metabelian, supersoluble, monomial
Aliases: D6⋊2Dic6, Dic3.10D12, C62.63C23, (S3×C6)⋊2Q8, D6⋊C4.3S3, C6.17(S3×D4), C6.28(S3×Q8), Dic3⋊C4⋊5S3, (C2×C12).22D6, C2.20(S3×D12), C6.18(C2×D12), C3⋊2(C4.D12), D6⋊Dic3.6C2, (C6×C12).5C22, (C3×Dic3).8D4, C2.16(S3×Dic6), C6.16(C2×Dic6), Dic3⋊Dic3⋊6C2, C12⋊Dic3⋊4C2, (C2×Dic3).25D6, (C22×S3).39D6, C6.63(D4⋊2S3), C32⋊10(C22⋊Q8), C3⋊1(Dic3.D4), C2.11(D6.4D6), (C6×Dic3).38C22, (C2×C4).27S32, (C3×D6⋊C4).1C2, (C3×C6).50(C2×D4), (C3×C6).30(C2×Q8), (C2×S3×Dic3).3C2, C22.108(C2×S32), (S3×C2×C6).22C22, (C3×Dic3⋊C4)⋊17C2, (C3×C6).66(C4○D4), (C2×C32⋊2Q8)⋊11C2, (C2×C6).82(C22×S3), (C2×C3⋊Dic3).47C22, SmallGroup(288,541)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊2Dic6
G = < a,b,c,d | a6=b2=c12=1, d2=c6, bab=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd-1=c-1 >
Subgroups: 586 in 161 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, Q8, C23, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C3×S3, C3×C6, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊Q8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, S3×Dic3, C32⋊2Q8, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, Dic3.D4, C4.D12, D6⋊Dic3, Dic3⋊Dic3, C3×Dic3⋊C4, C3×D6⋊C4, C12⋊Dic3, C2×S3×Dic3, C2×C32⋊2Q8, D6⋊2Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, D12, C22×S3, C22⋊Q8, S32, C2×Dic6, C2×D12, S3×D4, D4⋊2S3, S3×Q8, C2×S32, Dic3.D4, C4.D12, S3×Dic6, S3×D12, D6.4D6, D6⋊2Dic6
►On 96 points
(1 15 9 23 5 19)(2 20 6 24 10 16)(3 17 11 13 7 21)(4 22 8 14 12 18)(25 40 33 48 29 44)(26 45 30 37 34 41)(27 42 35 38 31 46)(28 47 32 39 36 43)(49 78 53 82 57 74)(50 75 58 83 54 79)(51 80 55 84 59 76)(52 77 60 73 56 81)(61 85 69 93 65 89)(62 90 66 94 70 86)(63 87 71 95 67 91)(64 92 68 96 72 88)
(1 45)(2 27)(3 47)(4 29)(5 37)(6 31)(7 39)(8 33)(9 41)(10 35)(11 43)(12 25)(13 36)(14 40)(15 26)(16 42)(17 28)(18 44)(19 30)(20 46)(21 32)(22 48)(23 34)(24 38)(49 91)(50 68)(51 93)(52 70)(53 95)(54 72)(55 85)(56 62)(57 87)(58 64)(59 89)(60 66)(61 84)(63 74)(65 76)(67 78)(69 80)(71 82)(73 90)(75 92)(77 94)(79 96)(81 86)(83 88)
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 53 7 59)(2 52 8 58)(3 51 9 57)(4 50 10 56)(5 49 11 55)(6 60 12 54)(13 84 19 78)(14 83 20 77)(15 82 21 76)(16 81 22 75)(17 80 23 74)(18 79 24 73)(25 72 31 66)(26 71 32 65)(27 70 33 64)(28 69 34 63)(29 68 35 62)(30 67 36 61)(37 91 43 85)(38 90 44 96)(39 89 45 95)(40 88 46 94)(41 87 47 93)(42 86 48 92)
G:=sub<Sym(96)| (1,15,9,23,5,19)(2,20,6,24,10,16)(3,17,11,13,7,21)(4,22,8,14,12,18)(25,40,33,48,29,44)(26,45,30,37,34,41)(27,42,35,38,31,46)(28,47,32,39,36,43)(49,78,53,82,57,74)(50,75,58,83,54,79)(51,80,55,84,59,76)(52,77,60,73,56,81)(61,85,69,93,65,89)(62,90,66,94,70,86)(63,87,71,95,67,91)(64,92,68,96,72,88), (1,45)(2,27)(3,47)(4,29)(5,37)(6,31)(7,39)(8,33)(9,41)(10,35)(11,43)(12,25)(13,36)(14,40)(15,26)(16,42)(17,28)(18,44)(19,30)(20,46)(21,32)(22,48)(23,34)(24,38)(49,91)(50,68)(51,93)(52,70)(53,95)(54,72)(55,85)(56,62)(57,87)(58,64)(59,89)(60,66)(61,84)(63,74)(65,76)(67,78)(69,80)(71,82)(73,90)(75,92)(77,94)(79,96)(81,86)(83,88), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,53,7,59)(2,52,8,58)(3,51,9,57)(4,50,10,56)(5,49,11,55)(6,60,12,54)(13,84,19,78)(14,83,20,77)(15,82,21,76)(16,81,22,75)(17,80,23,74)(18,79,24,73)(25,72,31,66)(26,71,32,65)(27,70,33,64)(28,69,34,63)(29,68,35,62)(30,67,36,61)(37,91,43,85)(38,90,44,96)(39,89,45,95)(40,88,46,94)(41,87,47,93)(42,86,48,92)>;
