metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42:19D6, C6.212+ 1+4, C4:C4:50D6, (C4xD4):22S3, (D4xC12):24C2, D6:D4:7C2, C22:C4:49D6, (C22xC4):17D6, Dic3:D4:10C2, C23:2D6:21C2, C12:7D4:11C2, (C4xC12):28C22, D6:C4:31C22, D6.D4:8C2, (C2xD4).221D6, C42:3S3:10C2, C42:7S3:28C2, C2.17(D4oD12), (C2xC6).104C24, C4:Dic3:10C22, (C2xDic6):7C22, C23.14D6:27C2, C2.22(D4:6D6), (C2xC12).162C23, Dic3:C4:33C22, (C22xC12):11C22, Dic3.D4:9C2, C23.11D6:9C2, C3:2(C22.32C24), (C4xDic3):53C22, (C6xD4).308C22, (C2xD12).27C22, C22.6(C4oD12), C23.28D6:2C2, C6.D4:10C22, (C22xS3).38C23, (S3xC23).42C22, (C22xC6).174C23, C22.129(S3xC23), C23.111(C22xS3), (C2xDic3).45C23, (C22xDic3).99C22, C4:C4:S3:8C2, (C2xD6:C4):35C2, (C4xC3:D4):46C2, (S3xC2xC4):49C22, C6.46(C2xC4oD4), (C3xC4:C4):62C22, C2.53(C2xC4oD12), (C2xC3:D4):5C22, (C2xC6).17(C4oD4), (C3xC22:C4):58C22, (C2xC4).162(C22xS3), SmallGroup(192,1119)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42:19D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 744 in 250 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, C12, D6, C2xC6, C2xC6, C2xC6, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C24, Dic6, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C22xS3, C22xS3, C22xC6, C2xC22:C4, C4xD4, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C4.4D4, C42:2C2, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C4xC12, C3xC22:C4, C3xC4:C4, C2xDic6, S3xC2xC4, C2xD12, C22xDic3, C2xC3:D4, C22xC12, C6xD4, S3xC23, C22.32C24, C42:7S3, C42:3S3, Dic3.D4, D6:D4, Dic3:D4, C23.11D6, D6.D4, C4:C4:S3, C2xD6:C4, C4xC3:D4, C23.28D6, C12:7D4, C23:2D6, C23.14D6, D4xC12, C42:19D6
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C24, C22xS3, C2xC4oD4, 2+ 1+4, C4oD12, S3xC23, C22.32C24, C2xC4oD12, D4:6D6, D4oD12, C42:19D6
►On 48 points
(1 39 19 42)(2 37 20 40)(3 41 21 38)(4 31 17 34)(5 35 18 32)(6 33 16 36)(7 25 10 28)(8 29 11 26)(9 27 12 30)(13 46 22 43)(14 44 23 47)(15 48 24 45)
(1 16 24 9)(2 17 22 7)(3 18 23 8)(4 13 10 20)(5 14 11 21)(6 15 12 19)(25 37 34 43)(26 38 35 44)(27 39 36 45)(28 40 31 46)(29 41 32 47)(30 42 33 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 20)(2 19)(3 21)(4 6)(7 9)(10 12)(13 24)(14 23)(15 22)(16 17)(25 33)(26 32)(27 31)(28 36)(29 35)(30 34)(37 45)(38 44)(39 43)(40 48)(41 47)(42 46)
G:=sub<Sym(48)| (1,39,19,42)(2,37,20,40)(3,41,21,38)(4,31,17,34)(5,35,18,32)(6,33,16,36)(7,25,10,28)(8,29,11,26)(9,27,12,30)(13,46,22,43)(14,44,23,47)(15,48,24,45), (1,16,24,9)(2,17,22,7)(3,18,23,8)(4,13,10,20)(5,14,11,21)(6,15,12,19)(25,37,34,43)(26,38,35,44)(27,39,36,45)(28,40,31,46)(29,41,32,47)(30,42,33,48), (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), (1,20)(2,19)(3,21)(4,6)(7,9)(10,12)(13,24)(14,23)(15,22)(16,17)(25,33)(26,32)(27,31)(28,36)(29,35)(30,34)(37,45)(38,44)(39,43)(40,48)(41,47)(42,46)>;
G:=Group( (1,39,19,42)(2,37,20,40)(3,41,21,38)(4,31,17,34)(5,35,18,32)(6,33,16,36)(7,25,10,28)(8,29,11,26)(9,27,12,30)(13,46,22,43)(14,44,23,47)(15,48,24,45), (1,16,24,9)(2,17,22,7)(3,18,23,8)(4,13,10,20)(5,14,11,21)(6,15,12,19)(25,37,34,43)(26,38,35,44)(27,39,36,45)(28,40,31,46)(29,41,32,47)(30,42,33,48), (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), (1,20)(2,19)(3,21)(4,6)(7,9)(10,12)(13,24)(14,23)(15,22)(16,17)(25,33)(26,32)(27,31)(28,36)(29,35)(30,34)(37,45)(38,44)(39,43)(40,48)(41,47)(42,46) );
G=PermutationGroup([[(1,39,19,42),(2,37,20,40),(3,41,21,38),(4,31,17,34),(5,35,18,32),(6,33,16,36),(7,25,10,28),(8,29,11,26),(9,27,12,30),(13,46,22,43),(14,44,23,47),(15,48,24,45)], [(1,16,24,9),(2,17,22,7),(3,18,23,8),(4,13,10,20),(5,14,11,21),(6,15,12,19),(25,37,34,43),(26,38,35,44),(27,39,36,45),(28,40,31,46),(29,41,32,47),(30,42,33,48)], [(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)], [(1,20),(2,19),(3,21),(4,6),(7,9),(10,12),(13,24),(14,23),(15,22),(16,17),(25,33),(26,32),(27,31),(28,36),(29,35),(30,34),(37,45),(38,44),(39,43),(40,48),(41,47),(42,46)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | C4oD4 | C4oD12 | 2+ 1+4 | D4:6D6 | D4oD12 |
| kernel | C42:19D6 | C42:7S3 | C42:3S3 | Dic3.D4 | D6:D4 | Dic3:D4 | C23.11D6 | D6.D4 | C4:C4:S3 | C2xD6:C4 | C4xC3:D4 | C23.28D6 | C12:7D4 | C23:2D6 | C23.14D6 | D4xC12 | C4xD4 | C42 | C22:C4 | C4:C4 | C22xC4 | C2xD4 | C2xC6 | C22 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 2 | 2 | 2 |
Matrix representation of C42:19D6 ►in GL6(F13)
| 11 | 9 | 0 | 0 | 0 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 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 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [11,4,0,0,0,0,9,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,12,0,0],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,10,7,0,0,0,0,6,3,0,0,0,0,0,0,10,7,0,0,0,0,6,3],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,1,1] >;
C42:19D6 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{19}D_6 % in TeX
G:=Group("C4^2:19D6"); // GroupNames label
G:=SmallGroup(192,1119);
// by ID
G=gap.SmallGroup(192,1119);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,100,675,570,6278]);
// 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^-1,d*a*d=a^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations