direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C6×C4.4D4, C4.13(C6×D4), (C2×C42)⋊12C6, C42⋊21(C2×C6), (C4×C12)⋊58C22, C12.320(C2×D4), (C2×C12).430D4, C24.14(C2×C6), (C6×Q8)⋊50C22, (C22×Q8)⋊12C6, C22.62(C6×D4), (C2×C6).346C24, (C22×D4).12C6, C6.185(C22×D4), C23.6(C22×C6), (C2×C12).659C23, (C6×D4).317C22, (C23×C6).13C22, C22.20(C23×C6), (C22×C6).85C23, (C22×C12).508C22, C2.9(D4×C2×C6), (C2×C4×C12)⋊22C2, (Q8×C2×C6)⋊16C2, (D4×C2×C6).24C2, C2.9(C6×C4○D4), (C2×Q8)⋊12(C2×C6), (C2×C4).86(C3×D4), (C6×C22⋊C4)⋊31C2, (C2×C22⋊C4)⋊11C6, C22⋊C4⋊14(C2×C6), (C2×D4).62(C2×C6), C6.228(C2×C4○D4), (C2×C6).683(C2×D4), (C2×C4).58(C22×C6), C22.32(C3×C4○D4), (C2×C6).232(C4○D4), (C3×C22⋊C4)⋊68C22, (C22×C4).107(C2×C6), SmallGroup(192,1415)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C4.4D4
G = < a,b,c,d | a6=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 530 in 330 conjugacy classes, 178 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C22×C6, C2×C42, C2×C22⋊C4, C4.4D4, C22×D4, C22×Q8, C4×C12, C3×C22⋊C4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C6×Q8, C23×C6, C2×C4.4D4, C2×C4×C12, C6×C22⋊C4, C3×C4.4D4, D4×C2×C6, Q8×C2×C6, C6×C4.4D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C24, C3×D4, C22×C6, C4.4D4, C22×D4, C2×C4○D4, C6×D4, C3×C4○D4, C23×C6, C2×C4.4D4, C3×C4.4D4, D4×C2×C6, C6×C4○D4, C6×C4.4D4
►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 14 26 21)(2 15 27 22)(3 16 28 23)(4 17 29 24)(5 18 30 19)(6 13 25 20)(7 31 91 42)(8 32 92 37)(9 33 93 38)(10 34 94 39)(11 35 95 40)(12 36 96 41)(43 73 55 85)(44 74 56 86)(45 75 57 87)(46 76 58 88)(47 77 59 89)(48 78 60 90)(49 67 61 79)(50 68 62 80)(51 69 63 81)(52 70 64 82)(53 71 65 83)(54 72 66 84)
(1 89 37 83)(2 90 38 84)(3 85 39 79)(4 86 40 80)(5 87 41 81)(6 88 42 82)(7 52 13 46)(8 53 14 47)(9 54 15 48)(10 49 16 43)(11 50 17 44)(12 51 18 45)(19 57 96 63)(20 58 91 64)(21 59 92 65)(22 60 93 66)(23 55 94 61)(24 56 95 62)(25 76 31 70)(26 77 32 71)(27 78 33 72)(28 73 34 67)(29 74 35 68)(30 75 36 69)
(1 53 26 65)(2 54 27 66)(3 49 28 61)(4 50 29 62)(5 51 30 63)(6 52 25 64)(7 88 91 76)(8 89 92 77)(9 90 93 78)(10 85 94 73)(11 86 95 74)(12 87 96 75)(13 82 20 70)(14 83 21 71)(15 84 22 72)(16 79 23 67)(17 80 24 68)(18 81 19 69)(31 58 42 46)(32 59 37 47)(33 60 38 48)(34 55 39 43)(35 56 40 44)(36 57 41 45)
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), (1,14,26,21)(2,15,27,22)(3,16,28,23)(4,17,29,24)(5,18,30,19)(6,13,25,20)(7,31,91,42)(8,32,92,37)(9,33,93,38)(10,34,94,39)(11,35,95,40)(12,36,96,41)(43,73,55,85)(44,74,56,86)(45,75,57,87)(46,76,58,88)(47,77,59,89)(48,78,60,90)(49,67,61,79)(50,68,62,80)(51,69,63,81)(52,70,64,82)(53,71,65,83)(54,72,66,84), (1,89,37,83)(2,90,38,84)(3,85,39,79)(4,86,40,80)(5,87,41,81)(6,88,42,82)(7,52,13,46)(8,53,14,47)(9,54,15,48)(10,49,16,43)(11,50,17,44)(12,51,18,45)(19,57,96,63)(20,58,91,64)(21,59,92,65)(22,60,93,66)(23,55,94,61)(24,56,95,62)(25,76,31,70)(26,77,32,71)(27,78,33,72)(28,73,34,67)(29,74,35,68)(30,75,36,69), (1,53,26,65)(2,54,27,66)(3,49,28,61)(4,50,29,62)(5,51,30,63)(6,52,25,64)(7,88,91,76)(8,89,92,77)(9,90,93,78)(10,85,94,73)(11,86,95,74)(12,87,96,75)(13,82,20,70)(14,83,21,71)(15,84,22,72)(16,79,23,67)(17,80,24,68)(18,81,19,69)(31,58,42,46)(32,59,37,47)(33,60,38,48)(34,55,39,43)(35,56,40,44)(36,57,41,45)>;
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), (1,14,26,21)(2,15,27,22)(3,16,28,23)(4,17,29,24)(5,18,30,19)(6,13,25,20)(7,31,91,42)(8,32,92,37)(9,33,93,38)(10,34,94,39)(11,35,95,40)(12,36,96,41)(43,73,55,85)(44,74,56,86)(45,75,57,87)(46,76,58,88)(47,77,59,89)(48,78,60,90)(49,67,61,79)(50,68,62,80)(51,69,63,81)(52,70,64,82)(53,71,65,83)(54,72,66,84), (1,89,37,83)(2,90,38,84)(3,85,39,79)(4,86,40,80)(5,87,41,81)(6,88,42,82)(7,52,13,46)(8,53,14,47)(9,54,15,48)(10,49,16,43)(11,50,17,44)(12,51,18,45)(19,57,96,63)(20,58,91,64)(21,59,92,65)(22,60,93,66)(23,55,94,61)(24,56,95,62)(25,76,31,70)(26,77,32,71)(27,78,33,72)(28,73,34,67)(29,74,35,68)(30,75,36,69), (1,53,26,65)(2,54,27,66)(3,49,28,61)(4,50,29,62)(5,51,30,63)(6,52,25,64)(7,88,91,76)(8,89,92,77)(9,90,93,78)(10,85,94,73)(11,86,95,74)(12,87,96,75)(13,82,20,70)(14,83,21,71)(15,84,22,72)(16,79,23,67)(17,80,24,68)(18,81,19,69)(31,58,42,46)(32,59,37,47)(33,60,38,48)(34,55,39,43)(35,56,40,44)(36,57,41,45) );
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)], [(1,14,26,21),(2,15,27,22),(3,16,28,23),(4,17,29,24),(5,18,30,19),(6,13,25,20),(7,31,91,42),(8,32,92,37),(9,33,93,38),(10,34,94,39),(11,35,95,40),(12,36,96,41),(43,73,55,85),(44,74,56,86),(45,75,57,87),(46,76,58,88),(47,77,59,89),(48,78,60,90),(49,67,61,79),(50,68,62,80),(51,69,63,81),(52,70,64,82),(53,71,65,83),(54,72,66,84)], [(1,89,37,83),(2,90,38,84),(3,85,39,79),(4,86,40,80),(5,87,41,81),(6,88,42,82),(7,52,13,46),(8,53,14,47),(9,54,15,48),(10,49,16,43),(11,50,17,44),(12,51,18,45),(19,57,96,63),(20,58,91,64),(21,59,92,65),(22,60,93,66),(23,55,94,61),(24,56,95,62),(25,76,31,70),(26,77,32,71),(27,78,33,72),(28,73,34,67),(29,74,35,68),(30,75,36,69)], [(1,53,26,65),(2,54,27,66),(3,49,28,61),(4,50,29,62),(5,51,30,63),(6,52,25,64),(7,88,91,76),(8,89,92,77),(9,90,93,78),(10,85,94,73),(11,86,95,74),(12,87,96,75),(13,82,20,70),(14,83,21,71),(15,84,22,72),(16,79,23,67),(17,80,24,68),(18,81,19,69),(31,58,42,46),(32,59,37,47),(33,60,38,48),(34,55,39,43),(35,56,40,44),(36,57,41,45)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | ··· | 4L | 4M | 4N | 4O | 4P | 6A | ··· | 6N | 6O | ··· | 6V | 12A | ··· | 12X | 12Y | ··· | 12AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | C4○D4 | C3×D4 | C3×C4○D4 |
| kernel | C6×C4.4D4 | C2×C4×C12 | C6×C22⋊C4 | C3×C4.4D4 | D4×C2×C6 | Q8×C2×C6 | C2×C4.4D4 | C2×C42 | C2×C22⋊C4 | C4.4D4 | C22×D4 | C22×Q8 | C2×C12 | C2×C6 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 8 | 1 | 1 | 2 | 2 | 8 | 16 | 2 | 2 | 4 | 8 | 8 | 16 |
Matrix representation of C6×C4.4D4 ►in GL5(𝔽13)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 5 | 0 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 2 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(5,GF(13))| [4,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,3,0,0,0,0,0,3],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,5,0],[12,0,0,0,0,0,1,12,0,0,0,2,12,0,0,0,0,0,0,12,0,0,0,12,0],[12,0,0,0,0,0,12,0,0,0,0,11,1,0,0,0,0,0,8,0,0,0,0,0,5] >;
C6×C4.4D4 in GAP, Magma, Sage, TeX
C_6\times C_4._4D_4
% in TeX
G:=Group("C6xC4.4D4"); // GroupNames label
G:=SmallGroup(192,1415);
// by ID
G=gap.SmallGroup(192,1415);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,680,2102,268]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^4=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations