direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C6×C4⋊C4, (C2×C12)⋊8C4, C4⋊2(C2×C12), (C2×C4)⋊3C12, C12⋊9(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
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, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C4⋊C4, C22×C4, C22×C4, C2×C12, C2×C12, C22×C6, C2×C4⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C6×C4⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C6×C4⋊C4
►Regular action on 96 points
(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⋊C4⋊5Dic3 (C2×C4).44D12 (C2×C12).54D4 (C2×Dic3).Q8 C4⋊C4⋊6Dic3 (C2×C12).288D4 (C2×C12).55D4 C4⋊(D6⋊C4) (C2×D12)⋊10C4 D6⋊C4⋊6C4 D6⋊C4⋊7C4 (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 | ··· | 2G | 3A | 3B | 4A | ··· | 4L | 6A | ··· | 6N | 12A | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | Q8 | C3×D4 | C3×Q8 |
| kernel | C6×C4⋊C4 | C3×C4⋊C4 | C22×C12 | C2×C4⋊C4 | C2×C12 | C4⋊C4 | C22×C4 | C2×C4 | C2×C6 | C2×C6 | C22 | C22 |
| # reps | 1 | 4 | 3 | 2 | 8 | 8 | 6 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C6×C4⋊C4 ►in GL4(𝔽13) generated by
| 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 | 0 |
| 0 | 0 | 5 | 5 |
| 5 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 1 |
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