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 |
(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