metabelian, supersoluble, monomial
Aliases: D12.30D6, C62.41D4, Dic6.31D6, C4○D12⋊2S3, C32⋊8(C4○D8), (C3×C12).116D4, (C2×C12).111D6, C32⋊2D8⋊10C2, C3⋊6(Q8.13D6), C12.98(C3⋊D4), (C6×C12).71C22, (C3×C12).58C23, C12.78(C22×S3), C32⋊2Q16⋊10C2, Dic6⋊S3⋊15C2, C4.20(D6⋊S3), (C3×D12).31C22, C22.1(D6⋊S3), C32⋊4C8.24C22, (C3×Dic6).32C22, C4.73(C2×S32), (C2×C4).108S32, (C3×C4○D12)⋊1C2, (C3×C6).62(C2×D4), C6.70(C2×C3⋊D4), (C2×C32⋊4C8)⋊6C2, C2.5(C2×D6⋊S3), (C2×C6).16(C3⋊D4), SmallGroup(288,470)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.30D6
G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a3b, dcd-1=c5 >
Subgroups: 434 in 135 conjugacy classes, 44 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C4○D8, C3×Dic3, C3×C12, S3×C6, C62, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C4○D12, C3×C4○D4, C32⋊4C8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, Q8.13D6, C32⋊2D8, Dic6⋊S3, C32⋊2Q16, C2×C32⋊4C8, C3×C4○D12, D12.30D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C4○D8, S32, C2×C3⋊D4, D6⋊S3, C2×S32, Q8.13D6, C2×D6⋊S3, D12.30D6
►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 28)(2 27)(3 26)(4 25)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 48)(24 47)
(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 14 7 20)(2 21 8 15)(3 16 9 22)(4 23 10 17)(5 18 11 24)(6 13 12 19)(25 45 31 39)(26 40 32 46)(27 47 33 41)(28 42 34 48)(29 37 35 43)(30 44 36 38)
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), (1,28)(2,27)(3,26)(4,25)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,48)(24,47), (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), (1,14,7,20)(2,21,8,15)(3,16,9,22)(4,23,10,17)(5,18,11,24)(6,13,12,19)(25,45,31,39)(26,40,32,46)(27,47,33,41)(28,42,34,48)(29,37,35,43)(30,44,36,38)>;
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), (1,28)(2,27)(3,26)(4,25)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,48)(24,47), (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), (1,14,7,20)(2,21,8,15)(3,16,9,22)(4,23,10,17)(5,18,11,24)(6,13,12,19)(25,45,31,39)(26,40,32,46)(27,47,33,41)(28,42,34,48)(29,37,35,43)(30,44,36,38) );
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)], [(1,28),(2,27),(3,26),(4,25),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,46),(14,45),(15,44),(16,43),(17,42),(18,41),(19,40),(20,39),(21,38),(22,37),(23,48),(24,47)], [(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)], [(1,14,7,20),(2,21,8,15),(3,16,9,22),(4,23,10,17),(5,18,11,24),(6,13,12,19),(25,45,31,39),(26,40,32,46),(27,47,33,41),(28,42,34,48),(29,37,35,43),(30,44,36,38)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6G | 6H | 6I | 6J | 6K | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 12 | 12 | 2 | 2 | 4 | 1 | 1 | 2 | 12 | 12 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C3⋊D4 | C3⋊D4 | C4○D8 | S32 | D6⋊S3 | C2×S32 | D6⋊S3 | Q8.13D6 | D12.30D6 |
| kernel | D12.30D6 | C32⋊2D8 | Dic6⋊S3 | C32⋊2Q16 | C2×C32⋊4C8 | C3×C4○D12 | C4○D12 | C3×C12 | C62 | Dic6 | D12 | C2×C12 | C12 | C2×C6 | C32 | C2×C4 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 4 |
Matrix representation of D12.30D6 ►in GL4(𝔽5) generated by
| 2 | 3 | 3 | 2 |
| 3 | 1 | 2 | 0 |
| 3 | 1 | 3 | 0 |
| 3 | 0 | 2 | 4 |
| 4 | 2 | 0 | 4 |
| 4 | 4 | 4 | 2 |
| 3 | 0 | 2 | 4 |
| 3 | 1 | 3 | 0 |
| 4 | 0 | 2 | 0 |
| 3 | 0 | 1 | 3 |
| 4 | 0 | 3 | 0 |
| 0 | 2 | 1 | 2 |
| 3 | 4 | 2 | 2 |
| 2 | 0 | 4 | 2 |
| 1 | 4 | 4 | 0 |
| 0 | 0 | 0 | 3 |
G:=sub<GL(4,GF(5))| [2,3,3,3,3,1,1,0,3,2,3,2,2,0,0,4],[4,4,3,3,2,4,0,1,0,4,2,3,4,2,4,0],[4,3,4,0,0,0,0,2,2,1,3,1,0,3,0,2],[3,2,1,0,4,0,4,0,2,4,4,0,2,2,0,3] >;
D12.30D6 in GAP, Magma, Sage, TeX
D_{12}._{30}D_6 % in TeX
G:=Group("D12.30D6"); // GroupNames label
G:=SmallGroup(288,470);
// by ID
G=gap.SmallGroup(288,470);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,100,675,346,80,1356,9414]);
// 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,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^5>;
// generators/relations