metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊21D4, C6.1172+ 1+4, C4⋊C4⋊10D6, (C2×Q8)⋊19D6, C22⋊Q8⋊7S3, C3⋊7(D4⋊5D4), D6.20(C2×D4), C4.111(S3×D4), C12⋊D4⋊25C2, D6⋊D4⋊16C2, (C6×Q8)⋊7C22, D6⋊C4⋊20C22, C12.234(C2×D4), C22⋊C4.57D6, Dic3⋊5D4⋊25C2, C6.76(C22×D4), D6.D4⋊17C2, C2.34(D4○D12), (C2×D12)⋊25C22, (C22×D12)⋊16C2, (C2×C6).174C24, (C2×C12).54C23, (C22×C4).252D6, C12.23D4⋊12C2, Dic3⋊C4⋊53C22, C22⋊3(Q8⋊3S3), (C4×Dic3)⋊28C22, (S3×C23).52C22, (C22×C6).202C23, C23.199(C22×S3), C22.195(S3×C23), (C22×S3).196C23, (C22×C12).254C22, (C2×Dic3).233C23, C6.D4.115C22, C2.49(C2×S3×D4), (C2×C6)⋊7(C4○D4), (C4×C3⋊D4)⋊22C2, (S3×C22⋊C4)⋊8C2, (S3×C2×C4)⋊18C22, (C3×C4⋊C4)⋊19C22, (C2×Q8⋊3S3)⋊7C2, C6.114(C2×C4○D4), (C3×C22⋊Q8)⋊10C2, (C2×C4).47(C22×S3), C2.17(C2×Q8⋊3S3), (C2×C3⋊D4).122C22, (C3×C22⋊C4).29C22, SmallGroup(192,1189)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊21D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=dad=a5, cbc-1=dbd=a10b, dcd=c-1 >
Subgroups: 1040 in 334 conjugacy classes, 107 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C2×D12, Q8⋊3S3, C2×C3⋊D4, C22×C12, C6×Q8, S3×C23, D4⋊5D4, S3×C22⋊C4, D6⋊D4, Dic3⋊5D4, D6.D4, C12⋊D4, C12⋊D4, C4×C3⋊D4, C12.23D4, C3×C22⋊Q8, C22×D12, C2×Q8⋊3S3, D12⋊21D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, 2+ 1+4, S3×D4, Q8⋊3S3, S3×C23, D4⋊5D4, C2×S3×D4, C2×Q8⋊3S3, D4○D12, D12⋊21D4
►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 3)(4 12)(5 11)(6 10)(7 9)(13 23)(14 22)(15 21)(16 20)(17 19)(25 31)(26 30)(27 29)(32 36)(33 35)(37 45)(38 44)(39 43)(40 42)(46 48)
(1 36 40 14)(2 29 41 19)(3 34 42 24)(4 27 43 17)(5 32 44 22)(6 25 45 15)(7 30 46 20)(8 35 47 13)(9 28 48 18)(10 33 37 23)(11 26 38 16)(12 31 39 21)
(1 20)(2 13)(3 18)(4 23)(5 16)(6 21)(7 14)(8 19)(9 24)(10 17)(11 22)(12 15)(25 39)(26 44)(27 37)(28 42)(29 47)(30 40)(31 45)(32 38)(33 43)(34 48)(35 41)(36 46)
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,31)(26,30)(27,29)(32,36)(33,35)(37,45)(38,44)(39,43)(40,42)(46,48), (1,36,40,14)(2,29,41,19)(3,34,42,24)(4,27,43,17)(5,32,44,22)(6,25,45,15)(7,30,46,20)(8,35,47,13)(9,28,48,18)(10,33,37,23)(11,26,38,16)(12,31,39,21), (1,20)(2,13)(3,18)(4,23)(5,16)(6,21)(7,14)(8,19)(9,24)(10,17)(11,22)(12,15)(25,39)(26,44)(27,37)(28,42)(29,47)(30,40)(31,45)(32,38)(33,43)(34,48)(35,41)(36,46)>;
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,31)(26,30)(27,29)(32,36)(33,35)(37,45)(38,44)(39,43)(40,42)(46,48), (1,36,40,14)(2,29,41,19)(3,34,42,24)(4,27,43,17)(5,32,44,22)(6,25,45,15)(7,30,46,20)(8,35,47,13)(9,28,48,18)(10,33,37,23)(11,26,38,16)(12,31,39,21), (1,20)(2,13)(3,18)(4,23)(5,16)(6,21)(7,14)(8,19)(9,24)(10,17)(11,22)(12,15)(25,39)(26,44)(27,37)(28,42)(29,47)(30,40)(31,45)(32,38)(33,43)(34,48)(35,41)(36,46) );
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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,23),(14,22),(15,21),(16,20),(17,19),(25,31),(26,30),(27,29),(32,36),(33,35),(37,45),(38,44),(39,43),(40,42),(46,48)], [(1,36,40,14),(2,29,41,19),(3,34,42,24),(4,27,43,17),(5,32,44,22),(6,25,45,15),(7,30,46,20),(8,35,47,13),(9,28,48,18),(10,33,37,23),(11,26,38,16),(12,31,39,21)], [(1,20),(2,13),(3,18),(4,23),(5,16),(6,21),(7,14),(8,19),(9,24),(10,17),(11,22),(12,15),(25,39),(26,44),(27,37),(28,42),(29,47),(30,40),(31,45),(32,38),(33,43),(34,48),(35,41),(36,46)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | C4○D4 | 2+ 1+4 | S3×D4 | Q8⋊3S3 | D4○D12 |
| kernel | D12⋊21D4 | S3×C22⋊C4 | D6⋊D4 | Dic3⋊5D4 | D6.D4 | C12⋊D4 | C4×C3⋊D4 | C12.23D4 | C3×C22⋊Q8 | C22×D12 | C2×Q8⋊3S3 | C22⋊Q8 | D12 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 1 | 1 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of D12⋊21D4 ►in GL6(𝔽13)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
D12⋊21D4 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{21}D_4 % in TeX
G:=Group("D12:21D4"); // GroupNames label
G:=SmallGroup(192,1189);
// by ID
G=gap.SmallGroup(192,1189);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,1571,297,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^5,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations