metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊5D12, C42⋊17D6, C6.172+ 1+4, C4⋊C4⋊48D6, (C3×D4)⋊10D4, (C4×D4)⋊16S3, D6⋊7(C4○D4), (C4×D12)⋊30C2, D6⋊D4⋊6C2, (D4×C12)⋊18C2, C12⋊7D4⋊9C2, C3⋊3(D4⋊5D4), C22⋊C4⋊47D6, C12.54(C2×D4), C4.22(C2×D12), (C22×C4)⋊16D6, C12⋊D4⋊15C2, (C4×C12)⋊20C22, D6⋊C4⋊52C22, C4.D12⋊14C2, (C2×D4).248D6, (C2×D12)⋊6C22, (C2×C6).98C24, C4⋊Dic3⋊8C22, C22.1(C2×D12), C6.16(C22×D4), C42⋊7S3⋊18C2, C2.18(C22×D12), C2.18(D4⋊6D6), (C2×C12).159C23, (C22×C12)⋊10C22, (C2×Dic6)⋊17C22, (C6×D4).259C22, C23.21D6⋊5C2, (C22×S3).33C23, (S3×C23).40C22, C22.123(S3×C23), C23.182(C22×S3), (C22×C6).168C23, (C2×Dic3).42C23, (C22×Dic3)⋊9C22, (C2×S3×D4)⋊4C2, (S3×C2×C4)⋊3C22, (C2×C6).1(C2×D4), (C2×D6⋊C4)⋊21C2, C2.22(S3×C4○D4), (C2×D4⋊2S3)⋊3C2, (C3×C4⋊C4)⋊60C22, C6.139(C2×C4○D4), (C2×C3⋊D4)⋊4C22, (C3×C22⋊C4)⋊50C22, (C2×C4).160(C22×S3), SmallGroup(192,1113)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊5D12
G = < a,b,c,d | a4=b2=c12=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 1016 in 334 conjugacy classes, 113 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C4×D4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C2×D12, S3×D4, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, D4⋊5D4, C4×D12, C42⋊7S3, D6⋊D4, C23.21D6, C12⋊D4, C4.D12, C2×D6⋊C4, C12⋊7D4, D4×C12, C2×S3×D4, C2×D4⋊2S3, D4⋊5D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, D12, C22×S3, C22×D4, C2×C4○D4, 2+ 1+4, C2×D12, S3×C23, D4⋊5D4, C22×D12, D4⋊6D6, S3×C4○D4, D4⋊5D12
►On 48 points
(1 18 28 46)(2 47 29 19)(3 20 30 48)(4 37 31 21)(5 22 32 38)(6 39 33 23)(7 24 34 40)(8 41 35 13)(9 14 36 42)(10 43 25 15)(11 16 26 44)(12 45 27 17)
(1 46)(2 19)(3 48)(4 21)(5 38)(6 23)(7 40)(8 13)(9 42)(10 15)(11 44)(12 17)(14 36)(16 26)(18 28)(20 30)(22 32)(24 34)(25 43)(27 45)(29 47)(31 37)(33 39)(35 41)
(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 45)(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 48)(11 47)(12 46)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(21 36)(22 35)(23 34)(24 33)
G:=sub<Sym(48)| (1,18,28,46)(2,47,29,19)(3,20,30,48)(4,37,31,21)(5,22,32,38)(6,39,33,23)(7,24,34,40)(8,41,35,13)(9,14,36,42)(10,43,25,15)(11,16,26,44)(12,45,27,17), (1,46)(2,19)(3,48)(4,21)(5,38)(6,23)(7,40)(8,13)(9,42)(10,15)(11,44)(12,17)(14,36)(16,26)(18,28)(20,30)(22,32)(24,34)(25,43)(27,45)(29,47)(31,37)(33,39)(35,41), (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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,48)(11,47)(12,46)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,36)(22,35)(23,34)(24,33)>;
G:=Group( (1,18,28,46)(2,47,29,19)(3,20,30,48)(4,37,31,21)(5,22,32,38)(6,39,33,23)(7,24,34,40)(8,41,35,13)(9,14,36,42)(10,43,25,15)(11,16,26,44)(12,45,27,17), (1,46)(2,19)(3,48)(4,21)(5,38)(6,23)(7,40)(8,13)(9,42)(10,15)(11,44)(12,17)(14,36)(16,26)(18,28)(20,30)(22,32)(24,34)(25,43)(27,45)(29,47)(31,37)(33,39)(35,41), (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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,48)(11,47)(12,46)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,36)(22,35)(23,34)(24,33) );
G=PermutationGroup([[(1,18,28,46),(2,47,29,19),(3,20,30,48),(4,37,31,21),(5,22,32,38),(6,39,33,23),(7,24,34,40),(8,41,35,13),(9,14,36,42),(10,43,25,15),(11,16,26,44),(12,45,27,17)], [(1,46),(2,19),(3,48),(4,21),(5,38),(6,23),(7,40),(8,13),(9,42),(10,15),(11,44),(12,17),(14,36),(16,26),(18,28),(20,30),(22,32),(24,34),(25,43),(27,45),(29,47),(31,37),(33,39),(35,41)], [(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)], [(1,45),(2,44),(3,43),(4,42),(5,41),(6,40),(7,39),(8,38),(9,37),(10,48),(11,47),(12,46),(13,32),(14,31),(15,30),(16,29),(17,28),(18,27),(19,26),(20,25),(21,36),(22,35),(23,34),(24,33)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 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 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 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 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 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 | S3 | D4 | D6 | D6 | D6 | D6 | D6 | C4○D4 | D12 | 2+ 1+4 | D4⋊6D6 | S3×C4○D4 |
| kernel | D4⋊5D12 | C4×D12 | C42⋊7S3 | D6⋊D4 | C23.21D6 | C12⋊D4 | C4.D12 | C2×D6⋊C4 | C12⋊7D4 | D4×C12 | C2×S3×D4 | C2×D4⋊2S3 | C4×D4 | C3×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D6 | D4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of D4⋊5D12 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 3 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 3 |
| 0 | 0 | 0 | 0 | 5 | 8 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 | 5 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,3,8],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,5,0,0,0,0,3,8],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,8,0,0,0,0,0,5],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,2,1] >;
D4⋊5D12 in GAP, Magma, Sage, TeX
D_4\rtimes_5D_{12} % in TeX
G:=Group("D4:5D12"); // GroupNames label
G:=SmallGroup(192,1113);
// by ID
G=gap.SmallGroup(192,1113);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^12=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations