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G = C2×C4×Dic3order 96 = 25·3

Direct product of C2×C4 and Dic3

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

Aliases: C2×C4×Dic3, C6⋊C42, C23.33D6, (C2×C12)⋊7C4, C128(C2×C4), C32(C2×C42), C42(C4×Dic3), (C2×C4).101D6, C6.22(C22×C4), (C22×C4).12S3, C22.15(C4×S3), (C2×C6).40C23, (C22×C12).13C2, C2.2(C22×Dic3), (C2×C12).113C22, C22.19(C22×S3), (C22×C6).32C22, (C22×Dic3).7C2, C22.13(C2×Dic3), (C2×Dic3).50C22, C2.3(S3×C2×C4), (C2×C4)(C4×Dic3), (C2×C6).33(C2×C4), SmallGroup(96,129)

Series: Derived Chief Lower central Upper central

C1C3 — C2×C4×Dic3
C1C3C6C2×C6C2×Dic3C22×Dic3 — C2×C4×Dic3
C3 — C2×C4×Dic3
C1C22×C4

Generators and relations for C2×C4×Dic3
 G = < a,b,c,d | a2=b4=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 162 in 108 conjugacy classes, 81 normal (11 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C4 [×8], C22, C22 [×6], C6, C6 [×6], C2×C4 [×6], C2×C4 [×12], C23, Dic3 [×8], C12 [×4], C2×C6, C2×C6 [×6], C42 [×4], C22×C4, C22×C4 [×2], C2×Dic3 [×12], C2×C12 [×6], C22×C6, C2×C42, C4×Dic3 [×4], C22×Dic3 [×2], C22×C12, C2×C4×Dic3
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], S3, C2×C4 [×18], C23, Dic3 [×4], D6 [×3], C42 [×4], C22×C4 [×3], C4×S3 [×4], C2×Dic3 [×6], C22×S3, C2×C42, C4×Dic3 [×4], S3×C2×C4 [×2], C22×Dic3, C2×C4×Dic3

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

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

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

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

C2×C4×Dic3 is a maximal subgroup of
C12.2C42  (C2×C24)⋊5C4  C12.3C42  (C2×C12)⋊Q8  C6.(C4×Q8)  Dic3.5C42  Dic3⋊C42  C3⋊(C428C4)  C3⋊(C425C4)  C6.(C4×D4)  C2.(C4×D12)  C2.(C4×Dic6)  Dic3⋊C4⋊C4  D6⋊C42  D6⋊C45C4  D6⋊C43C4  Dic3.5M4(2)  Dic3.M4(2)  C426Dic3  C24.14D6  C24.15D6  C24.19D6  C24.24D6  C12⋊(C4⋊C4)  C4.(D6⋊C4)  (C4×Dic3)⋊8C4  Dic3⋊(C4⋊C4)  (C4×Dic3)⋊9C4  C6.67(C4×D4)  C4⋊C45Dic3  C4⋊C46Dic3  (C2×D12)⋊10C4  D6⋊C47C4  Dic34M4(2)  C24.30D6  (C6×Q8)⋊7C4  S3×C2×C42  C42.88D6  C42.188D6  C42.102D6  C12⋊(C4○D4)  C4⋊C4.178D6  (Q8×Dic3)⋊C2  C4⋊C4.187D6  C4⋊C4.197D6  (C2×C12)⋊17D4
C2×C4×Dic3 is a maximal quotient of
C426Dic3  C12.5C42  C12.12C42  C12.7C42

48 conjugacy classes

class 1 2A···2G 3 4A···4H4I···4X6A···6G12A···12H
order12···234···44···46···612···12
size11···121···13···32···22···2

48 irreducible representations

dim11111122222
type+++++-++
imageC1C2C2C2C4C4S3Dic3D6D6C4×S3
kernelC2×C4×Dic3C4×Dic3C22×Dic3C22×C12C2×Dic3C2×C12C22×C4C2×C4C2×C4C23C22
# reps142116814218

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

10000
012000
001200
00010
00001
,
50000
05000
001200
000120
000012
,
10000
012000
00100
000121
000120
,
120000
05000
001200
0001010
00073

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

C2×C4×Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC4xDic3");
// GroupNames label

G:=SmallGroup(96,129);
// by ID

G=gap.SmallGroup(96,129);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,86,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^6=1,d^2=c^3,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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