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## G = C2×Dic12order 96 = 25·3

### Direct product of C2 and Dic12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×Dic12
 Chief series C1 — C3 — C6 — C12 — Dic6 — C2×Dic6 — C2×Dic12
 Lower central C3 — C6 — C12 — C2×Dic12
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×Dic12
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 [×2], C3, C4 [×2], C4 [×4], C22, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×6], Dic3 [×4], C12 [×2], C2×C6, C2×C8, Q16 [×4], C2×Q8 [×2], C24 [×2], Dic6 [×4], Dic6 [×2], C2×Dic3 [×2], C2×C12, C2×Q16, Dic12 [×4], C2×C24, C2×Dic6 [×2], C2×Dic12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], Q16 [×2], C2×D4, D12 [×2], C22×S3, C2×Q16, Dic12 [×2], C2×D12, C2×Dic12

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 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

Smallest permutation representation of C2×Dic12
Regular action on 96 points
Generators in S96
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 65)(25 93)(26 94)(27 95)(28 96)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 91)(48 92)
(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 41 13 29)(2 40 14 28)(3 39 15 27)(4 38 16 26)(5 37 17 25)(6 36 18 48)(7 35 19 47)(8 34 20 46)(9 33 21 45)(10 32 22 44)(11 31 23 43)(12 30 24 42)(49 78 61 90)(50 77 62 89)(51 76 63 88)(52 75 64 87)(53 74 65 86)(54 73 66 85)(55 96 67 84)(56 95 68 83)(57 94 69 82)(58 93 70 81)(59 92 71 80)(60 91 72 79)

G:=sub<Sym(96)| (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,93)(26,94)(27,95)(28,96)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88)(45,89)(46,90)(47,91)(48,92), (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), (1,41,13,29)(2,40,14,28)(3,39,15,27)(4,38,16,26)(5,37,17,25)(6,36,18,48)(7,35,19,47)(8,34,20,46)(9,33,21,45)(10,32,22,44)(11,31,23,43)(12,30,24,42)(49,78,61,90)(50,77,62,89)(51,76,63,88)(52,75,64,87)(53,74,65,86)(54,73,66,85)(55,96,67,84)(56,95,68,83)(57,94,69,82)(58,93,70,81)(59,92,71,80)(60,91,72,79)>;

G:=Group( (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,93)(26,94)(27,95)(28,96)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88)(45,89)(46,90)(47,91)(48,92), (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), (1,41,13,29)(2,40,14,28)(3,39,15,27)(4,38,16,26)(5,37,17,25)(6,36,18,48)(7,35,19,47)(8,34,20,46)(9,33,21,45)(10,32,22,44)(11,31,23,43)(12,30,24,42)(49,78,61,90)(50,77,62,89)(51,76,63,88)(52,75,64,87)(53,74,65,86)(54,73,66,85)(55,96,67,84)(56,95,68,83)(57,94,69,82)(58,93,70,81)(59,92,71,80)(60,91,72,79) );

G=PermutationGroup([(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,49),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,56),(16,57),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,65),(25,93),(26,94),(27,95),(28,96),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,81),(38,82),(39,83),(40,84),(41,85),(42,86),(43,87),(44,88),(45,89),(46,90),(47,91),(48,92)], [(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)], [(1,41,13,29),(2,40,14,28),(3,39,15,27),(4,38,16,26),(5,37,17,25),(6,36,18,48),(7,35,19,47),(8,34,20,46),(9,33,21,45),(10,32,22,44),(11,31,23,43),(12,30,24,42),(49,78,61,90),(50,77,62,89),(51,76,63,88),(52,75,64,87),(53,74,65,86),(54,73,66,85),(55,96,67,84),(56,95,68,83),(57,94,69,82),(58,93,70,81),(59,92,71,80),(60,91,72,79)])

Matrix representation of C2×Dic12 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] >;

C2×Dic12 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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