metabelian, supersoluble, monomial
Aliases: D12.33D6, Dic6.34D6, C32⋊12- 1+4, C62.129C23, C4○D12⋊5S3, C3⋊D4.3D6, C3⋊1(Q8○D12), (S3×Dic6)⋊7C2, (C4×S3).13D6, C6.4(S3×C23), (C3×C6).4C24, D12⋊S3⋊7C2, (C2×Dic6)⋊13S3, (C6×Dic6)⋊19C2, D6.6D6⋊7C2, D6.3D6⋊1C2, (C2×C12).166D6, (S3×C6).2C23, C3⋊D12.C22, D6.3(C22×S3), C12.59D6⋊7C2, (C2×Dic3).49D6, Dic3.D6⋊10C2, (S3×C12).30C22, (C6×C12).159C22, (C3×C12).113C23, C12.130(C22×S3), C3⋊1(Q8.15D6), (C3×D12).42C22, C3⋊Dic3.15C23, (S3×Dic3).1C22, (C3×Dic3).3C23, Dic3.2(C22×S3), C32⋊2Q8.4C22, C6.D6.5C22, C32⋊7D4.2C22, C12⋊S3.33C22, (C3×Dic6).43C22, (C6×Dic3).45C22, C32⋊4Q8.34C22, C4.61(C2×S32), (C2×C4).35S32, C22.6(C2×S32), C2.7(C22×S32), (C3×C4○D12)⋊9C2, (C4×C3⋊S3).41C22, (C2×C3⋊S3).17C23, (C2×C6).12(C22×S3), (C3×C3⋊D4).2C22, SmallGroup(288,945)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.33D6
G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=a6c5 >
Subgroups: 1050 in 312 conjugacy classes, 108 normal (36 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, D6, C2×C6, C2×C6, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, 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, C2×C3⋊S3, C62, C2×Dic6, C2×Dic6, C4○D12, C4○D12, D4⋊2S3, S3×Q8, Q8⋊3S3, C6×Q8, C3×C4○D4, S3×Dic3, C6.D6, C3⋊D12, C32⋊2Q8, C3×Dic6, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, Q8.15D6, Q8○D12, S3×Dic6, D12⋊S3, Dic3.D6, D6.6D6, D6.3D6, C6×Dic6, C3×C4○D12, C12.59D6, D12.33D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2- 1+4, S32, S3×C23, C2×S32, Q8.15D6, Q8○D12, C22×S32, D12.33D6
►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 40)(2 39)(3 38)(4 37)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 29)(14 28)(15 27)(16 26)(17 25)(18 36)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)
(1 17 11 15 9 13 7 23 5 21 3 19)(2 18 12 16 10 14 8 24 6 22 4 20)(25 42 27 44 29 46 31 48 33 38 35 40)(26 43 28 45 30 47 32 37 34 39 36 41)
(1 27 7 33)(2 34 8 28)(3 29 9 35)(4 36 10 30)(5 31 11 25)(6 26 12 32)(13 38 19 44)(14 45 20 39)(15 40 21 46)(16 47 22 41)(17 42 23 48)(18 37 24 43)
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), (1,40)(2,39)(3,38)(4,37)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,29)(14,28)(15,27)(16,26)(17,25)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30), (1,17,11,15,9,13,7,23,5,21,3,19)(2,18,12,16,10,14,8,24,6,22,4,20)(25,42,27,44,29,46,31,48,33,38,35,40)(26,43,28,45,30,47,32,37,34,39,36,41), (1,27,7,33)(2,34,8,28)(3,29,9,35)(4,36,10,30)(5,31,11,25)(6,26,12,32)(13,38,19,44)(14,45,20,39)(15,40,21,46)(16,47,22,41)(17,42,23,48)(18,37,24,43)>;
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), (1,40)(2,39)(3,38)(4,37)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,29)(14,28)(15,27)(16,26)(17,25)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30), (1,17,11,15,9,13,7,23,5,21,3,19)(2,18,12,16,10,14,8,24,6,22,4,20)(25,42,27,44,29,46,31,48,33,38,35,40)(26,43,28,45,30,47,32,37,34,39,36,41), (1,27,7,33)(2,34,8,28)(3,29,9,35)(4,36,10,30)(5,31,11,25)(6,26,12,32)(13,38,19,44)(14,45,20,39)(15,40,21,46)(16,47,22,41)(17,42,23,48)(18,37,24,43) );
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)], [(1,40),(2,39),(3,38),(4,37),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,29),(14,28),(15,27),(16,26),(17,25),(18,36),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30)], [(1,17,11,15,9,13,7,23,5,21,3,19),(2,18,12,16,10,14,8,24,6,22,4,20),(25,42,27,44,29,46,31,48,33,38,35,40),(26,43,28,45,30,47,32,37,34,39,36,41)], [(1,27,7,33),(2,34,8,28),(3,29,9,35),(4,36,10,30),(5,31,11,25),(6,26,12,32),(13,38,19,44),(14,45,20,39),(15,40,21,46),(16,47,22,41),(17,42,23,48),(18,37,24,43)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 12A | 12B | 12C | ··· | 12I | 12J | ··· | 12O |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 6 | 6 | 18 | 18 | 2 | 2 | 4 | 2 | 2 | 6 | ··· | 6 | 18 | 18 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 4 | ··· | 4 | 12 | ··· | 12 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | D6 | D6 | D6 | 2- 1+4 | S32 | C2×S32 | C2×S32 | Q8.15D6 | Q8○D12 | D12.33D6 |
| kernel | D12.33D6 | S3×Dic6 | D12⋊S3 | Dic3.D6 | D6.6D6 | D6.3D6 | C6×Dic6 | C3×C4○D12 | C12.59D6 | C2×Dic6 | C4○D12 | Dic6 | C4×S3 | D12 | C2×Dic3 | C3⋊D4 | C2×C12 | C32 | C2×C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 5 | 2 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 4 |
Matrix representation of D12.33D6 ►in GL4(𝔽13) generated by
| 8 | 8 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 1 |
| 12 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 7 | 10 | 0 | 0 |
| 3 | 10 | 0 | 0 |
| 0 | 0 | 10 | 3 |
| 0 | 0 | 10 | 7 |
| 0 | 0 | 3 | 10 |
| 0 | 0 | 3 | 6 |
| 7 | 10 | 0 | 0 |
| 3 | 10 | 0 | 0 |
G:=sub<GL(4,GF(13))| [8,5,0,0,8,0,0,0,0,0,5,8,0,0,5,0],[0,0,12,0,0,0,12,1,12,0,0,0,12,1,0,0],[7,3,0,0,10,10,0,0,0,0,10,10,0,0,3,7],[0,0,7,3,0,0,10,10,3,3,0,0,10,6,0,0] >;
D12.33D6 in GAP, Magma, Sage, TeX
D_{12}._{33}D_6 % in TeX
G:=Group("D12.33D6"); // GroupNames label
G:=SmallGroup(288,945);
// by ID
G=gap.SmallGroup(288,945);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,1356,9414]);
// 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,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=a^6*c^5>;
// generators/relations