metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.1Q16, Dic6⋊2C4, C12.44D4, C6.1SD16, C2.1Dic12, C22.7D12, C4.7(C4×S3), (C2×C8).2S3, (C2×C24).2C2, (C2×C6).12D4, (C2×C4).67D6, C12.17(C2×C4), C2.7(D6⋊C4), C3⋊2(Q8⋊C4), C4⋊Dic3.1C2, C2.1(C24⋊C2), C4.19(C3⋊D4), C6.5(C22⋊C4), (C2×Dic6).1C2, (C2×C12).80C22, SmallGroup(96,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.Dic12
G = < a,b,c | a6=b8=1, c2=a3, ab=ba, cac-1=a-1, cbc-1=a3b3 >
Character table of C2.Dic12
| 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 | i | -1 | -i | -1 | 1 | -1 | i | i | -i | -i | 1 | -1 | -1 | 1 | i | -i | -i | i | -i | -i | i | i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -i | -1 | i | -1 | 1 | -1 | -i | -i | i | i | 1 | -1 | -1 | 1 | -i | i | i | -i | i | i | -i | -i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -i | 1 | i | -1 | 1 | -1 | i | i | -i | -i | 1 | -1 | -1 | 1 | i | -i | -i | i | -i | -i | i | i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | i | 1 | -i | -1 | 1 | -1 | -i | -i | i | i | 1 | -1 | -1 | 1 | -i | i | i | -i | i | i | -i | -i | linear of order 4 |
| 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 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√2 | √2 | -√2 | √2 | -√3 | -√3 | √3 | √3 | ζ83ζ3+ζ83+ζ8ζ3 | ζ83ζ3+ζ83+ζ8ζ3 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ3+ζ85ζ3+ζ85 | ζ87ζ3+ζ85ζ3+ζ85 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | symplectic lifted from Dic12, Schur index 2 |
| ρ18 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √2 | -√2 | √2 | -√2 | -√3 | -√3 | √3 | √3 | ζ83ζ32+ζ8ζ32+ζ8 | ζ83ζ32+ζ8ζ32+ζ8 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ87ζ32+ζ87+ζ85ζ32 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | symplectic lifted from Dic12, Schur index 2 |
| ρ19 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √2 | -√2 | √2 | -√2 | √3 | √3 | -√3 | -√3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ87ζ32+ζ87+ζ85ζ32 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ32+ζ8ζ32+ζ8 | ζ83ζ32+ζ8ζ32+ζ8 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | symplectic lifted from Dic12, Schur index 2 |
| ρ20 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√2 | √2 | -√2 | √2 | √3 | √3 | -√3 | -√3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ87ζ3+ζ85ζ3+ζ85 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ3+ζ83+ζ8ζ3 | ζ83ζ3+ζ83+ζ8ζ3 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | symplectic lifted from Dic12, Schur index 2 |
| ρ21 | 2 | -2 | 2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 2i | 2i | -2i | -2i | -1 | 1 | 1 | -1 | -i | i | i | -i | i | i | -i | -i | complex lifted from C4×S3 |
| ρ22 | 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 | complex lifted from C3⋊D4 |
| ρ23 | 2 | -2 | 2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -2i | -2i | 2i | 2i | -1 | 1 | 1 | -1 | i | -i | -i | i | -i | -i | i | i | complex lifted from C4×S3 |
