metabelian, supersoluble, monomial
Aliases: Dic6.9D6, C3⋊S3⋊3Q16, C3⋊C8.16D6, C3⋊3(S3×Q16), Q8.14S32, C3⋊Q16⋊2S3, C6.63(S3×D4), C32⋊8(C2×Q16), (C3×Q8).31D6, C3⋊Dic3.23D4, (C3×C12).21C23, C12.21(C22×S3), C32⋊3Q16⋊10C2, C2.23(Dic3⋊D6), Dic3.D6.4C2, C12.29D6.1C2, (Q8×C32).3C22, (C3×Dic6).17C22, C32⋊4Q8.12C22, C4.21(C2×S32), (Q8×C3⋊S3).2C2, (C2×C3⋊S3).60D4, (C3×C3⋊Q16)⋊2C2, (C3×C3⋊C8).7C22, (C3×C6).136(C2×D4), (C4×C3⋊S3).19C22, SmallGroup(288,592)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6.9D6
G = < a,b,c,d | a12=1, b2=c6=a6, d2=a3, bab-1=a-1, cac-1=a7, ad=da, cbc-1=a9b, dbd-1=a3b, dcd-1=a3c5 >
Subgroups: 546 in 139 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, Q8, Q8, C32, Dic3, C12, C12, D6, C2×C8, Q16, C2×Q8, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, Dic6, C4×S3, C3×Q8, C3×Q8, C2×Q16, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, S3×C8, Dic12, C3⋊Q16, C3⋊Q16, C3×Q16, S3×Q8, C3×C3⋊C8, C6.D6, C32⋊2Q8, C3×Dic6, C32⋊4Q8, C32⋊4Q8, C4×C3⋊S3, C4×C3⋊S3, Q8×C32, S3×Q16, C12.29D6, C32⋊3Q16, C3×C3⋊Q16, Dic3.D6, Q8×C3⋊S3, Dic6.9D6
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C22×S3, C2×Q16, S32, S3×D4, C2×S32, S3×Q16, Dic3⋊D6, Dic6.9D6
►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 31 7 25)(2 30 8 36)(3 29 9 35)(4 28 10 34)(5 27 11 33)(6 26 12 32)(13 40 19 46)(14 39 20 45)(15 38 21 44)(16 37 22 43)(17 48 23 42)(18 47 24 41)
(1 17 3 19 5 21 7 23 9 13 11 15)(2 24 4 14 6 16 8 18 10 20 12 22)(25 45 35 43 33 41 31 39 29 37 27 47)(26 40 36 38 34 48 32 46 30 44 28 42)
(1 41 4 44 7 47 10 38)(2 42 5 45 8 48 11 39)(3 43 6 46 9 37 12 40)(13 26 16 29 19 32 22 35)(14 27 17 30 20 33 23 36)(15 28 18 31 21 34 24 25)
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), (1,31,7,25)(2,30,8,36)(3,29,9,35)(4,28,10,34)(5,27,11,33)(6,26,12,32)(13,40,19,46)(14,39,20,45)(15,38,21,44)(16,37,22,43)(17,48,23,42)(18,47,24,41), (1,17,3,19,5,21,7,23,9,13,11,15)(2,24,4,14,6,16,8,18,10,20,12,22)(25,45,35,43,33,41,31,39,29,37,27,47)(26,40,36,38,34,48,32,46,30,44,28,42), (1,41,4,44,7,47,10,38)(2,42,5,45,8,48,11,39)(3,43,6,46,9,37,12,40)(13,26,16,29,19,32,22,35)(14,27,17,30,20,33,23,36)(15,28,18,31,21,34,24,25)>;
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), (1,31,7,25)(2,30,8,36)(3,29,9,35)(4,28,10,34)(5,27,11,33)(6,26,12,32)(13,40,19,46)(14,39,20,45)(15,38,21,44)(16,37,22,43)(17,48,23,42)(18,47,24,41), (1,17,3,19,5,21,7,23,9,13,11,15)(2,24,4,14,6,16,8,18,10,20,12,22)(25,45,35,43,33,41,31,39,29,37,27,47)(26,40,36,38,34,48,32,46,30,44,28,42), (1,41,4,44,7,47,10,38)(2,42,5,45,8,48,11,39)(3,43,6,46,9,37,12,40)(13,26,16,29,19,32,22,35)(14,27,17,30,20,33,23,36)(15,28,18,31,21,34,24,25) );
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)], [(1,31,7,25),(2,30,8,36),(3,29,9,35),(4,28,10,34),(5,27,11,33),(6,26,12,32),(13,40,19,46),(14,39,20,45),(15,38,21,44),(16,37,22,43),(17,48,23,42),(18,47,24,41)], [(1,17,3,19,5,21,7,23,9,13,11,15),(2,24,4,14,6,16,8,18,10,20,12,22),(25,45,35,43,33,41,31,39,29,37,27,47),(26,40,36,38,34,48,32,46,30,44,28,42)], [(1,41,4,44,7,47,10,38),(2,42,5,45,8,48,11,39),(3,43,6,46,9,37,12,40),(13,26,16,29,19,32,22,35),(14,27,17,30,20,33,23,36),(15,28,18,31,21,34,24,25)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | ··· | 12G | 12H | 12I | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 4 | 2 | 4 | 12 | 12 | 18 | 36 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | ··· | 8 | 24 | 24 | 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 | Q16 | S32 | S3×D4 | C2×S32 | S3×Q16 | Dic3⋊D6 | Dic6.9D6 |
| kernel | Dic6.9D6 | C12.29D6 | C32⋊3Q16 | C3×C3⋊Q16 | Dic3.D6 | Q8×C3⋊S3 | C3⋊Q16 | C3⋊Dic3 | C2×C3⋊S3 | C3⋊C8 | Dic6 | C3×Q8 | 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 Dic6.9D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 62 | 0 | 0 |
| 0 | 0 | 62 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 61 | 0 | 0 |
| 0 | 0 | 61 | 72 | 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 | 16 | 57 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,62,0,0,0,0,62,43,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,61,0,0,0,0,61,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,57,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
Dic6.9D6 in GAP, Magma, Sage, TeX
{\rm Dic}_6._9D_6 % in TeX
G:=Group("Dic6.9D6"); // GroupNames label
G:=SmallGroup(288,592);
// by ID
G=gap.SmallGroup(288,592);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,254,135,100,675,346,185,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=1,b^2=c^6=a^6,d^2=a^3,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^9*b,d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^5>;
// generators/relations