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G = C22×Dic6order 96 = 25·3

Direct product of C22 and Dic6

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

Aliases: C22×Dic6, C6.1C24, C23.38D6, C12.34C23, Dic3.1C23, (C2×C6)⋊4Q8, C61(C2×Q8), C31(C22×Q8), (C2×C4).86D6, C2.3(S3×C23), C4.32(C22×S3), (C2×C6).62C23, (C22×C4).10S3, (C22×C12).8C2, (C2×C12).95C22, (C22×C6).43C22, C22.28(C22×S3), (C22×Dic3).6C2, (C2×Dic3).42C22, SmallGroup(96,205)

Series: Derived Chief Lower central Upper central

C1C6 — C22×Dic6
C1C3C6Dic3C2×Dic3C22×Dic3 — C22×Dic6
C3C6 — C22×Dic6
C1C23C22×C4

Generators and relations for C22×Dic6
 G = < a,b,c,d | a2=b2=c12=1, d2=c6, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 258 in 156 conjugacy classes, 105 normal (9 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C4 [×8], C22 [×7], C6, C6 [×6], C2×C4 [×6], C2×C4 [×12], Q8 [×16], C23, Dic3 [×8], C12 [×4], C2×C6 [×7], C22×C4, C22×C4 [×2], C2×Q8 [×12], Dic6 [×16], C2×Dic3 [×12], C2×C12 [×6], C22×C6, C22×Q8, C2×Dic6 [×12], C22×Dic3 [×2], C22×C12, C22×Dic6
Quotients: C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C24, Dic6 [×4], C22×S3 [×7], C22×Q8, C2×Dic6 [×6], S3×C23, C22×Dic6

Smallest permutation representation of C22×Dic6
Regular action on 96 points
Generators in S96
(1 71)(2 72)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 81)(14 82)(15 83)(16 84)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 96)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(57 92)(58 93)(59 94)(60 95)
(1 47)(2 48)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 88)(14 89)(15 90)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 85)(23 86)(24 87)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(49 77)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(57 73)(58 74)(59 75)(60 76)
(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 93 7 87)(2 92 8 86)(3 91 9 85)(4 90 10 96)(5 89 11 95)(6 88 12 94)(13 46 19 40)(14 45 20 39)(15 44 21 38)(16 43 22 37)(17 42 23 48)(18 41 24 47)(25 84 31 78)(26 83 32 77)(27 82 33 76)(28 81 34 75)(29 80 35 74)(30 79 36 73)(49 62 55 68)(50 61 56 67)(51 72 57 66)(52 71 58 65)(53 70 59 64)(54 69 60 63)

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

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

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

C22×Dic6 is a maximal subgroup of
(C2×C12)⋊Q8  (C2×C4)⋊Dic6  (C22×S3)⋊Q8  Dic614D4  Dic6.32D4  (C2×Dic6)⋊7C4  C232Dic6  C4.(D6⋊C4)  (C2×Dic3)⋊Q8  C4⋊C4.237D6  Dic617D4  Dic6.37D4  C23.51D12  C42.87D6  C42.92D6  Dic623D4  Dic619D4  Dic621D4  C6.792- 1+4  C6.1052- 1+4  C22×S3×Q8
C22×Dic6 is a maximal quotient of
C42.274D6  C233Dic6  C6.72+ 1+4  C42.88D6  C42.90D6  D45Dic6  D46Dic6  Q86Dic6  Q87Dic6

36 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E···4L6A···6G12A···12H
order12···2344444···46···612···12
size11···1222226···62···22···2

36 irreducible representations

dim111122222
type+++++-++-
imageC1C2C2C2S3Q8D6D6Dic6
kernelC22×Dic6C2×Dic6C22×Dic3C22×C12C22×C4C2×C6C2×C4C23C22
# reps1122114618

Matrix representation of C22×Dic6 in GL5(𝔽13)

120000
012000
001200
000120
000012
,
120000
01000
00100
00010
00001
,
120000
011200
01000
00063
000103
,
120000
011200
001200
00042
000119

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,1,1,0,0,0,12,0,0,0,0,0,0,6,10,0,0,0,3,3],[12,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,4,11,0,0,0,2,9] >;

C22×Dic6 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_6
% in TeX

G:=Group("C2^2xDic6");
// GroupNames label

G:=SmallGroup(96,205);
// by ID

G=gap.SmallGroup(96,205);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,579,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^12=1,d^2=c^6,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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