direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C6×C8.C4, C4.77(C6×D4), (C2×C24).29C4, C24.79(C2×C4), C8.17(C2×C12), (C2×C8).11C12, C12.71(C4⋊C4), (C2×C12).79Q8, C12.482(C2×D4), (C2×C12).538D4, C22.1(C6×Q8), (C22×C8).14C6, (C22×C6).21Q8, C23.10(C3×Q8), C4.28(C22×C12), (C22×C24).24C2, M4(2).9(C2×C6), C12.186(C22×C4), (C2×C12).902C23, (C2×C24).410C22, (C2×M4(2)).15C6, (C6×M4(2)).33C2, (C22×C12).590C22, (C3×M4(2)).43C22, C2.15(C6×C4⋊C4), C6.71(C2×C4⋊C4), C4.22(C3×C4⋊C4), (C2×C8).90(C2×C6), (C2×C4).74(C3×D4), (C2×C4).21(C3×Q8), (C2×C6).14(C2×Q8), (C2×C6).65(C4⋊C4), (C2×C4).77(C2×C12), C22.11(C3×C4⋊C4), (C2×C12).338(C2×C4), (C2×C4).77(C22×C6), (C22×C4).126(C2×C6), SmallGroup(192,862)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C8.C4
G = < a,b,c | a6=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 130 in 106 conjugacy classes, 82 normal (42 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C8, C8, C2×C4, C23, C12, C2×C6, C2×C6, C2×C8, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C24, C24, C2×C12, C22×C6, C8.C4, C22×C8, C2×M4(2), C2×C24, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C22×C12, C2×C8.C4, C3×C8.C4, C22×C24, C6×M4(2), C6×C8.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, C8.C4, C2×C4⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C2×C8.C4, C3×C8.C4, C6×C4⋊C4, C6×C8.C4
►On 96 points
(1 47 10 18 39 55)(2 48 11 19 40 56)(3 41 12 20 33 49)(4 42 13 21 34 50)(5 43 14 22 35 51)(6 44 15 23 36 52)(7 45 16 24 37 53)(8 46 9 17 38 54)(25 86 96 61 72 77)(26 87 89 62 65 78)(27 88 90 63 66 79)(28 81 91 64 67 80)(29 82 92 57 68 73)(30 83 93 58 69 74)(31 84 94 59 70 75)(32 85 95 60 71 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 85 24 65 5 81 20 69)(2 84 17 72 6 88 21 68)(3 83 18 71 7 87 22 67)(4 82 19 70 8 86 23 66)(9 61 52 27 13 57 56 31)(10 60 53 26 14 64 49 30)(11 59 54 25 15 63 50 29)(12 58 55 32 16 62 51 28)(33 74 47 95 37 78 43 91)(34 73 48 94 38 77 44 90)(35 80 41 93 39 76 45 89)(36 79 42 92 40 75 46 96)
G:=sub<Sym(96)| (1,47,10,18,39,55)(2,48,11,19,40,56)(3,41,12,20,33,49)(4,42,13,21,34,50)(5,43,14,22,35,51)(6,44,15,23,36,52)(7,45,16,24,37,53)(8,46,9,17,38,54)(25,86,96,61,72,77)(26,87,89,62,65,78)(27,88,90,63,66,79)(28,81,91,64,67,80)(29,82,92,57,68,73)(30,83,93,58,69,74)(31,84,94,59,70,75)(32,85,95,60,71,76), (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), (1,85,24,65,5,81,20,69)(2,84,17,72,6,88,21,68)(3,83,18,71,7,87,22,67)(4,82,19,70,8,86,23,66)(9,61,52,27,13,57,56,31)(10,60,53,26,14,64,49,30)(11,59,54,25,15,63,50,29)(12,58,55,32,16,62,51,28)(33,74,47,95,37,78,43,91)(34,73,48,94,38,77,44,90)(35,80,41,93,39,76,45,89)(36,79,42,92,40,75,46,96)>;
G:=Group( (1,47,10,18,39,55)(2,48,11,19,40,56)(3,41,12,20,33,49)(4,42,13,21,34,50)(5,43,14,22,35,51)(6,44,15,23,36,52)(7,45,16,24,37,53)(8,46,9,17,38,54)(25,86,96,61,72,77)(26,87,89,62,65,78)(27,88,90,63,66,79)(28,81,91,64,67,80)(29,82,92,57,68,73)(30,83,93,58,69,74)(31,84,94,59,70,75)(32,85,95,60,71,76), (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), (1,85,24,65,5,81,20,69)(2,84,17,72,6,88,21,68)(3,83,18,71,7,87,22,67)(4,82,19,70,8,86,23,66)(9,61,52,27,13,57,56,31)(10,60,53,26,14,64,49,30)(11,59,54,25,15,63,50,29)(12,58,55,32,16,62,51,28)(33,74,47,95,37,78,43,91)(34,73,48,94,38,77,44,90)(35,80,41,93,39,76,45,89)(36,79,42,92,40,75,46,96) );
G=PermutationGroup([[(1,47,10,18,39,55),(2,48,11,19,40,56),(3,41,12,20,33,49),(4,42,13,21,34,50),(5,43,14,22,35,51),(6,44,15,23,36,52),(7,45,16,24,37,53),(8,46,9,17,38,54),(25,86,96,61,72,77),(26,87,89,62,65,78),(27,88,90,63,66,79),(28,81,91,64,67,80),(29,82,92,57,68,73),(30,83,93,58,69,74),(31,84,94,59,70,75),(32,85,95,60,71,76)], [(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)], [(1,85,24,65,5,81,20,69),(2,84,17,72,6,88,21,68),(3,83,18,71,7,87,22,67),(4,82,19,70,8,86,23,66),(9,61,52,27,13,57,56,31),(10,60,53,26,14,64,49,30),(11,59,54,25,15,63,50,29),(12,58,55,32,16,62,51,28),(33,74,47,95,37,78,43,91),(34,73,48,94,38,77,44,90),(35,80,41,93,39,76,45,89),(36,79,42,92,40,75,46,96)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 8I | ··· | 8P | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | ··· | 24P | 24Q | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | Q8 | Q8 | C3×D4 | C3×Q8 | C3×Q8 | C8.C4 | C3×C8.C4 |
| kernel | C6×C8.C4 | C3×C8.C4 | C22×C24 | C6×M4(2) | C2×C8.C4 | C2×C24 | C8.C4 | C22×C8 | C2×M4(2) | C2×C8 | C2×C12 | C2×C12 | C22×C6 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 8 | 8 | 2 | 4 | 16 | 2 | 1 | 1 | 4 | 2 | 2 | 8 | 16 |
Matrix representation of C6×C8.C4 ►in GL3(𝔽73) generated by
| 9 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 72 | 0 | 0 |
| 0 | 51 | 0 |
| 0 | 22 | 63 |
| 1 | 0 | 0 |
| 0 | 28 | 2 |
| 0 | 23 | 45 |
G:=sub<GL(3,GF(73))| [9,0,0,0,1,0,0,0,1],[72,0,0,0,51,22,0,0,63],[1,0,0,0,28,23,0,2,45] >;
C6×C8.C4 in GAP, Magma, Sage, TeX
C_6\times C_8.C_4
% in TeX
G:=Group("C6xC8.C4"); // GroupNames label
G:=SmallGroup(192,862);
// by ID
G=gap.SmallGroup(192,862);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,176,4204,172,124]);
// Polycyclic
G:=Group<a,b,c|a^6=b^8=1,c^4=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations