metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.4C42, C23.10Dic6, M4(2).2Dic3, C4.4(C4×Dic3), C4.50(D6⋊C4), (C2×C4).129D12, (C2×C12).111D4, (C22×C6).8Q8, C4.Dic3.4C4, C6.6(C8.C4), C3⋊2(C4.C42), (C22×C4).342D6, (C2×M4(2)).9S3, (C3×M4(2)).4C4, C12.95(C22⋊C4), (C6×M4(2)).13C2, C22.4(C4⋊Dic3), C4.29(C6.D4), C2.3(C12.53D4), C6.17(C2.C42), C2.17(C6.C42), C22.22(Dic3⋊C4), (C22×C12).127C22, (C2×C3⋊C8).7C4, (C2×C6).9(C4⋊C4), (C22×C3⋊C8).2C2, (C2×C12).64(C2×C4), (C2×C4).141(C4×S3), (C2×C4).40(C2×Dic3), (C2×C4).270(C3⋊D4), (C2×C4.Dic3).12C2, SmallGroup(192,117)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.4C42
G = < a,b,c | a12=1, b4=c4=a6, bab-1=a5, cac-1=a7, cbc-1=a3b >
Subgroups: 168 in 90 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C8, C2×C4, C23, C12, C2×C6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C3⋊C8, C24, C2×C12, C22×C6, C22×C8, C2×M4(2), C2×M4(2), C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C4.Dic3, C2×C24, C3×M4(2), C3×M4(2), C22×C12, C4.C42, C22×C3⋊C8, C2×C4.Dic3, C6×M4(2), C12.4C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C8.C4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4.C42, C12.53D4, C6.C42, C12.4C42
►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 49 77 61 7 55 83 67)(2 54 78 66 8 60 84 72)(3 59 79 71 9 53 73 65)(4 52 80 64 10 58 74 70)(5 57 81 69 11 51 75 63)(6 50 82 62 12 56 76 68)(13 94 37 35 19 88 43 29)(14 87 38 28 20 93 44 34)(15 92 39 33 21 86 45 27)(16 85 40 26 22 91 46 32)(17 90 41 31 23 96 47 25)(18 95 42 36 24 89 48 30)
(1 32 83 85 7 26 77 91)(2 27 84 92 8 33 78 86)(3 34 73 87 9 28 79 93)(4 29 74 94 10 35 80 88)(5 36 75 89 11 30 81 95)(6 31 76 96 12 25 82 90)(13 61 37 49 19 67 43 55)(14 68 38 56 20 62 44 50)(15 63 39 51 21 69 45 57)(16 70 40 58 22 64 46 52)(17 65 41 53 23 71 47 59)(18 72 42 60 24 66 48 54)
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), (1,49,77,61,7,55,83,67)(2,54,78,66,8,60,84,72)(3,59,79,71,9,53,73,65)(4,52,80,64,10,58,74,70)(5,57,81,69,11,51,75,63)(6,50,82,62,12,56,76,68)(13,94,37,35,19,88,43,29)(14,87,38,28,20,93,44,34)(15,92,39,33,21,86,45,27)(16,85,40,26,22,91,46,32)(17,90,41,31,23,96,47,25)(18,95,42,36,24,89,48,30), (1,32,83,85,7,26,77,91)(2,27,84,92,8,33,78,86)(3,34,73,87,9,28,79,93)(4,29,74,94,10,35,80,88)(5,36,75,89,11,30,81,95)(6,31,76,96,12,25,82,90)(13,61,37,49,19,67,43,55)(14,68,38,56,20,62,44,50)(15,63,39,51,21,69,45,57)(16,70,40,58,22,64,46,52)(17,65,41,53,23,71,47,59)(18,72,42,60,24,66,48,54)>;
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), (1,49,77,61,7,55,83,67)(2,54,78,66,8,60,84,72)(3,59,79,71,9,53,73,65)(4,52,80,64,10,58,74,70)(5,57,81,69,11,51,75,63)(6,50,82,62,12,56,76,68)(13,94,37,35,19,88,43,29)(14,87,38,28,20,93,44,34)(15,92,39,33,21,86,45,27)(16,85,40,26,22,91,46,32)(17,90,41,31,23,96,47,25)(18,95,42,36,24,89,48,30), (1,32,83,85,7,26,77,91)(2,27,84,92,8,33,78,86)(3,34,73,87,9,28,79,93)(4,29,74,94,10,35,80,88)(5,36,75,89,11,30,81,95)(6,31,76,96,12,25,82,90)(13,61,37,49,19,67,43,55)(14,68,38,56,20,62,44,50)(15,63,39,51,21,69,45,57)(16,70,40,58,22,64,46,52)(17,65,41,53,23,71,47,59)(18,72,42,60,24,66,48,54) );
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)], [(1,49,77,61,7,55,83,67),(2,54,78,66,8,60,84,72),(3,59,79,71,9,53,73,65),(4,52,80,64,10,58,74,70),(5,57,81,69,11,51,75,63),(6,50,82,62,12,56,76,68),(13,94,37,35,19,88,43,29),(14,87,38,28,20,93,44,34),(15,92,39,33,21,86,45,27),(16,85,40,26,22,91,46,32),(17,90,41,31,23,96,47,25),(18,95,42,36,24,89,48,30)], [(1,32,83,85,7,26,77,91),(2,27,84,92,8,33,78,86),(3,34,73,87,9,28,79,93),(4,29,74,94,10,35,80,88),(5,36,75,89,11,30,81,95),(6,31,76,96,12,25,82,90),(13,61,37,49,19,67,43,55),(14,68,38,56,20,62,44,50),(15,63,39,51,21,69,45,57),(16,70,40,58,22,64,46,52),(17,65,41,53,23,71,47,59),(18,72,42,60,24,66,48,54)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 8M | 8N | 8O | 8P | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 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 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | Q8 | Dic3 | D6 | C4×S3 | D12 | C3⋊D4 | Dic6 | C8.C4 | C12.53D4 |
| kernel | C12.4C42 | C22×C3⋊C8 | C2×C4.Dic3 | C6×M4(2) | C2×C3⋊C8 | C4.Dic3 | C3×M4(2) | C2×M4(2) | C2×C12 | C22×C6 | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 3 | 1 | 2 | 1 | 4 | 2 | 4 | 2 | 8 | 4 |
Matrix representation of C12.4C42 ►in GL4(𝔽73) generated by
| 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 1 | 27 |
| 71 | 53 | 0 | 0 |
| 55 | 2 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 67 | 51 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 27 | 71 |
| 0 | 0 | 13 | 46 |
G:=sub<GL(4,GF(73))| [0,72,0,0,1,72,0,0,0,0,46,1,0,0,0,27],[71,55,0,0,53,2,0,0,0,0,10,67,0,0,0,51],[46,0,0,0,0,46,0,0,0,0,27,13,0,0,71,46] >;
C12.4C42 in GAP, Magma, Sage, TeX
C_{12}._4C_4^2 % in TeX
G:=Group("C12.4C4^2"); // GroupNames label
G:=SmallGroup(192,117);
// by ID
G=gap.SmallGroup(192,117);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,365,36,184,570,136,6278]);
// 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,c*b*c^-1=a^3*b>;
// generators/relations