direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4xC12, C42:7C6, C4:C4:7C6, C4:1(C2xC12), C12:6(C2xC4), C12o2(C4:C4), C2.3(C6xD4), (C4xC12):11C2, C22:C4:6C6, (C22xC4):4C6, (C2xD4).7C6, C6.66(C2xD4), C22:2(C2xC12), (C22xC12):4C2, (C6xD4).14C2, C12o2(C22:C4), C6.39(C4oD4), (C2xC6).73C23, C2.4(C22xC12), C23.10(C2xC6), C6.32(C22xC4), C22.7(C22xC6), (C2xC12).121C22, (C22xC6).26C22, C4o2(C3xC4:C4), C4:C4o(C2xC12), (C2xC4)o(C6xD4), C12o2(C3xC4:C4), (C2xC6):4(C2xC4), (C2xC12)o(C6xD4), (C2xD4)o(C2xC12), (C3xC4:C4):16C2, C22:C4o(C2xC12), C4o2(C3xC22:C4), C2.2(C3xC4oD4), C12o2(C3xC22:C4), (C2xC4).15(C2xC6), (C3xC22:C4):14C2, (C2xC4)o(C3xC4:C4), (C2xC12)o(C3xC4:C4), (C2xC4)o(C3xC22:C4), (C2xC12)o(C3xC22:C4), SmallGroup(96,165)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4xC12
G = < a,b,c | a12=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 124 in 94 conjugacy classes, 64 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2xC4, C2xC4, C2xC4, D4, C23, C12, C12, C2xC6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C22xC4, C2xD4, C2xC12, C2xC12, C2xC12, C3xD4, C22xC6, C4xD4, C4xC12, C3xC22:C4, C3xC4:C4, C22xC12, C6xD4, D4xC12
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, D4, C23, C12, C2xC6, C22xC4, C2xD4, C4oD4, C2xC12, C3xD4, C22xC6, C4xD4, C22xC12, C6xD4, C3xC4oD4, D4xC12
►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 41 24 30)(2 42 13 31)(3 43 14 32)(4 44 15 33)(5 45 16 34)(6 46 17 35)(7 47 18 36)(8 48 19 25)(9 37 20 26)(10 38 21 27)(11 39 22 28)(12 40 23 29)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 37)(33 38)(34 39)(35 40)(36 41)
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), (1,41,24,30)(2,42,13,31)(3,43,14,32)(4,44,15,33)(5,45,16,34)(6,46,17,35)(7,47,18,36)(8,48,19,25)(9,37,20,26)(10,38,21,27)(11,39,22,28)(12,40,23,29), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,37)(33,38)(34,39)(35,40)(36,41)>;
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), (1,41,24,30)(2,42,13,31)(3,43,14,32)(4,44,15,33)(5,45,16,34)(6,46,17,35)(7,47,18,36)(8,48,19,25)(9,37,20,26)(10,38,21,27)(11,39,22,28)(12,40,23,29), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,37)(33,38)(34,39)(35,40)(36,41) );
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)], [(1,41,24,30),(2,42,13,31),(3,43,14,32),(4,44,15,33),(5,45,16,34),(6,46,17,35),(7,47,18,36),(8,48,19,25),(9,37,20,26),(10,38,21,27),(11,39,22,28),(12,40,23,29)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,37),(33,38),(34,39),(35,40),(36,41)]])
D4xC12 is a maximal subgroup of
C12.57D8 C12.50D8 C12.38SD16 D4.3Dic6 C42.47D6 C12:3M4(2) C42.48D6 C12:7D8 D4.1D12 C42.51D6 D4.2D12 C42.102D6 D4:5Dic6 C42.104D6 C42.105D6 C42.106D6 D4:6Dic6 C42:13D6 C42.108D6 C42:14D6 C42.228D6 D12:23D4 D12:24D4 Dic6:23D4 Dic6:24D4 D4:5D12 D4:6D12 C42:18D6 C42.229D6 C42.113D6 C42.114D6 C42:19D6 C42.115D6 C42.116D6 C42.117D6 C42.118D6 C42.119D6
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6F | 6G | ··· | 6N | 12A | ··· | 12H | 12I | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | D4 | C4oD4 | C3xD4 | C3xC4oD4 |
| kernel | D4xC12 | C4xC12 | C3xC22:C4 | C3xC4:C4 | C22xC12 | C6xD4 | C4xD4 | C3xD4 | C42 | C22:C4 | C4:C4 | C22xC4 | C2xD4 | D4 | C12 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 8 | 2 | 4 | 2 | 4 | 2 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of D4xC12 ►in GL3(F13) generated by
| 6 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 1 | 0 | 0 |
| 0 | 1 | 2 |
| 0 | 12 | 12 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 12 | 12 |
G:=sub<GL(3,GF(13))| [6,0,0,0,10,0,0,0,10],[1,0,0,0,1,12,0,2,12],[1,0,0,0,1,12,0,0,12] >;
D4xC12 in GAP, Magma, Sage, TeX
D_4\times C_{12} % in TeX
G:=Group("D4xC12"); // GroupNames label
G:=SmallGroup(96,165);
// by ID
G=gap.SmallGroup(96,165);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,288,313,230]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations