metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.10C42, C23.15Dic6, (C2xC24).12C4, (C22xC8).6S3, (C2xC8).7Dic3, C4.49(D6:C4), (C2xC12).479D4, (C2xC4).165D12, (C22xC24).3C2, C4.10(C4xDic3), C4.Dic3.2C4, C6.3(C8.C4), (C22xC6).22Q8, C3:1(C4.C42), (C22xC4).430D6, C2.3(C24.C4), C12.63(C22:C4), C4.16(C6.D4), C22.19(C4:Dic3), C2.12(C6.C42), C6.12(C2.C42), C22.13(Dic3:C4), (C22xC12).533C22, (C2xC6).39(C4:C4), (C2xC4).100(C4xS3), (C2xC12).294(C2xC4), (C2xC4).74(C2xDic3), (C2xC4.Dic3).2C2, (C2xC4).234(C3:D4), SmallGroup(192,111)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.10C42
G = < a,b,c | a12=1, b4=c4=a6, bab-1=a-1, ac=ca, cbc-1=a3b >
Subgroups: 168 in 90 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2xC4, C2xC4, C23, C12, C12, C2xC6, C2xC6, C2xC6, C2xC8, C2xC8, M4(2), C22xC4, C3:C8, C24, C2xC12, C2xC12, C22xC6, C22xC8, C2xM4(2), C2xC3:C8, C4.Dic3, C4.Dic3, C2xC24, C2xC24, C22xC12, C4.C42, C2xC4.Dic3, C22xC24, C12.10C42
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Q8, Dic3, D6, C42, C22:C4, C4:C4, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2.C42, C8.C4, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C4.C42, C24.C4, C6.C42, C12.10C42
►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 13 4 22 7 19 10 16)(2 24 5 21 8 18 11 15)(3 23 6 20 9 17 12 14)(25 39 34 42 31 45 28 48)(26 38 35 41 32 44 29 47)(27 37 36 40 33 43 30 46)(49 63 58 66 55 69 52 72)(50 62 59 65 56 68 53 71)(51 61 60 64 57 67 54 70)(73 92 76 89 79 86 82 95)(74 91 77 88 80 85 83 94)(75 90 78 87 81 96 84 93)
(1 44 72 80 7 38 66 74)(2 45 61 81 8 39 67 75)(3 46 62 82 9 40 68 76)(4 47 63 83 10 41 69 77)(5 48 64 84 11 42 70 78)(6 37 65 73 12 43 71 79)(13 32 55 94 19 26 49 88)(14 33 56 95 20 27 50 89)(15 34 57 96 21 28 51 90)(16 35 58 85 22 29 52 91)(17 36 59 86 23 30 53 92)(18 25 60 87 24 31 54 93)
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), (1,13,4,22,7,19,10,16)(2,24,5,21,8,18,11,15)(3,23,6,20,9,17,12,14)(25,39,34,42,31,45,28,48)(26,38,35,41,32,44,29,47)(27,37,36,40,33,43,30,46)(49,63,58,66,55,69,52,72)(50,62,59,65,56,68,53,71)(51,61,60,64,57,67,54,70)(73,92,76,89,79,86,82,95)(74,91,77,88,80,85,83,94)(75,90,78,87,81,96,84,93), (1,44,72,80,7,38,66,74)(2,45,61,81,8,39,67,75)(3,46,62,82,9,40,68,76)(4,47,63,83,10,41,69,77)(5,48,64,84,11,42,70,78)(6,37,65,73,12,43,71,79)(13,32,55,94,19,26,49,88)(14,33,56,95,20,27,50,89)(15,34,57,96,21,28,51,90)(16,35,58,85,22,29,52,91)(17,36,59,86,23,30,53,92)(18,25,60,87,24,31,54,93)>;
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), (1,13,4,22,7,19,10,16)(2,24,5,21,8,18,11,15)(3,23,6,20,9,17,12,14)(25,39,34,42,31,45,28,48)(26,38,35,41,32,44,29,47)(27,37,36,40,33,43,30,46)(49,63,58,66,55,69,52,72)(50,62,59,65,56,68,53,71)(51,61,60,64,57,67,54,70)(73,92,76,89,79,86,82,95)(74,91,77,88,80,85,83,94)(75,90,78,87,81,96,84,93), (1,44,72,80,7,38,66,74)(2,45,61,81,8,39,67,75)(3,46,62,82,9,40,68,76)(4,47,63,83,10,41,69,77)(5,48,64,84,11,42,70,78)(6,37,65,73,12,43,71,79)(13,32,55,94,19,26,49,88)(14,33,56,95,20,27,50,89)(15,34,57,96,21,28,51,90)(16,35,58,85,22,29,52,91)(17,36,59,86,23,30,53,92)(18,25,60,87,24,31,54,93) );
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)], [(1,13,4,22,7,19,10,16),(2,24,5,21,8,18,11,15),(3,23,6,20,9,17,12,14),(25,39,34,42,31,45,28,48),(26,38,35,41,32,44,29,47),(27,37,36,40,33,43,30,46),(49,63,58,66,55,69,52,72),(50,62,59,65,56,68,53,71),(51,61,60,64,57,67,54,70),(73,92,76,89,79,86,82,95),(74,91,77,88,80,85,83,94),(75,90,78,87,81,96,84,93)], [(1,44,72,80,7,38,66,74),(2,45,61,81,8,39,67,75),(3,46,62,82,9,40,68,76),(4,47,63,83,10,41,69,77),(5,48,64,84,11,42,70,78),(6,37,65,73,12,43,71,79),(13,32,55,94,19,26,49,88),(14,33,56,95,20,27,50,89),(15,34,57,96,21,28,51,90),(16,35,58,85,22,29,52,91),(17,36,59,86,23,30,53,92),(18,25,60,87,24,31,54,93)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6G | 8A | ··· | 8H | 8I | ··· | 8P | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | D4 | Q8 | Dic3 | D6 | C4xS3 | D12 | C3:D4 | Dic6 | C8.C4 | C24.C4 |
| kernel | C12.10C42 | C2xC4.Dic3 | C22xC24 | C4.Dic3 | C2xC24 | C22xC8 | C2xC12 | C22xC6 | C2xC8 | C22xC4 | C2xC4 | C2xC4 | C2xC4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 1 | 3 | 1 | 2 | 1 | 4 | 2 | 4 | 2 | 8 | 16 |
Matrix representation of C12.10C42 ►in GL3(F73) generated by
| 1 | 0 | 0 |
| 0 | 24 | 0 |
| 0 | 0 | 70 |
| 46 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 27 | 0 |
| 46 | 0 | 0 |
| 0 | 63 | 0 |
| 0 | 0 | 51 |
G:=sub<GL(3,GF(73))| [1,0,0,0,24,0,0,0,70],[46,0,0,0,0,27,0,1,0],[46,0,0,0,63,0,0,0,51] >;
C12.10C42 in GAP, Magma, Sage, TeX
C_{12}._{10}C_4^2 % in TeX
G:=Group("C12.10C4^2"); // GroupNames label
G:=SmallGroup(192,111);
// by ID
G=gap.SmallGroup(192,111);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,176,184,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^4=c^4=a^6,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b>;
// generators/relations