| 45 | 50 | 0 | 0 |
| 50 | 28 | 0 | 0 |
| 0 | 0 | 5 | 18 |
| 0 | 0 | 23 | 68 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[57,57,0,0,16,57,0,0,0,0,7,66,0,0,7,14],[45,50,0,0,50,28,0,0,0,0,5,23,0,0,18,68] >;
C2xDic12 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{12} % in TeX
G:=Group("C2xDic12"); // GroupNames label
G:=SmallGroup(96,112);
// by ID
G=gap.SmallGroup(96,112);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,218,122,579,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^2=b^24=1,c^2=b^12,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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