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, (D4×C12)⋊18C2, D6⋊D4⋊6C2, 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, C22.D12⋊5C2, (C22×S3).33C23, (S3×C23).40C22, C23.182(C22×S3), C22.123(S3×C23), (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
Subgroups: 1016 in 334 conjugacy classes, 113 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×2], C4 [×8], C22, C22 [×4], C22 [×25], S3 [×5], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×14], Dic3 [×4], C12 [×2], C12 [×4], D6 [×2], D6 [×19], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×4], D12 [×6], C2×Dic3 [×2], C2×Dic3 [×2], C2×Dic3 [×4], C3⋊D4 [×8], C2×C12 [×3], C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C22×S3 [×2], C22×S3 [×2], C22×S3 [×10], C22×C6 [×2], C2×C22⋊C4 [×2], C4×D4, C4×D4, C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4⋊Dic3, C4⋊Dic3 [×2], D6⋊C4 [×2], D6⋊C4 [×8], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4 [×2], C2×D12 [×2], C2×D12 [×2], S3×D4 [×4], D4⋊2S3 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×4], C22×C12 [×2], C6×D4, S3×C23 [×2], D4⋊5D4, C4×D12, C42⋊7S3, D6⋊D4 [×2], C22.D12 [×2], C12⋊D4, C4.D12, C2×D6⋊C4 [×2], C12⋊7D4 [×2], D4×C12, C2×S3×D4, C2×D4⋊2S3, D4⋊5D12
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C2×D12 [×6], S3×C23, D4⋊5D4, C22×D12, D4⋊6D6, S3×C4○D4, D4⋊5D12
Generators and relations
G = < a,b,c,d | a4=b2=c12=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
►On 48 points
(1 16 44 27)(2 28 45 17)(3 18 46 29)(4 30 47 19)(5 20 48 31)(6 32 37 21)(7 22 38 33)(8 34 39 23)(9 24 40 35)(10 36 41 13)(11 14 42 25)(12 26 43 15)
(1 27)(2 17)(3 29)(4 19)(5 31)(6 21)(7 33)(8 23)(9 35)(10 13)(11 25)(12 15)(14 42)(16 44)(18 46)(20 48)(22 38)(24 40)(26 43)(28 45)(30 47)(32 37)(34 39)(36 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 26)(2 25)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 48)(24 47)
G:=sub<Sym(48)| (1,16,44,27)(2,28,45,17)(3,18,46,29)(4,30,47,19)(5,20,48,31)(6,32,37,21)(7,22,38,33)(8,34,39,23)(9,24,40,35)(10,36,41,13)(11,14,42,25)(12,26,43,15), (1,27)(2,17)(3,29)(4,19)(5,31)(6,21)(7,33)(8,23)(9,35)(10,13)(11,25)(12,15)(14,42)(16,44)(18,46)(20,48)(22,38)(24,40)(26,43)(28,45)(30,47)(32,37)(34,39)(36,41), (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), (1,26)(2,25)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,48)(24,47)>;
G:=Group( (1,16,44,27)(2,28,45,17)(3,18,46,29)(4,30,47,19)(5,20,48,31)(6,32,37,21)(7,22,38,33)(8,34,39,23)(9,24,40,35)(10,36,41,13)(11,14,42,25)(12,26,43,15), (1,27)(2,17)(3,29)(4,19)(5,31)(6,21)(7,33)(8,23)(9,35)(10,13)(11,25)(12,15)(14,42)(16,44)(18,46)(20,48)(22,38)(24,40)(26,43)(28,45)(30,47)(32,37)(34,39)(36,41), (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), (1,26)(2,25)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,48)(24,47) );
G=PermutationGroup([(1,16,44,27),(2,28,45,17),(3,18,46,29),(4,30,47,19),(5,20,48,31),(6,32,37,21),(7,22,38,33),(8,34,39,23),(9,24,40,35),(10,36,41,13),(11,14,42,25),(12,26,43,15)], [(1,27),(2,17),(3,29),(4,19),(5,31),(6,21),(7,33),(8,23),(9,35),(10,13),(11,25),(12,15),(14,42),(16,44),(18,46),(20,48),(22,38),(24,40),(26,43),(28,45),(30,47),(32,37),(34,39),(36,41)], [(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)], [(1,26),(2,25),(3,36),(4,35),(5,34),(6,33),(7,32),(8,31),(9,30),(10,29),(11,28),(12,27),(13,46),(14,45),(15,44),(16,43),(17,42),(18,41),(19,40),(20,39),(21,38),(22,37),(23,48),(24,47)])
Matrix representation ►G ⊆ 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] >;
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 | C22.D12 | 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 |
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