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G = C2×Dic3⋊C4order 96 = 25·3

Direct product of C2 and Dic3⋊C4

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

Aliases: C2×Dic3⋊C4, C23.34D6, C22.4Dic6, C61(C4⋊C4), (C2×C6).5Q8, C6.7(C2×Q8), C6.38(C2×D4), (C2×C6).35D4, (C2×C4).64D6, (C2×Dic3)⋊4C4, Dic34(C2×C4), (C22×C4).5S3, C2.2(C2×Dic6), C6.17(C22×C4), (C22×C12).4C2, C22.16(C4×S3), (C2×C6).41C23, (C2×C12).76C22, C22.19(C3⋊D4), (C22×C6).33C22, C22.20(C22×S3), (C22×Dic3).4C2, (C2×Dic3).33C22, C32(C2×C4⋊C4), C2.18(S3×C2×C4), C2.1(C2×C3⋊D4), (C2×C6).17(C2×C4), SmallGroup(96,130)

Series: Derived Chief Lower central Upper central

C1C6 — C2×Dic3⋊C4
C1C3C6C2×C6C2×Dic3C22×Dic3 — C2×Dic3⋊C4
C3C6 — C2×Dic3⋊C4
C1C23C22×C4

Generators and relations for C2×Dic3⋊C4
 G = < a,b,c,d | a2=b6=d4=1, c2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 162 in 92 conjugacy classes, 57 normal (17 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×8], C22, C22 [×6], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×12], C23, Dic3 [×4], Dic3 [×2], C12 [×2], C2×C6, C2×C6 [×6], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C2×Dic3 [×8], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×2], C22×C6, C2×C4⋊C4, Dic3⋊C4 [×4], C22×Dic3 [×2], C22×C12, C2×Dic3⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C4⋊C4, Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, C2×C3⋊D4, C2×Dic3⋊C4

Smallest permutation representation of C2×Dic3⋊C4
Regular action on 96 points
Generators in S96
(1 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 28)(8 29)(9 30)(10 25)(11 26)(12 27)(13 46)(14 47)(15 48)(16 43)(17 44)(18 45)(19 40)(20 41)(21 42)(22 37)(23 38)(24 39)(49 82)(50 83)(51 84)(52 79)(53 80)(54 81)(55 76)(56 77)(57 78)(58 73)(59 74)(60 75)(61 94)(62 95)(63 96)(64 91)(65 92)(66 93)(67 88)(68 89)(69 90)(70 85)(71 86)(72 87)
(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 56 4 59)(2 55 5 58)(3 60 6 57)(7 50 10 53)(8 49 11 52)(9 54 12 51)(13 68 16 71)(14 67 17 70)(15 72 18 69)(19 62 22 65)(20 61 23 64)(21 66 24 63)(25 80 28 83)(26 79 29 82)(27 84 30 81)(31 74 34 77)(32 73 35 76)(33 78 36 75)(37 92 40 95)(38 91 41 94)(39 96 42 93)(43 86 46 89)(44 85 47 88)(45 90 48 87)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 43 31 37)(26 44 32 38)(27 45 33 39)(28 46 34 40)(29 47 35 41)(30 48 36 42)(49 70 55 64)(50 71 56 65)(51 72 57 66)(52 67 58 61)(53 68 59 62)(54 69 60 63)(73 94 79 88)(74 95 80 89)(75 96 81 90)(76 91 82 85)(77 92 83 86)(78 93 84 87)

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

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

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

C2×Dic3⋊C4 is a maximal subgroup of
(C2×Dic3)⋊C8  (C2×C12)⋊Q8  C6.(C4×Q8)  Dic3⋊C42  C3⋊(C428C4)  C6.(C4×D4)  C2.(C4×D12)  C2.(C4×Dic6)  Dic3⋊C4⋊C4  (C2×C4)⋊Dic6  C6.(C4⋊Q8)  (C2×Dic3).9D4  (C2×C4).17D12  (C2×C4).Dic6  (C22×C4).85D6  D6⋊(C4⋊C4)  D6⋊C4⋊C4  D6⋊C45C4  C6.C22≀C2  C6.(C4⋊D4)  (C22×C4).37D6  C124(C4⋊C4)  (C2×C42).6S3  (C2×C42)⋊3S3  C24.55D6  C24.14D6  C24.15D6  C24.57D6  C24.17D6  C24.18D6  C24.20D6  C24.24D6  C24.25D6  C12⋊(C4⋊C4)  (C4×Dic3)⋊8C4  Dic3⋊(C4⋊C4)  C6.67(C4×D4)  (C2×Dic3)⋊Q8  (C2×C4).44D12  (C2×C12).54D4  (C2×Dic3).Q8  (C2×C12).288D4  D6⋊C46C4  (C2×C12).290D4  (C2×C12).56D4  C24.73D6  C24.31D6  C22.52(S3×Q8)  C2×C4×Dic6  C2×S3×C4⋊C4  C42.96D6  D45Dic6  C42.104D6  C42.108D6  C42.118D6  C6.322+ 1+4  C6.342+ 1+4  C6.702- 1+4  C6.752- 1+4  C6.522+ 1+4  C6.782- 1+4  C6.802- 1+4  C6.822- 1+4  C2×C4×C3⋊D4  C6.1042- 1+4
C2×Dic3⋊C4 is a maximal quotient of
C124(C4⋊C4)  C24.55D6  C24.57D6  C4⋊C4.225D6  C12⋊(C4⋊C4)  (C4×Dic3)⋊8C4  (C4×Dic3)⋊9C4  C4⋊C4.232D6  C4⋊C4.234D6  Dic3⋊C8⋊C2  Dic34M4(2)  C12.88(C2×Q8)  C23.8Dic6  C24.73D6

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

dim1111122222222
type++++++-++-
imageC1C2C2C2C4S3D4Q8D6D6Dic6C4×S3C3⋊D4
kernelC2×Dic3⋊C4Dic3⋊C4C22×Dic3C22×C12C2×Dic3C22×C4C2×C6C2×C6C2×C4C23C22C22C22
# reps1421812221444

Matrix representation of C2×Dic3⋊C4 in GL5(𝔽13)

120000
012000
001200
00010
00001
,
10000
0121200
01000
00011
000120
,
10000
010700
010300
00088
00005
,
80000
012000
001200
000119
00042

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,1,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,1,0],[1,0,0,0,0,0,10,10,0,0,0,7,3,0,0,0,0,0,8,0,0,0,0,8,5],[8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,11,4,0,0,0,9,2] >;

C2×Dic3⋊C4 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3\rtimes C_4
% in TeX

G:=Group("C2xDic3:C4");
// GroupNames label

G:=SmallGroup(96,130);
// by ID

G=gap.SmallGroup(96,130);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,362,50,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=d^4=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations

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