| ρ24 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√-2 | √-2 | √-2 | -√-2 | -√3 | √3 | -√3 | √3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ3+ζ83-ζ8ζ3 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | complex lifted from C24⋊C2 |
| ρ25 | 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 | complex lifted from SD16 |
| ρ26 | 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 | complex lifted from C3⋊D4 |
| ρ27 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √-2 | -√-2 | -√-2 | √-2 | √3 | -√3 | √3 | -√3 | ζ83ζ3+ζ83-ζ8ζ3 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ32+ζ87-ζ85ζ32 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | complex lifted from C24⋊C2 |
| 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 | ζ83ζ3+ζ83-ζ8ζ3 | ζ87ζ3+ζ87-ζ85ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ32+ζ87-ζ85ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | complex lifted from C24⋊C2 |
| ρ29 | 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 | ζ87ζ32+ζ87-ζ85ζ32 | ζ83ζ32+ζ83-ζ8ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ3+ζ83-ζ8ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | complex lifted from C24⋊C2 |
| ρ30 | 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 | complex lifted from SD16 |
►Regular action on 96 points
Generators in S96
(1 53 74 87 10 35)(2 54 75 88 11 36)(3 55 76 81 12 37)(4 56 77 82 13 38)(5 49 78 83 14 39)(6 50 79 84 15 40)(7 51 80 85 16 33)(8 52 73 86 9 34)(17 62 90 66 25 43)(18 63 91 67 26 44)(19 64 92 68 27 45)(20 57 93 69 28 46)(21 58 94 70 29 47)(22 59 95 71 30 48)(23 60 96 72 31 41)(24 61 89 65 32 42)
(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 62 87 25)(2 28 88 57)(3 60 81 31)(4 26 82 63)(5 58 83 29)(6 32 84 61)(7 64 85 27)(8 30 86 59)(9 22 52 71)(10 66 53 17)(11 20 54 69)(12 72 55 23)(13 18 56 67)(14 70 49 21)(15 24 50 65)(16 68 51 19)(33 92 80 45)(34 48 73 95)(35 90 74 43)(36 46 75 93)(37 96 76 41)(38 44 77 91)(39 94 78 47)(40 42 79 89)
G:=sub<Sym(96)| (1,53,74,87,10,35)(2,54,75,88,11,36)(3,55,76,81,12,37)(4,56,77,82,13,38)(5,49,78,83,14,39)(6,50,79,84,15,40)(7,51,80,85,16,33)(8,52,73,86,9,34)(17,62,90,66,25,43)(18,63,91,67,26,44)(19,64,92,68,27,45)(20,57,93,69,28,46)(21,58,94,70,29,47)(22,59,95,71,30,48)(23,60,96,72,31,41)(24,61,89,65,32,42), (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), (1,62,87,25)(2,28,88,57)(3,60,81,31)(4,26,82,63)(5,58,83,29)(6,32,84,61)(7,64,85,27)(8,30,86,59)(9,22,52,71)(10,66,53,17)(11,20,54,69)(12,72,55,23)(13,18,56,67)(14,70,49,21)(15,24,50,65)(16,68,51,19)(33,92,80,45)(34,48,73,95)(35,90,74,43)(36,46,75,93)(37,96,76,41)(38,44,77,91)(39,94,78,47)(40,42,79,89)>;
