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G = C2×Dic12order 96 = 25·3

Direct product of C2 and Dic12

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Dic12, C61Q16, C8.16D6, C4.8D12, C12.31D4, C12.31C23, C24.18C22, C22.14D12, Dic6.7C22, C31(C2×Q16), (C2×C8).4S3, (C2×C24).6C2, (C2×C4).82D6, (C2×C6).19D4, C6.12(C2×D4), C2.14(C2×D12), C4.29(C22×S3), (C2×Dic6).4C2, (C2×C12).90C22, SmallGroup(96,112)

Series: Derived Chief Lower central Upper central

C1C12 — C2×Dic12
C1C3C6C12Dic6C2×Dic6 — C2×Dic12
C3C6C12 — C2×Dic12
C1C22C2×C4C2×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 12A2B2C34A4B4C4D4E4F6A6B6C8A8B8C8D12A12B12C12D24A24B24C24D24E24F24G24H
 size 1111222121212122222222222222222222
ρ1111111111111111111111111111111    trivial
ρ211111111-11-1111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ31-1-1111-111-1-1-11-1-111-1-1-1111-111-1-1-11    linear of order 2
ρ41-1-1111-11-1-11-11-11-1-11-1-111-11-1-1111-1    linear of order 2
ρ51-1-1111-1-111-1-11-11-1-11-1-111-11-1-1111-1    linear of order 2
ρ61-1-1111-1-1-111-11-1-111-1-1-1111-111-1-1-11    linear of order 2
ρ71111111-11-11111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ81111111-1-1-1-11111111111111111111    linear of order 2
ρ92222-1220000-1-1-12222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ102-2-22-12-200001-11-222-211-1-1-11-1-1111-1    orthogonal lifted from D6
ρ1122222-2-200002220000-2-2-2-200000000    orthogonal lifted from D4
ρ122-2-22-12-200001-112-2-2211-1-11-111-1-1-11    orthogonal lifted from D6
ρ132222-1220000-1-1-1-2-2-2-2-1-1-1-111111111    orthogonal lifted from D6
ρ142-2-222-220000-22-2000022-2-200000000    orthogonal lifted from D4
ρ152-2-22-1-2200001-110000-1-1113-3-33-333-3    orthogonal lifted from D12
ρ162-2-22-1-2200001-110000-1-111-333-33-3-33    orthogonal lifted from D12
ρ172222-1-2-20000-1-1-100001111-3-33-3-3333    orthogonal lifted from D12
ρ182222-1-2-20000-1-1-10000111133-333-3-3-3    orthogonal lifted from D12
ρ1922-2-22000000-2-22-22-22000022-2-2-2-222    symplectic lifted from Q16, Schur index 2
ρ202-22-220000002-2-222-2-200002-2-2-222-22    symplectic lifted from Q16, Schur index 2
ρ212-22-220000002-2-2-2-2220000-2222-2-22-2    symplectic lifted from Q16, Schur index 2
ρ2222-2-22000000-2-222-22-20000-2-22222-2-2    symplectic lifted from Q16, Schur index 2
ρ232-22-2-1000000-111-2-2223-33-3ζ83ζ38ζ38ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ83ζ38ζ38ζ87ζ38785ζ3ζ87ζ3285ζ3285ζ87ζ38785ζ3    symplectic lifted from Dic12, Schur index 2
ρ242-22-2-1000000-11122-2-23-33-3ζ83ζ32838ζ32ζ83ζ38ζ38ζ87ζ38785ζ3ζ83ζ38ζ38ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ87ζ38785ζ3ζ87ζ3285ζ3285    symplectic lifted from Dic12, Schur index 2
ρ2522-2-2-100000011-1-22-22-333-3ζ83ζ32838ζ32ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ38ζ38ζ83ζ38ζ38ζ87ζ38785ζ3ζ87ζ3285ζ3285ζ87ζ3285ζ3285    symplectic lifted from Dic12, Schur index 2
ρ2622-2-2-100000011-12-22-2-333-3ζ83ζ38ζ38ζ83ζ38ζ38ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ87ζ38785ζ3ζ87ζ38785ζ3    symplectic lifted from Dic12, Schur index 2
ρ2722-2-2-100000011-1-22-223-3-33ζ87ζ3285ζ3285ζ87ζ3285ζ3285ζ83ζ38ζ38ζ87ζ38785ζ3ζ87ζ38785ζ3ζ83ζ38ζ38ζ83ζ32838ζ32ζ83ζ32838ζ32    symplectic lifted from Dic12, Schur index 2
ρ282-22-2-1000000-111-2-222-33-33ζ87ζ38785ζ3ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ87ζ38785ζ3ζ83ζ38ζ38ζ83ζ32838ζ32ζ83ζ38ζ38    symplectic lifted from Dic12, Schur index 2
ρ292-22-2-1000000-11122-2-2-33-33ζ87ζ3285ζ3285ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ38785ζ3ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ83ζ38ζ38ζ83ζ32838ζ32    symplectic lifted from Dic12, Schur index 2
ρ3022-2-2-100000011-12-22-23-3-33ζ87ζ38785ζ3ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ83ζ38ζ38ζ83ζ38ζ38    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)])

C2×Dic12 is a maximal subgroup of
C6.Q32  C24.8D4  C2.Dic24  C12.4D8  C8.8D12  C124Q16  C8.D12  Dic12⋊C4  D12.32D4  Dic6.32D4  Dic6.D4  D4.D12  Dic3⋊Q16  D6⋊Q16  C42.36D6  C4⋊Dic12  Dic129C4  C8.2D12  Dic35Q16  D62Q16  C24.18D4  C16.D6  C24.82D4  C24.4D4  Q8.10D12  C24.22D4  C24.31D4  C24.26D4  D8.9D6  D4.13D12  C2×S3×Q16  D8.10D6
C2×Dic12 is a maximal quotient of
C12.14Q16  C248Q8  C124Q16  C23.40D12  Dic6.32D4  C4⋊Dic12  Dic63Q8  C24.82D4

Matrix representation of C2×Dic12 in GL4(𝔽73) generated by

72000
07200
00720
00072
,
571600
575700
0077
006614
,
455000
502800
00518
002368
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

Export

Character table of C2×Dic12 in TeX

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