metabelian, supersoluble, monomial
Aliases: D12.9D6, Q8:4S32, C3:C8:17D6, (C3xQ8):5D6, C3:S3:5SD16, C3:4(S3xSD16), C6.61(S3xD4), Q8:2S3:2S3, D6:D6.4C2, C3:Dic3.22D4, C32:12(C2xSD16), C12.29D6:5C2, C12.17(C22xS3), (C3xC12).17C23, D12.S3:13C2, C2.21(Dic3:D6), (Q8xC32):3C22, C32:4Q8:6C22, (C3xD12).17C22, C4.17(C2xS32), (Q8xC3:S3):1C2, (C3xC3:C8):17C22, (C2xC3:S3).59D4, (C3xQ8:2S3):6C2, (C3xC6).132(C2xD4), (C4xC3:S3).17C22, SmallGroup(288,588)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.9D6
G = < a,b,c,d | a12=b2=1, c6=a6, d2=a9, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a3b, dcd-1=a3c5 >
Subgroups: 738 in 155 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2xC4, D4, Q8, Q8, C23, C32, Dic3, C12, C12, D6, C2xC6, C2xC8, SD16, C2xD4, C2xQ8, C3xS3, C3:S3, C3xC6, C3:C8, C24, Dic6, C4xS3, D12, C3:D4, C3xD4, C3xQ8, C3xQ8, C22xS3, C2xSD16, C3:Dic3, C3:Dic3, C3xC12, C3xC12, S32, S3xC6, C2xC3:S3, S3xC8, C24:C2, D4.S3, Q8:2S3, C3xSD16, S3xD4, S3xQ8, C3xC3:C8, D6:S3, C3xD12, C32:4Q8, C32:4Q8, C4xC3:S3, C4xC3:S3, Q8xC32, C2xS32, S3xSD16, C12.29D6, D12.S3, C3xQ8:2S3, D6:D6, Q8xC3:S3, D12.9D6
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2xD4, C22xS3, C2xSD16, S32, S3xD4, C2xS32, S3xSD16, Dic3:D6, D12.9D6
►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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)
(1 21 11 19 9 17 7 15 5 13 3 23)(2 16 12 14 10 24 8 22 6 20 4 18)(25 44 27 46 29 48 31 38 33 40 35 42)(26 39 28 41 30 43 32 45 34 47 36 37)
(1 42 10 39 7 48 4 45)(2 43 11 40 8 37 5 46)(3 44 12 41 9 38 6 47)(13 36 22 33 19 30 16 27)(14 25 23 34 20 31 17 28)(15 26 24 35 21 32 18 29)
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43), (1,21,11,19,9,17,7,15,5,13,3,23)(2,16,12,14,10,24,8,22,6,20,4,18)(25,44,27,46,29,48,31,38,33,40,35,42)(26,39,28,41,30,43,32,45,34,47,36,37), (1,42,10,39,7,48,4,45)(2,43,11,40,8,37,5,46)(3,44,12,41,9,38,6,47)(13,36,22,33,19,30,16,27)(14,25,23,34,20,31,17,28)(15,26,24,35,21,32,18,29)>;
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43), (1,21,11,19,9,17,7,15,5,13,3,23)(2,16,12,14,10,24,8,22,6,20,4,18)(25,44,27,46,29,48,31,38,33,40,35,42)(26,39,28,41,30,43,32,45,34,47,36,37), (1,42,10,39,7,48,4,45)(2,43,11,40,8,37,5,46)(3,44,12,41,9,38,6,47)(13,36,22,33,19,30,16,27)(14,25,23,34,20,31,17,28)(15,26,24,35,21,32,18,29) );
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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,42),(14,41),(15,40),(16,39),(17,38),(18,37),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43)], [(1,21,11,19,9,17,7,15,5,13,3,23),(2,16,12,14,10,24,8,22,6,20,4,18),(25,44,27,46,29,48,31,38,33,40,35,42),(26,39,28,41,30,43,32,45,34,47,36,37)], [(1,42,10,39,7,48,4,45),(2,43,11,40,8,37,5,46),(3,44,12,41,9,38,6,47),(13,36,22,33,19,30,16,27),(14,25,23,34,20,31,17,28),(15,26,24,35,21,32,18,29)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | ··· | 12G | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 9 | 9 | 12 | 12 | 2 | 2 | 4 | 2 | 4 | 18 | 36 | 2 | 2 | 4 | 24 | 24 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | ··· | 8 | 12 | 12 | 12 | 12 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | SD16 | S32 | S3xD4 | C2xS32 | S3xSD16 | Dic3:D6 | D12.9D6 |
| kernel | D12.9D6 | C12.29D6 | D12.S3 | C3xQ8:2S3 | D6:D6 | Q8xC3:S3 | Q8:2S3 | C3:Dic3 | C2xC3:S3 | C3:C8 | D12 | C3xQ8 | C3:S3 | Q8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 1 | 4 | 2 | 1 |
Matrix representation of D12.9D6 ►in GL6(F73)
| 72 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 3 | 0 | 0 |
| 0 | 0 | 48 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 62 | 68 | 0 | 0 |
| 0 | 0 | 24 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 21 | 0 | 0 |
| 0 | 0 | 33 | 67 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 55 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,48,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,0,1,0,0,0,0,0,0,62,24,0,0,0,0,68,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,33,0,0,0,0,21,67,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,4,0,0,0,0,55,0,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;
D12.9D6 in GAP, Magma, Sage, TeX
D_{12}._9D_6 % in TeX
G:=Group("D12.9D6"); // GroupNames label
G:=SmallGroup(288,588);
// by ID
G=gap.SmallGroup(288,588);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,135,100,675,346,185,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,d^2=a^9,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^5>;
// generators/relations