direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C12, C42⋊7C6, C4⋊C4⋊7C6, C4⋊1(C2×C12), C12⋊6(C2×C4), C12○2(C4⋊C4), C2.3(C6×D4), (C4×C12)⋊11C2, C22⋊C4⋊6C6, (C22×C4)⋊4C6, (C2×D4).7C6, C6.66(C2×D4), C22⋊2(C2×C12), (C22×C12)⋊4C2, (C6×D4).14C2, C12○2(C22⋊C4), C6.39(C4○D4), (C2×C6).73C23, C2.4(C22×C12), C23.10(C2×C6), C6.32(C22×C4), C22.7(C22×C6), (C2×C12).121C22, (C22×C6).26C22, C4○2(C3×C4⋊C4), C4⋊C4○(C2×C12), (C2×C4)○(C6×D4), C12○2(C3×C4⋊C4), (C2×C6)⋊4(C2×C4), (C2×C12)○(C6×D4), (C2×D4)○(C2×C12), (C3×C4⋊C4)⋊16C2, C22⋊C4○(C2×C12), C4○2(C3×C22⋊C4), C2.2(C3×C4○D4), C12○2(C3×C22⋊C4), (C2×C4).15(C2×C6), (C3×C22⋊C4)⋊14C2, (C2×C4)○(C3×C4⋊C4), (C2×C12)○(C3×C4⋊C4), (C2×C4)○(C3×C22⋊C4), (C2×C12)○(C3×C22⋊C4), SmallGroup(96,165)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C12
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, C2×C4, C2×C4, C2×C4, D4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C4×D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, D4×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22×C4, C2×D4, C4○D4, C2×C12, C3×D4, C22×C6, C4×D4, C22×C12, C6×D4, C3×C4○D4, D4×C12
►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)]])
D4×C12 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 | C4○D4 | C3×D4 | C3×C4○D4 |
| kernel | D4×C12 | C4×C12 | C3×C22⋊C4 | C3×C4⋊C4 | C22×C12 | C6×D4 | C4×D4 | C3×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | 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 D4×C12 ►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] >;
D4×C12 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