metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.12C42, (C2×C24)⋊16C4, (C2×C8)⋊9Dic3, C24.85(C2×C4), C24⋊C4⋊32C2, (C2×C8).328D6, (C8×Dic3)⋊25C2, C6.15(C8○D4), C2.4(C8○D12), C4.Dic3⋊11C4, C4⋊Dic3.24C4, (C22×C8).18S3, C6.24(C2×C42), (C2×C6).26C42, C23.36(C4×S3), C4.12(C4×Dic3), C8.27(C2×Dic3), C3⋊4(C8○2M4(2)), (C22×C24).30C2, (C22×C4).436D6, (C2×C12).856C23, C12.139(C22×C4), (C2×C24).434C22, C6.D4.14C4, C4.33(C22×Dic3), C22.12(C4×Dic3), (C22×C12).537C22, (C4×Dic3).281C22, C23.26D6.26C2, C3⋊C8.15(C2×C4), C4.113(S3×C2×C4), C22.59(S3×C2×C4), C2.12(C2×C4×Dic3), (C2×C4).112(C4×S3), (C2×C12).234(C2×C4), (C2×C3⋊C8).319C22, (C22×C6).91(C2×C4), (C2×C4).80(C2×Dic3), (C2×C4).798(C22×S3), (C2×C6).126(C22×C4), (C2×Dic3).65(C2×C4), (C2×C4.Dic3).30C2, SmallGroup(192,660)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.12C42
G = < a,b,c | a12=b4=1, c4=a6, bab-1=a-1, ac=ca, bc=cb >
Subgroups: 216 in 130 conjugacy classes, 87 normal (27 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, 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, C4.Dic3, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C2×C24, C22×C12, C8○2M4(2), C8×Dic3, C24⋊C4, C2×C4.Dic3, C23.26D6, C22×C24, C12.12C42
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), C8○D12, C2×C4×Dic3, C12.12C42
►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 67 16 55)(2 66 17 54)(3 65 18 53)(4 64 19 52)(5 63 20 51)(6 62 21 50)(7 61 22 49)(8 72 23 60)(9 71 24 59)(10 70 13 58)(11 69 14 57)(12 68 15 56)(25 94 46 76)(26 93 47 75)(27 92 48 74)(28 91 37 73)(29 90 38 84)(30 89 39 83)(31 88 40 82)(32 87 41 81)(33 86 42 80)(34 85 43 79)(35 96 44 78)(36 95 45 77)
(1 25 10 34 7 31 4 28)(2 26 11 35 8 32 5 29)(3 27 12 36 9 33 6 30)(13 43 22 40 19 37 16 46)(14 44 23 41 20 38 17 47)(15 45 24 42 21 39 18 48)(49 82 52 73 55 76 58 79)(50 83 53 74 56 77 59 80)(51 84 54 75 57 78 60 81)(61 88 64 91 67 94 70 85)(62 89 65 92 68 95 71 86)(63 90 66 93 69 96 72 87)
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), (1,67,16,55)(2,66,17,54)(3,65,18,53)(4,64,19,52)(5,63,20,51)(6,62,21,50)(7,61,22,49)(8,72,23,60)(9,71,24,59)(10,70,13,58)(11,69,14,57)(12,68,15,56)(25,94,46,76)(26,93,47,75)(27,92,48,74)(28,91,37,73)(29,90,38,84)(30,89,39,83)(31,88,40,82)(32,87,41,81)(33,86,42,80)(34,85,43,79)(35,96,44,78)(36,95,45,77), (1,25,10,34,7,31,4,28)(2,26,11,35,8,32,5,29)(3,27,12,36,9,33,6,30)(13,43,22,40,19,37,16,46)(14,44,23,41,20,38,17,47)(15,45,24,42,21,39,18,48)(49,82,52,73,55,76,58,79)(50,83,53,74,56,77,59,80)(51,84,54,75,57,78,60,81)(61,88,64,91,67,94,70,85)(62,89,65,92,68,95,71,86)(63,90,66,93,69,96,72,87)>;
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), (1,67,16,55)(2,66,17,54)(3,65,18,53)(4,64,19,52)(5,63,20,51)(6,62,21,50)(7,61,22,49)(8,72,23,60)(9,71,24,59)(10,70,13,58)(11,69,14,57)(12,68,15,56)(25,94,46,76)(26,93,47,75)(27,92,48,74)(28,91,37,73)(29,90,38,84)(30,89,39,83)(31,88,40,82)(32,87,41,81)(33,86,42,80)(34,85,43,79)(35,96,44,78)(36,95,45,77), (1,25,10,34,7,31,4,28)(2,26,11,35,8,32,5,29)(3,27,12,36,9,33,6,30)(13,43,22,40,19,37,16,46)(14,44,23,41,20,38,17,47)(15,45,24,42,21,39,18,48)(49,82,52,73,55,76,58,79)(50,83,53,74,56,77,59,80)(51,84,54,75,57,78,60,81)(61,88,64,91,67,94,70,85)(62,89,65,92,68,95,71,86)(63,90,66,93,69,96,72,87) );
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)], [(1,67,16,55),(2,66,17,54),(3,65,18,53),(4,64,19,52),(5,63,20,51),(6,62,21,50),(7,61,22,49),(8,72,23,60),(9,71,24,59),(10,70,13,58),(11,69,14,57),(12,68,15,56),(25,94,46,76),(26,93,47,75),(27,92,48,74),(28,91,37,73),(29,90,38,84),(30,89,39,83),(31,88,40,82),(32,87,41,81),(33,86,42,80),(34,85,43,79),(35,96,44,78),(36,95,45,77)], [(1,25,10,34,7,31,4,28),(2,26,11,35,8,32,5,29),(3,27,12,36,9,33,6,30),(13,43,22,40,19,37,16,46),(14,44,23,41,20,38,17,47),(15,45,24,42,21,39,18,48),(49,82,52,73,55,76,58,79),(50,83,53,74,56,77,59,80),(51,84,54,75,57,78,60,81),(61,88,64,91,67,94,70,85),(62,89,65,92,68,95,71,86),(63,90,66,93,69,96,72,87)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 6A | ··· | 6G | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | ··· | 8T | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 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 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | Dic3 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | C8○D12 |
| kernel | C12.12C42 | C8×Dic3 | C24⋊C4 | C2×C4.Dic3 | C23.26D6 | C22×C24 | C4.Dic3 | C4⋊Dic3 | C6.D4 | C2×C24 | C22×C8 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 4 | 8 | 1 | 4 | 2 | 1 | 6 | 2 | 8 | 16 |
Matrix representation of C12.12C42 ►in GL4(𝔽73) generated by
| 27 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 49 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 63 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
G:=sub<GL(4,GF(73))| [27,0,0,0,0,46,0,0,0,0,49,0,0,0,0,3],[0,72,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[63,0,0,0,0,63,0,0,0,0,10,0,0,0,0,10] >;
C12.12C42 in GAP, Magma, Sage, TeX
C_{12}._{12}C_4^2 % in TeX
G:=Group("C12.12C4^2"); // GroupNames label
G:=SmallGroup(192,660);
// by ID
G=gap.SmallGroup(192,660);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,477,100,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=1,c^4=a^6,b*a*b^-1=a^-1,a*c=c*a,b*c=c*b>;
// generators/relations