G:=Group( (1,53,74,87,10,35)(2,54,75,88,11,36)(3,55,76,81,12,37)(4,56,77,82,13,38)(5,49,78,83,14,39)(6,50,79,84,15,40)(7,51,80,85,16,33)(8,52,73,86,9,34)(17,62,90,66,25,43)(18,63,91,67,26,44)(19,64,92,68,27,45)(20,57,93,69,28,46)(21,58,94,70,29,47)(22,59,95,71,30,48)(23,60,96,72,31,41)(24,61,89,65,32,42), (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), (1,62,87,25)(2,28,88,57)(3,60,81,31)(4,26,82,63)(5,58,83,29)(6,32,84,61)(7,64,85,27)(8,30,86,59)(9,22,52,71)(10,66,53,17)(11,20,54,69)(12,72,55,23)(13,18,56,67)(14,70,49,21)(15,24,50,65)(16,68,51,19)(33,92,80,45)(34,48,73,95)(35,90,74,43)(36,46,75,93)(37,96,76,41)(38,44,77,91)(39,94,78,47)(40,42,79,89) );
G=PermutationGroup([[(1,53,74,87,10,35),(2,54,75,88,11,36),(3,55,76,81,12,37),(4,56,77,82,13,38),(5,49,78,83,14,39),(6,50,79,84,15,40),(7,51,80,85,16,33),(8,52,73,86,9,34),(17,62,90,66,25,43),(18,63,91,67,26,44),(19,64,92,68,27,45),(20,57,93,69,28,46),(21,58,94,70,29,47),(22,59,95,71,30,48),(23,60,96,72,31,41),(24,61,89,65,32,42)], [(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)], [(1,62,87,25),(2,28,88,57),(3,60,81,31),(4,26,82,63),(5,58,83,29),(6,32,84,61),(7,64,85,27),(8,30,86,59),(9,22,52,71),(10,66,53,17),(11,20,54,69),(12,72,55,23),(13,18,56,67),(14,70,49,21),(15,24,50,65),(16,68,51,19),(33,92,80,45),(34,48,73,95),(35,90,74,43),(36,46,75,93),(37,96,76,41),(38,44,77,91),(39,94,78,47),(40,42,79,89)]])
C2.Dic12 is a maximal subgroup of
C12.14Q16 C4×C24⋊C2 C42.264D6 C4×Dic12 C42.14D6 C42.16D6 C42.20D6 Dic12⋊C4 C23.39D12 C23.40D12 C23.15D12 D12.32D4 D12⋊14D4 Dic6⋊14D4 Dic6.32D4 D4.S3⋊C4 Dic3⋊6SD16 C12⋊Q8⋊C2 (C2×C8).200D6 D4⋊(C4×S3) D4⋊2S3⋊C4 D6⋊SD16 C3⋊C8⋊1D4 C3⋊Q16⋊C4 Dic3⋊4Q16 Dic3.1Q16 (C2×Q8).36D6 S3×Q8⋊C4 (S3×Q8)⋊C4 D6⋊1Q16 C3⋊C8.D4 Dic6.3Q8 C12⋊SD16 D12.19D4 C42.36D6 C4⋊Dic12 Dic6⋊3Q8 Dic6⋊4Q8 Dic6⋊Q8 Dic6.Q8 D6.2SD16 C6.(C4○D8) Dic3.Q16 Dic6.2Q8 D6.2Q16 C2.D8⋊7S3 C23.28D12 C24⋊30D4 C24.82D4 C23.51D12 C23.54D12 C24⋊2D4 C24.4D4 (C6×D8).C2 Dic6⋊D4 Dic3⋊3SD16 (C3×Q8).D4 D6⋊8SD16 Dic6.16D4 Dic3⋊3Q16 D6⋊5Q16 C36.45D4 C6.Dic12 C12.73D12 C6.4Dic12 C10.Dic12 Dic30⋊15C4 Dic30⋊8C4 Dic30⋊C4
C2.Dic12 is a maximal quotient of
C4.8Dic12 C23.35D12 C4.Dic12 C12.47D8 C12.9C42 C36.45D4 C6.Dic12 C12.73D12 C6.4Dic12 C10.Dic12 Dic30⋊15C4 Dic30⋊8C4 Dic30⋊C4
Matrix representation of C2.Dic12 ►in GL4(𝔽73) generated by
| 1 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 |
| 0 | 0 | 1 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 36 | 11 |
| 0 | 0 | 62 | 25 |
| 34 | 8 | 0 | 0 |
| 47 | 39 | 0 | 0 |
| 0 | 0 | 3 | 45 |
| 0 | 0 | 42 | 70 |
G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,72,1,0,0,72,0],[46,0,0,0,0,46,0,0,0,0,36,62,0,0,11,25],[34,47,0,0,8,39,0,0,0,0,3,42,0,0,45,70] >;
C2.Dic12 in GAP, Magma, Sage, TeX
C_2.{\rm Dic}_{12} % in TeX
G:=Group("C2.Dic12"); // GroupNames label
G:=SmallGroup(96,23);
// by ID
G=gap.SmallGroup(96,23);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,73,79,362,86,2309]);
// Polycyclic
G:=Group<a,b,c|a^6=b^8=1,c^2=a^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^3*b^3>;
// generators/relations
Export
Subgroup lattice of C2.Dic12 in TeX
Character table of C2.Dic12 in TeX