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G = C6×C4⋊C4order 96 = 25·3

Direct product of C6 and C4⋊C4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C6×C4⋊C4, (C2×C12)⋊8C4, C42(C2×C12), (C2×C4)⋊3C12, C129(C2×C4), C2.2(C6×D4), (C2×C6).8Q8, C2.1(C6×Q8), (C2×C6).51D4, C6.65(C2×D4), C6.18(C2×Q8), (C22×C4).5C6, C22.3(C3×Q8), (C22×C12).5C2, C2.2(C22×C12), (C2×C6).71C23, C6.30(C22×C4), C23.16(C2×C6), C22.13(C3×D4), C22.11(C2×C12), C22.5(C22×C6), (C2×C12).120C22, (C22×C6).49C22, (C2×C4).13(C2×C6), (C2×C6).40(C2×C4), SmallGroup(96,163)

Series: Derived Chief Lower central Upper central

C1C2 — C6×C4⋊C4
C1C2C22C2×C6C2×C12C3×C4⋊C4 — C6×C4⋊C4
C1C2 — C6×C4⋊C4
C1C22×C6 — C6×C4⋊C4

Generators and relations for C6×C4⋊C4
 G = < a,b,c | a6=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 108 in 92 conjugacy classes, 76 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×4], C22, C22 [×6], C6 [×3], C6 [×4], C2×C4 [×10], C2×C4 [×4], C23, C12 [×4], C12 [×4], C2×C6, C2×C6 [×6], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C2×C12 [×10], C2×C12 [×4], C22×C6, C2×C4⋊C4, C3×C4⋊C4 [×4], C22×C12, C22×C12 [×2], C6×C4⋊C4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C12 [×4], C2×C6 [×7], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C2×C12 [×6], C3×D4 [×2], C3×Q8 [×2], C22×C6, C2×C4⋊C4, C3×C4⋊C4 [×4], C22×C12, C6×D4, C6×Q8, C6×C4⋊C4

Smallest permutation representation of C6×C4⋊C4
Regular action on 96 points
Generators in S96
(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 53 32 59)(2 54 33 60)(3 49 34 55)(4 50 35 56)(5 51 36 57)(6 52 31 58)(7 82 20 76)(8 83 21 77)(9 84 22 78)(10 79 23 73)(11 80 24 74)(12 81 19 75)(13 88 91 70)(14 89 92 71)(15 90 93 72)(16 85 94 67)(17 86 95 68)(18 87 96 69)(25 64 42 46)(26 65 37 47)(27 66 38 48)(28 61 39 43)(29 62 40 44)(30 63 41 45)
(1 83 37 89)(2 84 38 90)(3 79 39 85)(4 80 40 86)(5 81 41 87)(6 82 42 88)(7 46 13 52)(8 47 14 53)(9 48 15 54)(10 43 16 49)(11 44 17 50)(12 45 18 51)(19 63 96 57)(20 64 91 58)(21 65 92 59)(22 66 93 60)(23 61 94 55)(24 62 95 56)(25 70 31 76)(26 71 32 77)(27 72 33 78)(28 67 34 73)(29 68 35 74)(30 69 36 75)

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

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

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

C6×C4⋊C4 is a maximal subgroup of
(C2×C12)⋊C8  C12.C42  C12.(C4⋊C4)  C4⋊C4.225D6  C4○D12⋊C4  (C2×C6).40D8  C4⋊C4.228D6  C4⋊C4.230D6  C4⋊C4.231D6  C12⋊(C4⋊C4)  C4.(D6⋊C4)  (C4×Dic3)⋊8C4  Dic3⋊(C4⋊C4)  (C4×Dic3)⋊9C4  C6.67(C4×D4)  (C2×Dic3)⋊Q8  C4⋊C45Dic3  (C2×C4).44D12  (C2×C12).54D4  (C2×Dic3).Q8  C4⋊C46Dic3  (C2×C12).288D4  (C2×C12).55D4  C4⋊(D6⋊C4)  (C2×D12)⋊10C4  D6⋊C46C4  D6⋊C47C4  (C2×C4)⋊3D12  (C2×C12).289D4  (C2×C12).290D4  (C2×C12).56D4  C6.72+ 1+4  C6.82+ 1+4  C6.2- 1+4  C6.2+ 1+4  C6.102+ 1+4  C6.52- 1+4  C6.112+ 1+4  C6.62- 1+4  D4×C2×C12  Q8×C2×C12

60 conjugacy classes

class 1 2A···2G3A3B4A···4L6A···6N12A···12X
order12···2334···46···612···12
size11···1112···21···12···2

60 irreducible representations

dim111111112222
type++++-
imageC1C2C2C3C4C6C6C12D4Q8C3×D4C3×Q8
kernelC6×C4⋊C4C3×C4⋊C4C22×C12C2×C4⋊C4C2×C12C4⋊C4C22×C4C2×C4C2×C6C2×C6C22C22
# reps1432886162244

Matrix representation of C6×C4⋊C4 in GL4(𝔽13) generated by

12000
01200
0040
0004
,
1000
01200
0080
0055
,
5000
0100
001211
0001
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,12,0,0,0,0,8,5,0,0,0,5],[5,0,0,0,0,1,0,0,0,0,12,0,0,0,11,1] >;

C6×C4⋊C4 in GAP, Magma, Sage, TeX

C_6\times C_4\rtimes C_4
% in TeX

G:=Group("C6xC4:C4");
// GroupNames label

G:=SmallGroup(96,163);
// by ID

G=gap.SmallGroup(96,163);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,288,313,151]);
// Polycyclic

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

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