metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.5C42, C42.186D6, C4.Dic3⋊9C4, (C2×C6).5C42, C4.5(C4×Dic3), C6.35(C8○D4), C4⋊C4.10Dic3, C6.21(C2×C42), C3⋊3(C8○2M4(2)), (C22×C4).352D6, C22⋊C4.6Dic3, C22.5(C4×Dic3), C42⋊C2.14S3, (C4×C12).231C22, C12.136(C22×C4), (C2×C12).845C23, C42.S3⋊21C2, C2.1(D4.Dic3), C23.19(C2×Dic3), (C22×C12).147C22, C22.21(C22×Dic3), (C2×C3⋊C8)⋊7C4, (C4×C3⋊C8)⋊25C2, (C3×C4⋊C4).7C4, C3⋊C8.23(C2×C4), C4.110(S3×C2×C4), (C2×C4).80(C4×S3), (C22×C3⋊C8).6C2, C2.10(C2×C4×Dic3), (C2×C12).88(C2×C4), (C3×C22⋊C4).7C4, (C2×C3⋊C8).332C22, (C22×C6).56(C2×C4), (C2×C4).42(C2×Dic3), (C2×C6).182(C22×C4), (C2×C4).787(C22×S3), (C3×C42⋊C2).6C2, (C2×C4.Dic3).17C2, SmallGroup(192,556)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.5C42
G = < a,b,c | a12=c4=1, b4=a6, bab-1=a5, cac-1=a7, bc=cb >
Subgroups: 200 in 130 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C3⋊C8, 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×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C8○2M4(2), C4×C3⋊C8, C42.S3, C22×C3⋊C8, C2×C4.Dic3, C3×C42⋊C2, C12.5C42
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), C2×C4×Dic3, D4.Dic3, C12.5C42
►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 58 10 55 7 52 4 49)(2 51 11 60 8 57 5 54)(3 56 12 53 9 50 6 59)(13 46 22 43 19 40 16 37)(14 39 23 48 20 45 17 42)(15 44 24 41 21 38 18 47)(25 63 28 66 31 69 34 72)(26 68 29 71 32 62 35 65)(27 61 30 64 33 67 36 70)(73 85 76 88 79 91 82 94)(74 90 77 93 80 96 83 87)(75 95 78 86 81 89 84 92)
(1 28 42 75)(2 35 43 82)(3 30 44 77)(4 25 45 84)(5 32 46 79)(6 27 47 74)(7 34 48 81)(8 29 37 76)(9 36 38 83)(10 31 39 78)(11 26 40 73)(12 33 41 80)(13 88 57 71)(14 95 58 66)(15 90 59 61)(16 85 60 68)(17 92 49 63)(18 87 50 70)(19 94 51 65)(20 89 52 72)(21 96 53 67)(22 91 54 62)(23 86 55 69)(24 93 56 64)
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), (1,58,10,55,7,52,4,49)(2,51,11,60,8,57,5,54)(3,56,12,53,9,50,6,59)(13,46,22,43,19,40,16,37)(14,39,23,48,20,45,17,42)(15,44,24,41,21,38,18,47)(25,63,28,66,31,69,34,72)(26,68,29,71,32,62,35,65)(27,61,30,64,33,67,36,70)(73,85,76,88,79,91,82,94)(74,90,77,93,80,96,83,87)(75,95,78,86,81,89,84,92), (1,28,42,75)(2,35,43,82)(3,30,44,77)(4,25,45,84)(5,32,46,79)(6,27,47,74)(7,34,48,81)(8,29,37,76)(9,36,38,83)(10,31,39,78)(11,26,40,73)(12,33,41,80)(13,88,57,71)(14,95,58,66)(15,90,59,61)(16,85,60,68)(17,92,49,63)(18,87,50,70)(19,94,51,65)(20,89,52,72)(21,96,53,67)(22,91,54,62)(23,86,55,69)(24,93,56,64)>;
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), (1,58,10,55,7,52,4,49)(2,51,11,60,8,57,5,54)(3,56,12,53,9,50,6,59)(13,46,22,43,19,40,16,37)(14,39,23,48,20,45,17,42)(15,44,24,41,21,38,18,47)(25,63,28,66,31,69,34,72)(26,68,29,71,32,62,35,65)(27,61,30,64,33,67,36,70)(73,85,76,88,79,91,82,94)(74,90,77,93,80,96,83,87)(75,95,78,86,81,89,84,92), (1,28,42,75)(2,35,43,82)(3,30,44,77)(4,25,45,84)(5,32,46,79)(6,27,47,74)(7,34,48,81)(8,29,37,76)(9,36,38,83)(10,31,39,78)(11,26,40,73)(12,33,41,80)(13,88,57,71)(14,95,58,66)(15,90,59,61)(16,85,60,68)(17,92,49,63)(18,87,50,70)(19,94,51,65)(20,89,52,72)(21,96,53,67)(22,91,54,62)(23,86,55,69)(24,93,56,64) );
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)], [(1,58,10,55,7,52,4,49),(2,51,11,60,8,57,5,54),(3,56,12,53,9,50,6,59),(13,46,22,43,19,40,16,37),(14,39,23,48,20,45,17,42),(15,44,24,41,21,38,18,47),(25,63,28,66,31,69,34,72),(26,68,29,71,32,62,35,65),(27,61,30,64,33,67,36,70),(73,85,76,88,79,91,82,94),(74,90,77,93,80,96,83,87),(75,95,78,86,81,89,84,92)], [(1,28,42,75),(2,35,43,82),(3,30,44,77),(4,25,45,84),(5,32,46,79),(6,27,47,74),(7,34,48,81),(8,29,37,76),(9,36,38,83),(10,31,39,78),(11,26,40,73),(12,33,41,80),(13,88,57,71),(14,95,58,66),(15,90,59,61),(16,85,60,68),(17,92,49,63),(18,87,50,70),(19,94,51,65),(20,89,52,72),(21,96,53,67),(22,91,54,62),(23,86,55,69),(24,93,56,64)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 6A | 6B | 6C | 6D | 6E | 8A | ··· | 8H | 8I | ··· | 8T | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 4 | 4 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 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 | Dic3 | D6 | C4×S3 | C8○D4 | D4.Dic3 |
| kernel | C12.5C42 | C4×C3⋊C8 | C42.S3 | C22×C3⋊C8 | C2×C4.Dic3 | C3×C42⋊C2 | C2×C3⋊C8 | C4.Dic3 | C3×C22⋊C4 | C3×C4⋊C4 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C6 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 8 | 4 | 4 | 1 | 2 | 2 | 2 | 1 | 8 | 8 | 4 |
Matrix representation of C12.5C42 ►in GL4(𝔽73) generated by
| 27 | 0 | 0 | 0 |
| 1 | 46 | 0 | 0 |
| 0 | 0 | 1 | 72 |
| 0 | 0 | 1 | 0 |
| 63 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 |
| 0 | 0 | 46 | 27 |
| 0 | 0 | 0 | 27 |
| 46 | 71 | 0 | 0 |
| 72 | 27 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
G:=sub<GL(4,GF(73))| [27,1,0,0,0,46,0,0,0,0,1,1,0,0,72,0],[63,0,0,0,0,63,0,0,0,0,46,0,0,0,27,27],[46,72,0,0,71,27,0,0,0,0,27,0,0,0,0,27] >;
C12.5C42 in GAP, Magma, Sage, TeX
C_{12}._5C_4^2 % in TeX
G:=Group("C12.5C4^2"); // GroupNames label
G:=SmallGroup(192,556);
// by ID
G=gap.SmallGroup(192,556);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,387,100,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=c^4=1,b^4=a^6,b*a*b^-1=a^5,c*a*c^-1=a^7,b*c=c*b>;
// generators/relations