G:=Group( (1,15,9,23,5,19)(2,20,6,24,10,16)(3,17,11,13,7,21)(4,22,8,14,12,18)(25,40,33,48,29,44)(26,45,30,37,34,41)(27,42,35,38,31,46)(28,47,32,39,36,43)(49,78,53,82,57,74)(50,75,58,83,54,79)(51,80,55,84,59,76)(52,77,60,73,56,81)(61,85,69,93,65,89)(62,90,66,94,70,86)(63,87,71,95,67,91)(64,92,68,96,72,88), (1,45)(2,27)(3,47)(4,29)(5,37)(6,31)(7,39)(8,33)(9,41)(10,35)(11,43)(12,25)(13,36)(14,40)(15,26)(16,42)(17,28)(18,44)(19,30)(20,46)(21,32)(22,48)(23,34)(24,38)(49,91)(50,68)(51,93)(52,70)(53,95)(54,72)(55,85)(56,62)(57,87)(58,64)(59,89)(60,66)(61,84)(63,74)(65,76)(67,78)(69,80)(71,82)(73,90)(75,92)(77,94)(79,96)(81,86)(83,88), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,53,7,59)(2,52,8,58)(3,51,9,57)(4,50,10,56)(5,49,11,55)(6,60,12,54)(13,84,19,78)(14,83,20,77)(15,82,21,76)(16,81,22,75)(17,80,23,74)(18,79,24,73)(25,72,31,66)(26,71,32,65)(27,70,33,64)(28,69,34,63)(29,68,35,62)(30,67,36,61)(37,91,43,85)(38,90,44,96)(39,89,45,95)(40,88,46,94)(41,87,47,93)(42,86,48,92) );
G=PermutationGroup([[(1,15,9,23,5,19),(2,20,6,24,10,16),(3,17,11,13,7,21),(4,22,8,14,12,18),(25,40,33,48,29,44),(26,45,30,37,34,41),(27,42,35,38,31,46),(28,47,32,39,36,43),(49,78,53,82,57,74),(50,75,58,83,54,79),(51,80,55,84,59,76),(52,77,60,73,56,81),(61,85,69,93,65,89),(62,90,66,94,70,86),(63,87,71,95,67,91),(64,92,68,96,72,88)], [(1,45),(2,27),(3,47),(4,29),(5,37),(6,31),(7,39),(8,33),(9,41),(10,35),(11,43),(12,25),(13,36),(14,40),(15,26),(16,42),(17,28),(18,44),(19,30),(20,46),(21,32),(22,48),(23,34),(24,38),(49,91),(50,68),(51,93),(52,70),(53,95),(54,72),(55,85),(56,62),(57,87),(58,64),(59,89),(60,66),(61,84),(63,74),(65,76),(67,78),(69,80),(71,82),(73,90),(75,92),(77,94),(79,96),(81,86),(83,88)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,53,7,59),(2,52,8,58),(3,51,9,57),(4,50,10,56),(5,49,11,55),(6,60,12,54),(13,84,19,78),(14,83,20,77),(15,82,21,76),(16,81,22,75),(17,80,23,74),(18,79,24,73),(25,72,31,66),(26,71,32,65),(27,70,33,64),(28,69,34,63),(29,68,35,62),(30,67,36,61),(37,91,43,85),(38,90,44,96),(39,89,45,95),(40,88,46,94),(41,87,47,93),(42,86,48,92)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H | 12I | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 18 | 18 | 36 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | 12 | 4 | ··· | 4 | 12 | ··· | 12 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | + | - | - | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | Q8 | D6 | D6 | D6 | C4○D4 | D12 | Dic6 | S32 | S3×D4 | D4⋊2S3 | S3×Q8 | C2×S32 | S3×Dic6 | S3×D12 | D6.4D6 |
| kernel | D6⋊2Dic6 | D6⋊Dic3 | Dic3⋊Dic3 | C3×Dic3⋊C4 | C3×D6⋊C4 | C12⋊Dic3 | C2×S3×Dic3 | C2×C32⋊2Q8 | Dic3⋊C4 | D6⋊C4 | C3×Dic3 | S3×C6 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | Dic3 | D6 | C2×C4 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of D6⋊2Dic6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 11 | 6 | 0 | 0 | 0 | 0 |
| 6 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 7 | 11 | 0 | 0 | 0 | 0 |
| 11 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 3 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 2 |
| 0 | 0 | 0 | 0 | 4 | 2 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,6,0,0,0,0,6,2,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[7,11,0,0,0,0,11,6,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,10,10,0,0,0,0,3,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,4,0,0,0,0,2,2] >;
D6⋊2Dic6 in GAP, Magma, Sage, TeX
D_6\rtimes_2{\rm Dic}_6 % in TeX
G:=Group("D6:2Dic6"); // GroupNames label
G:=SmallGroup(288,541);
// by ID
G=gap.SmallGroup(288,541);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,422,135,142,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^12=1,d^2=c^6,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations