direct product, metabelian, supersoluble, monomial
Aliases: C6xC4oD12, C62.268C23, (C2xC12):32D6, (C6xD12):30C2, (C2xD12):14C6, D12:12(C2xC6), C6.4(C23xC6), (C22xC12):17S3, (C22xC12):12C6, (C6xC12):33C22, Dic6:11(C2xC6), (C2xDic6):15C6, (C6xDic6):31C2, C23.36(S3xC6), C6.72(S3xC23), (C3xC6).41C24, D6.1(C22xC6), (S3xC12):23C22, (C3xD12):40C22, (S3xC6).28C23, C12.43(C22xC6), (C22xC6).132D6, (C3xC12).183C23, C12.222(C22xS3), (C3xDic6):38C22, Dic3.2(C22xC6), (C2xC62).118C22, (C3xDic3).29C23, (C6xDic3).163C22, (S3xC2xC4):15C6, (C2xC6xC12):11C2, C3:1(C6xC4oD4), C6:1(C3xC4oD4), C4.43(S3xC2xC6), (S3xC2xC12):31C2, (C4xS3):6(C2xC6), (C2xC4):10(S3xC6), C3:D4:6(C2xC6), (C2xC12):13(C2xC6), C22.5(S3xC2xC6), C2.5(S3xC22xC6), (C3xC6):7(C4oD4), (C6xC3:D4):26C2, (C2xC3:D4):12C6, C32:13(C2xC4oD4), (C22xC4):10(C3xS3), (C3xC3:D4):19C22, (S3xC2xC6).109C22, (C22xC6).71(C2xC6), (C2xC6).70(C22xC6), (C22xS3).29(C2xC6), (C2xC6).164(C22xS3), (C2xDic3).44(C2xC6), SmallGroup(288,991)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6xC4oD12
G = < a,b,c,d | a6=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >
Subgroups: 698 in 355 conjugacy classes, 178 normal (34 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C32, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC6, C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C3xS3, C3xC6, C3xC6, C3xC6, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, C22xC6, C22xC6, C2xC4oD4, C3xDic3, C3xC12, S3xC6, S3xC6, C62, C62, C62, C2xDic6, S3xC2xC4, C2xD12, C4oD12, C2xC3:D4, C22xC12, C22xC12, C6xD4, C6xQ8, C3xC4oD4, C3xDic6, S3xC12, C3xD12, C6xDic3, C3xC3:D4, C6xC12, C6xC12, S3xC2xC6, C2xC62, C2xC4oD12, C6xC4oD4, C6xDic6, S3xC2xC12, C6xD12, C3xC4oD12, C6xC3:D4, C2xC6xC12, C6xC4oD12
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2xC6, C4oD4, C24, C3xS3, C22xS3, C22xC6, C2xC4oD4, S3xC6, C4oD12, C3xC4oD4, S3xC23, C23xC6, S3xC2xC6, C2xC4oD12, C6xC4oD4, C3xC4oD12, S3xC22xC6, C6xC4oD12
►On 48 points
(1 22 5 14 9 18)(2 23 6 15 10 19)(3 24 7 16 11 20)(4 13 8 17 12 21)(25 41 33 37 29 45)(26 42 34 38 30 46)(27 43 35 39 31 47)(28 44 36 40 32 48)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 40 43 46)(38 41 44 47)(39 42 45 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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 25)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)
G:=sub<Sym(48)| (1,22,5,14,9,18)(2,23,6,15,10,19)(3,24,7,16,11,20)(4,13,8,17,12,21)(25,41,33,37,29,45)(26,42,34,38,30,46)(27,43,35,39,31,47)(28,44,36,40,32,48), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,40,43,46)(38,41,44,47)(39,42,45,48), (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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,25)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)>;
G:=Group( (1,22,5,14,9,18)(2,23,6,15,10,19)(3,24,7,16,11,20)(4,13,8,17,12,21)(25,41,33,37,29,45)(26,42,34,38,30,46)(27,43,35,39,31,47)(28,44,36,40,32,48), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,40,43,46)(38,41,44,47)(39,42,45,48), (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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,25)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26) );
G=PermutationGroup([[(1,22,5,14,9,18),(2,23,6,15,10,19),(3,24,7,16,11,20),(4,13,8,17,12,21),(25,41,33,37,29,45),(26,42,34,38,30,46),(27,43,35,39,31,47),(28,44,36,40,32,48)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,22,19,16),(14,23,20,17),(15,24,21,18),(25,28,31,34),(26,29,32,35),(27,30,33,36),(37,40,43,46),(38,41,44,47),(39,42,45,48)], [(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)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,25),(14,36),(15,35),(16,34),(17,33),(18,32),(19,31),(20,30),(21,29),(22,28),(23,27),(24,26)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 6G | ··· | 6AE | 6AF | ··· | 6AM | 12A | ··· | 12H | 12I | ··· | 12AJ | 12AK | ··· | 12AR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | S3 | D6 | D6 | C4oD4 | C3xS3 | S3xC6 | S3xC6 | C4oD12 | C3xC4oD4 | C3xC4oD12 |
| kernel | C6xC4oD12 | C6xDic6 | S3xC2xC12 | C6xD12 | C3xC4oD12 | C6xC3:D4 | C2xC6xC12 | C2xC4oD12 | C2xDic6 | S3xC2xC4 | C2xD12 | C4oD12 | C2xC3:D4 | C22xC12 | C22xC12 | C2xC12 | C22xC6 | C3xC6 | C22xC4 | C2xC4 | C23 | C6 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 2 | 2 | 4 | 2 | 16 | 4 | 2 | 1 | 6 | 1 | 4 | 2 | 12 | 2 | 8 | 8 | 16 |
Matrix representation of C6xC4oD12 ►in GL3(F13) generated by
| 12 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 1 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 12 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 7 |
| 12 | 0 | 0 |
| 0 | 0 | 7 |
| 0 | 2 | 0 |
G:=sub<GL(3,GF(13))| [12,0,0,0,9,0,0,0,9],[1,0,0,0,5,0,0,0,5],[12,0,0,0,2,0,0,0,7],[12,0,0,0,0,2,0,7,0] >;
C6xC4oD12 in GAP, Magma, Sage, TeX
C_6\times C_4\circ D_{12} % in TeX
G:=Group("C6xC4oD12"); // GroupNames label
G:=SmallGroup(288,991);
// by ID
G=gap.SmallGroup(288,991);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,268,1571,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^4=d^2=1,c^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^5>;
// generators/relations