metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.21C42, C23.5Dic6, (C2xC24).3C4, (C2xC4).15D12, (C2xC8).3Dic3, C4.36(D6:C4), C3:(C4.10C42), (C2xC12).113D4, (C22xC6).9Q8, C4.26(C4xDic3), (C22xC4).81D6, C12.12(C22:C4), (C2xM4(2)).11S3, (C6xM4(2)).15C2, C4.20(C6.D4), C22.7(Dic3:C4), C22.11(C4:Dic3), C6.19(C2.C42), C2.19(C6.C42), (C22xC12).129C22, (C2xC3:C8).3C4, (C2xC6).10(C4:C4), (C2xC4).143(C4xS3), (C2xC12).66(C2xC4), (C2xC4).23(C3:D4), (C2xC4).78(C2xDic3), (C2xC4.Dic3).14C2, SmallGroup(192,119)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.21C42
G = < a,b,c | a12=1, b4=c4=a6, bab-1=a5, ac=ca, cbc-1=a9b >
Subgroups: 168 in 86 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2xC4, C2xC4, C23, C12, C12, C2xC6, C2xC6, C2xC6, C2xC8, C2xC8, M4(2), C22xC4, C3:C8, C24, C2xC12, C2xC12, C22xC6, C2xM4(2), C2xM4(2), C2xC3:C8, C4.Dic3, C2xC24, C3xM4(2), C22xC12, C4.10C42, C2xC4.Dic3, C6xM4(2), C12.21C42
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Q8, Dic3, D6, C42, C22:C4, C4:C4, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2.C42, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C4.10C42, C6.C42, C12.21C42
►On 48 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)
(1 28 4 31 7 34 10 25)(2 33 5 36 8 27 11 30)(3 26 6 29 9 32 12 35)(13 45 22 42 19 39 16 48)(14 38 23 47 20 44 17 41)(15 43 24 40 21 37 18 46)
(1 21 4 24 7 15 10 18)(2 22 5 13 8 16 11 19)(3 23 6 14 9 17 12 20)(25 37 34 46 31 43 28 40)(26 38 35 47 32 44 29 41)(27 39 36 48 33 45 30 42)
G:=sub<Sym(48)| (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), (1,28,4,31,7,34,10,25)(2,33,5,36,8,27,11,30)(3,26,6,29,9,32,12,35)(13,45,22,42,19,39,16,48)(14,38,23,47,20,44,17,41)(15,43,24,40,21,37,18,46), (1,21,4,24,7,15,10,18)(2,22,5,13,8,16,11,19)(3,23,6,14,9,17,12,20)(25,37,34,46,31,43,28,40)(26,38,35,47,32,44,29,41)(27,39,36,48,33,45,30,42)>;
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), (1,28,4,31,7,34,10,25)(2,33,5,36,8,27,11,30)(3,26,6,29,9,32,12,35)(13,45,22,42,19,39,16,48)(14,38,23,47,20,44,17,41)(15,43,24,40,21,37,18,46), (1,21,4,24,7,15,10,18)(2,22,5,13,8,16,11,19)(3,23,6,14,9,17,12,20)(25,37,34,46,31,43,28,40)(26,38,35,47,32,44,29,41)(27,39,36,48,33,45,30,42) );
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)], [(1,28,4,31,7,34,10,25),(2,33,5,36,8,27,11,30),(3,26,6,29,9,32,12,35),(13,45,22,42,19,39,16,48),(14,38,23,47,20,44,17,41),(15,43,24,40,21,37,18,46)], [(1,21,4,24,7,15,10,18),(2,22,5,13,8,16,11,19),(3,23,6,14,9,17,12,20),(25,37,34,46,31,43,28,40),(26,38,35,47,32,44,29,41),(27,39,36,48,33,45,30,42)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | D4 | Q8 | Dic3 | D6 | C4xS3 | D12 | C3:D4 | Dic6 | C4.10C42 | C12.21C42 |
| kernel | C12.21C42 | C2xC4.Dic3 | C6xM4(2) | C2xC3:C8 | C2xC24 | C2xM4(2) | C2xC12 | C22xC6 | C2xC8 | C22xC4 | C2xC4 | C2xC4 | C2xC4 | C23 | C3 | C1 |
| # reps | 1 | 2 | 1 | 8 | 4 | 1 | 3 | 1 | 2 | 1 | 4 | 2 | 4 | 2 | 2 | 4 |
Matrix representation of C12.21C42 ►in GL4(F73) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 29 | 0 | 24 | 0 |
| 29 | 0 | 0 | 24 |
| 32 | 0 | 71 | 0 |
| 0 | 0 | 72 | 1 |
| 24 | 0 | 41 | 0 |
| 51 | 46 | 41 | 0 |
| 1 | 71 | 0 | 0 |
| 60 | 72 | 0 | 0 |
| 10 | 41 | 0 | 46 |
| 0 | 41 | 1 | 0 |
G:=sub<GL(4,GF(73))| [3,0,29,29,0,3,0,0,0,0,24,0,0,0,0,24],[32,0,24,51,0,0,0,46,71,72,41,41,0,1,0,0],[1,60,10,0,71,72,41,41,0,0,0,1,0,0,46,0] >;
C12.21C42 in GAP, Magma, Sage, TeX
C_{12}._{21}C_4^2 % in TeX
G:=Group("C12.21C4^2"); // GroupNames label
G:=SmallGroup(192,119);
// by ID
G=gap.SmallGroup(192,119);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,184,1123,136,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^4=c^4=a^6,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=a^9*b>;
// generators/relations