metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42:11D6, C6.952+ 1+4, C4:C4:43D6, (C2xC4):5D12, (C2xC12):11D4, D6:D4:4C2, C4:D12:3C2, (C4xC12):1C22, D6:C4:3C22, C4.71(C2xD12), C12:D4:11C2, C42:C2:9S3, C42:7S3:3C2, C2.7(D4oD12), C12.287(C2xD4), (C2xD12):5C22, (C2xC6).69C24, C22:C4.93D6, C6.13(C22xD4), (C22xD12):14C2, C2.15(C22xD12), (C22xC4).206D6, C22.20(C2xD12), (C2xC12).144C23, C3:1(C22.29C24), (C2xDic6):51C22, C22.98(S3xC23), (C22xS3).19C23, (S3xC23).36C22, (C22xC6).139C23, C23.167(C22xS3), (C2xDic3).23C23, (C22xC12).229C22, (S3xC2xC4):1C22, (C2xC6).50(C2xD4), (C2xC4oD12):18C2, (C3xC4:C4):53C22, (C3xC42:C2):11C2, (C2xC4).149(C22xS3), (C2xC3:D4).100C22, (C3xC22:C4).101C22, SmallGroup(192,1084)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42:11D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=ab2, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 1128 in 334 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2xC6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C24, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C22xS3, C22xS3, C22xC6, C42:C2, C22wrC2, C4:D4, C4.4D4, C4:1D4, C22xD4, C2xC4oD4, D6:C4, C4xC12, C3xC22:C4, C3xC4:C4, C2xDic6, S3xC2xC4, C2xD12, C2xD12, C2xD12, C4oD12, C2xC3:D4, C22xC12, S3xC23, C22.29C24, C4:D12, C42:7S3, D6:D4, C12:D4, C3xC42:C2, C22xD12, C2xC4oD12, C42:11D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C24, D12, C22xS3, C22xD4, 2+ 1+4, C2xD12, S3xC23, C22.29C24, C22xD12, D4oD12, C42:11D6
►On 48 points
(1 26 4 19)(2 23 5 30)(3 28 6 21)(7 32 44 35)(8 40 45 37)(9 34 46 31)(10 42 47 39)(11 36 48 33)(12 38 43 41)(13 22 16 29)(14 27 17 20)(15 24 18 25)
(1 34 13 41)(2 35 14 42)(3 36 15 37)(4 31 16 38)(5 32 17 39)(6 33 18 40)(7 27 47 23)(8 28 48 24)(9 29 43 19)(10 30 44 20)(11 25 45 21)(12 26 46 22)
(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 40)(2 39)(3 38)(4 37)(5 42)(6 41)(7 27)(8 26)(9 25)(10 30)(11 29)(12 28)(13 33)(14 32)(15 31)(16 36)(17 35)(18 34)(19 45)(20 44)(21 43)(22 48)(23 47)(24 46)
G:=sub<Sym(48)| (1,26,4,19)(2,23,5,30)(3,28,6,21)(7,32,44,35)(8,40,45,37)(9,34,46,31)(10,42,47,39)(11,36,48,33)(12,38,43,41)(13,22,16,29)(14,27,17,20)(15,24,18,25), (1,34,13,41)(2,35,14,42)(3,36,15,37)(4,31,16,38)(5,32,17,39)(6,33,18,40)(7,27,47,23)(8,28,48,24)(9,29,43,19)(10,30,44,20)(11,25,45,21)(12,26,46,22), (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), (1,40)(2,39)(3,38)(4,37)(5,42)(6,41)(7,27)(8,26)(9,25)(10,30)(11,29)(12,28)(13,33)(14,32)(15,31)(16,36)(17,35)(18,34)(19,45)(20,44)(21,43)(22,48)(23,47)(24,46)>;
G:=Group( (1,26,4,19)(2,23,5,30)(3,28,6,21)(7,32,44,35)(8,40,45,37)(9,34,46,31)(10,42,47,39)(11,36,48,33)(12,38,43,41)(13,22,16,29)(14,27,17,20)(15,24,18,25), (1,34,13,41)(2,35,14,42)(3,36,15,37)(4,31,16,38)(5,32,17,39)(6,33,18,40)(7,27,47,23)(8,28,48,24)(9,29,43,19)(10,30,44,20)(11,25,45,21)(12,26,46,22), (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), (1,40)(2,39)(3,38)(4,37)(5,42)(6,41)(7,27)(8,26)(9,25)(10,30)(11,29)(12,28)(13,33)(14,32)(15,31)(16,36)(17,35)(18,34)(19,45)(20,44)(21,43)(22,48)(23,47)(24,46) );
G=PermutationGroup([[(1,26,4,19),(2,23,5,30),(3,28,6,21),(7,32,44,35),(8,40,45,37),(9,34,46,31),(10,42,47,39),(11,36,48,33),(12,38,43,41),(13,22,16,29),(14,27,17,20),(15,24,18,25)], [(1,34,13,41),(2,35,14,42),(3,36,15,37),(4,31,16,38),(5,32,17,39),(6,33,18,40),(7,27,47,23),(8,28,48,24),(9,29,43,19),(10,30,44,20),(11,25,45,21),(12,26,46,22)], [(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)], [(1,40),(2,39),(3,38),(4,37),(5,42),(6,41),(7,27),(8,26),(9,25),(10,30),(11,29),(12,28),(13,33),(14,32),(15,31),(16,36),(17,35),(18,34),(19,45),(20,44),(21,43),(22,48),(23,47),(24,46)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | D12 | 2+ 1+4 | D4oD12 |
| kernel | C42:11D6 | C4:D12 | C42:7S3 | D6:D4 | C12:D4 | C3xC42:C2 | C22xD12 | C2xC4oD12 | C42:C2 | C2xC12 | C42 | C22:C4 | C4:C4 | C22xC4 | C2xC4 | C6 | C2 |
| # reps | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 8 | 2 | 4 |
Matrix representation of C42:11D6 ►in GL6(F13)
| 3 | 6 | 0 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 11 | 0 |
| 0 | 0 | 0 | 1 | 0 | 11 |
| 0 | 0 | 1 | 0 | 12 | 0 |
| 0 | 0 | 0 | 1 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 |
| 0 | 0 | 7 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 12 |
| 0 | 0 | 12 | 1 | 1 | 12 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 |
| 0 | 0 | 3 | 3 | 10 | 10 |
| 0 | 0 | 6 | 10 | 7 | 3 |
G:=sub<GL(6,GF(13))| [3,7,0,0,0,0,6,10,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,11,0,12,0,0,0,0,11,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,10,7,0,0,0,0,6,3,0,0,0,0,0,0,10,7,0,0,0,0,6,3],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,12,0,0,1,1,1,1,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,3,6,3,6,0,0,3,10,3,10,0,0,0,0,10,7,0,0,0,0,10,3] >;
C42:11D6 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{11}D_6 % in TeX
G:=Group("C4^2:11D6"); // GroupNames label
G:=SmallGroup(192,1084);
// by ID
G=gap.SmallGroup(192,1084);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,570,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations