metabelian, supersoluble, monomial
Aliases: D12.34D6, Dic6.35D6, C32⋊22- 1+4, C62.130C23, C4○D12⋊6S3, C3⋊D4.2D6, C3⋊2(Q8○D12), (S3×Dic6)⋊8C2, (C4×S3).14D6, C6.5(S3×C23), (C3×C6).5C24, D12⋊5S3⋊7C2, D6.4D6⋊3C2, (C2×C12).167D6, (S3×C6).3C23, D6.4(C22×S3), (S3×C12).31C22, (C6×C12).160C22, (C3×C12).114C23, C12.131(C22×S3), D6⋊S3.4C22, (C3×D12).43C22, C3⋊Dic3.16C23, (C3×Dic3).4C23, Dic3.3(C22×S3), (S3×Dic3).2C22, C32⋊2Q8.5C22, (C3×Dic6).44C22, C32⋊4Q8.35C22, C4.62(C2×S32), (C2×C4).36S32, C2.8(C22×S32), C22.11(C2×S32), (C3×C4○D12)⋊10C2, (C2×C6).13(C22×S3), (C2×C32⋊4Q8)⋊20C2, (C3×C3⋊D4).3C22, (C2×C3⋊Dic3).103C22, SmallGroup(288,946)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.34D6
G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a6c5 >
Subgroups: 994 in 311 conjugacy classes, 108 normal (10 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, 2- 1+4, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C62, C2×Dic6, C4○D12, C4○D12, D4⋊2S3, S3×Q8, C3×C4○D4, S3×Dic3, D6⋊S3, C32⋊2Q8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, Q8○D12, S3×Dic6, D12⋊5S3, D6.4D6, C3×C4○D12, C2×C32⋊4Q8, D12.34D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2- 1+4, S32, S3×C23, C2×S32, Q8○D12, C22×S32, D12.34D6
►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 29)(2 28)(3 27)(4 26)(5 25)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 48)(23 47)(24 46)
(1 8 3 10 5 12 7 2 9 4 11 6)(13 18 23 16 21 14 19 24 17 22 15 20)(25 30 35 28 33 26 31 36 29 34 27 32)(37 44 39 46 41 48 43 38 45 40 47 42)
(1 20 7 14)(2 21 8 15)(3 22 9 16)(4 23 10 17)(5 24 11 18)(6 13 12 19)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 37 34 43)(29 38 35 44)(30 39 36 45)
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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,48)(23,47)(24,46), (1,8,3,10,5,12,7,2,9,4,11,6)(13,18,23,16,21,14,19,24,17,22,15,20)(25,30,35,28,33,26,31,36,29,34,27,32)(37,44,39,46,41,48,43,38,45,40,47,42), (1,20,7,14)(2,21,8,15)(3,22,9,16)(4,23,10,17)(5,24,11,18)(6,13,12,19)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,37,34,43)(29,38,35,44)(30,39,36,45)>;
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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,48)(23,47)(24,46), (1,8,3,10,5,12,7,2,9,4,11,6)(13,18,23,16,21,14,19,24,17,22,15,20)(25,30,35,28,33,26,31,36,29,34,27,32)(37,44,39,46,41,48,43,38,45,40,47,42), (1,20,7,14)(2,21,8,15)(3,22,9,16)(4,23,10,17)(5,24,11,18)(6,13,12,19)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,37,34,43)(29,38,35,44)(30,39,36,45) );
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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,45),(14,44),(15,43),(16,42),(17,41),(18,40),(19,39),(20,38),(21,37),(22,48),(23,47),(24,46)], [(1,8,3,10,5,12,7,2,9,4,11,6),(13,18,23,16,21,14,19,24,17,22,15,20),(25,30,35,28,33,26,31,36,29,34,27,32),(37,44,39,46,41,48,43,38,45,40,47,42)], [(1,20,7,14),(2,21,8,15),(3,22,9,16),(4,23,10,17),(5,24,11,18),(6,13,12,19),(25,46,31,40),(26,47,32,41),(27,48,33,42),(28,37,34,43),(29,38,35,44),(30,39,36,45)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | ··· | 6G | 6H | 6I | 6J | 6K | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 4 | 2 | 2 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | 2- 1+4 | S32 | C2×S32 | C2×S32 | Q8○D12 | D12.34D6 |
| kernel | D12.34D6 | S3×Dic6 | D12⋊5S3 | D6.4D6 | C3×C4○D12 | C2×C32⋊4Q8 | C4○D12 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C32 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 4 | 4 | 4 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 1 | 1 | 2 | 1 | 4 | 4 |
Matrix representation of D12.34D6 ►in GL4(𝔽13) generated by
| 7 | 3 | 0 | 0 |
| 10 | 10 | 0 | 0 |
| 0 | 0 | 7 | 3 |
| 0 | 0 | 10 | 10 |
| 0 | 0 | 8 | 5 |
| 0 | 0 | 0 | 5 |
| 5 | 8 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 6 | 10 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 10 | 6 |
| 0 | 0 | 10 | 10 |
| 0 | 0 | 3 | 7 |
| 6 | 10 | 0 | 0 |
| 3 | 3 | 0 | 0 |
G:=sub<GL(4,GF(13))| [7,10,0,0,3,10,0,0,0,0,7,10,0,0,3,10],[0,0,5,0,0,0,8,8,8,0,0,0,5,5,0,0],[6,3,0,0,10,3,0,0,0,0,3,10,0,0,3,6],[0,0,6,3,0,0,10,3,10,3,0,0,10,7,0,0] >;
D12.34D6 in GAP, Magma, Sage, TeX
D_{12}._{34}D_6 % in TeX
G:=Group("D12.34D6"); // GroupNames label
G:=SmallGroup(288,946);
// by ID
G=gap.SmallGroup(288,946);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^6=d^2=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^6*c^5>;
// generators/relations