metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.7C42, M4(2)⋊5Dic3, C24.46(C2×C4), C24⋊C4⋊27C2, (C2×C8).275D6, (C2×C6).7C42, C4.7(C4×Dic3), (C8×Dic3)⋊30C2, C6.31(C8○D4), (C3×M4(2))⋊4C4, C4⋊Dic3.21C4, C23.33(C4×S3), C6.26(C2×C42), C8.12(C2×Dic3), C3⋊5(C8○2M4(2)), C2.5(D12.C4), (C22×C4).363D6, (C6×M4(2)).8C2, C22.7(C4×Dic3), C12.177(C22×C4), (C2×C24).276C22, (C2×C12).865C23, C6.D4.11C4, (C2×M4(2)).19S3, C4.35(C22×Dic3), (C22×C12).181C22, (C4×Dic3).285C22, C23.26D6.18C2, (C2×C3⋊C8)⋊9C4, C3⋊C8.24(C2×C4), C4.115(S3×C2×C4), (C2×C4).83(C4×S3), C22.63(S3×C2×C4), C2.14(C2×C4×Dic3), (C22×C3⋊C8).11C2, (C2×C12).101(C2×C4), (C2×C3⋊C8).334C22, (C22×C6).65(C2×C4), (C2×C4).47(C2×Dic3), (C2×C4).807(C22×S3), (C2×C6).135(C22×C4), (C2×Dic3).67(C2×C4), SmallGroup(192,681)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.7C42
G = < a,b,c | a12=1, b4=c4=a6, bab-1=a5, cac-1=a7, bc=cb >
Subgroups: 216 in 130 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C22×C6, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C2×C3⋊C8, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×M4(2), C22×C12, C8○2M4(2), C8×Dic3, C24⋊C4, C22×C3⋊C8, C23.26D6, C6×M4(2), C12.7C42
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C42, C22×C4, C4×S3, C2×Dic3, C22×S3, C2×C42, C8○D4, C4×Dic3, S3×C2×C4, C22×Dic3, C8○2M4(2), D12.C4, C2×C4×Dic3, C12.7C42
►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 70 10 67 7 64 4 61)(2 63 11 72 8 69 5 66)(3 68 12 65 9 62 6 71)(13 32 22 29 19 26 16 35)(14 25 23 34 20 31 17 28)(15 30 24 27 21 36 18 33)(37 92 40 95 43 86 46 89)(38 85 41 88 44 91 47 94)(39 90 42 93 45 96 48 87)(49 84 52 75 55 78 58 81)(50 77 53 80 56 83 59 74)(51 82 54 73 57 76 60 79)
(1 77 24 87 7 83 18 93)(2 84 13 94 8 78 19 88)(3 79 14 89 9 73 20 95)(4 74 15 96 10 80 21 90)(5 81 16 91 11 75 22 85)(6 76 17 86 12 82 23 92)(25 37 62 57 31 43 68 51)(26 44 63 52 32 38 69 58)(27 39 64 59 33 45 70 53)(28 46 65 54 34 40 71 60)(29 41 66 49 35 47 72 55)(30 48 67 56 36 42 61 50)
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), (1,70,10,67,7,64,4,61)(2,63,11,72,8,69,5,66)(3,68,12,65,9,62,6,71)(13,32,22,29,19,26,16,35)(14,25,23,34,20,31,17,28)(15,30,24,27,21,36,18,33)(37,92,40,95,43,86,46,89)(38,85,41,88,44,91,47,94)(39,90,42,93,45,96,48,87)(49,84,52,75,55,78,58,81)(50,77,53,80,56,83,59,74)(51,82,54,73,57,76,60,79), (1,77,24,87,7,83,18,93)(2,84,13,94,8,78,19,88)(3,79,14,89,9,73,20,95)(4,74,15,96,10,80,21,90)(5,81,16,91,11,75,22,85)(6,76,17,86,12,82,23,92)(25,37,62,57,31,43,68,51)(26,44,63,52,32,38,69,58)(27,39,64,59,33,45,70,53)(28,46,65,54,34,40,71,60)(29,41,66,49,35,47,72,55)(30,48,67,56,36,42,61,50)>;
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), (1,70,10,67,7,64,4,61)(2,63,11,72,8,69,5,66)(3,68,12,65,9,62,6,71)(13,32,22,29,19,26,16,35)(14,25,23,34,20,31,17,28)(15,30,24,27,21,36,18,33)(37,92,40,95,43,86,46,89)(38,85,41,88,44,91,47,94)(39,90,42,93,45,96,48,87)(49,84,52,75,55,78,58,81)(50,77,53,80,56,83,59,74)(51,82,54,73,57,76,60,79), (1,77,24,87,7,83,18,93)(2,84,13,94,8,78,19,88)(3,79,14,89,9,73,20,95)(4,74,15,96,10,80,21,90)(5,81,16,91,11,75,22,85)(6,76,17,86,12,82,23,92)(25,37,62,57,31,43,68,51)(26,44,63,52,32,38,69,58)(27,39,64,59,33,45,70,53)(28,46,65,54,34,40,71,60)(29,41,66,49,35,47,72,55)(30,48,67,56,36,42,61,50) );
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)], [(1,70,10,67,7,64,4,61),(2,63,11,72,8,69,5,66),(3,68,12,65,9,62,6,71),(13,32,22,29,19,26,16,35),(14,25,23,34,20,31,17,28),(15,30,24,27,21,36,18,33),(37,92,40,95,43,86,46,89),(38,85,41,88,44,91,47,94),(39,90,42,93,45,96,48,87),(49,84,52,75,55,78,58,81),(50,77,53,80,56,83,59,74),(51,82,54,73,57,76,60,79)], [(1,77,24,87,7,83,18,93),(2,84,13,94,8,78,19,88),(3,79,14,89,9,73,20,95),(4,74,15,96,10,80,21,90),(5,81,16,91,11,75,22,85),(6,76,17,86,12,82,23,92),(25,37,62,57,31,43,68,51),(26,44,63,52,32,38,69,58),(27,39,64,59,33,45,70,53),(28,46,65,54,34,40,71,60),(29,41,66,49,35,47,72,55),(30,48,67,56,36,42,61,50)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 6A | 6B | 6C | 6D | 6E | 8A | ··· | 8H | 8I | ··· | 8P | 8Q | 8R | 8S | 8T | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D6 | Dic3 | D6 | C4×S3 | C4×S3 | C8○D4 | D12.C4 |
| kernel | C12.7C42 | C8×Dic3 | C24⋊C4 | C22×C3⋊C8 | C23.26D6 | C6×M4(2) | C2×C3⋊C8 | C4⋊Dic3 | C6.D4 | C3×M4(2) | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 4 | 8 | 1 | 2 | 4 | 1 | 6 | 2 | 8 | 4 |
Matrix representation of C12.7C42 ►in GL4(𝔽73) generated by
| 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 26 | 46 |
| 31 | 3 | 0 | 0 |
| 45 | 42 | 0 | 0 |
| 0 | 0 | 63 | 0 |
| 0 | 0 | 0 | 63 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 28 | 71 |
| 0 | 0 | 50 | 45 |
G:=sub<GL(4,GF(73))| [0,72,0,0,1,72,0,0,0,0,27,26,0,0,0,46],[31,45,0,0,3,42,0,0,0,0,63,0,0,0,0,63],[46,0,0,0,0,46,0,0,0,0,28,50,0,0,71,45] >;
C12.7C42 in GAP, Magma, Sage, TeX
C_{12}._7C_4^2 % in TeX
G:=Group("C12.7C4^2"); // GroupNames label
G:=SmallGroup(192,681);
// by ID
G=gap.SmallGroup(192,681);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,387,100,136,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^4=c^4=a^6,b*a*b^-1=a^5,c*a*c^-1=a^7,b*c=c*b>;
// generators/relations