metabelian, supersoluble, monomial
Aliases: D6⋊3Dic6, C62.66C23, (S3×C6)⋊3Q8, D6⋊C4.4S3, C6.50(S3×D4), C6.29(S3×Q8), Dic3⋊C4⋊6S3, (C2×C12).24D6, C3⋊3(D6⋊Q8), C3⋊Dic3.16D4, (C6×C12).6C22, C6.17(C2×Dic6), C2.17(S3×Dic6), C6.47(C4○D12), C2.9(Dic3⋊D6), D6⋊Dic3.14C2, (C2×Dic3).27D6, (C22×S3).40D6, Dic3⋊Dic3⋊17C2, C6.25(D4⋊2S3), C32⋊12(C22⋊Q8), C62.C22⋊10C2, C2.16(D12⋊5S3), C3⋊2(Dic3.D4), (C6×Dic3).81C22, (C2×C4).29S32, (C3×D6⋊C4).2C2, (C3×C6).97(C2×D4), (C3×C6).32(C2×Q8), (C2×S3×Dic3).8C2, C22.110(C2×S32), (S3×C2×C6).23C22, (C2×C32⋊4Q8)⋊1C2, (C3×Dic3⋊C4)⋊19C2, (C3×C6).39(C4○D4), (C2×C6).85(C22×S3), (C2×C3⋊Dic3).49C22, SmallGroup(288,544)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊3Dic6
G = < a,b,c,d | a6=b2=c12=1, d2=c6, bab=dad-1=a-1, ac=ca, cbc-1=a3b, dbd-1=a4b, dcd-1=c-1 >
Subgroups: 610 in 167 conjugacy classes, 50 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, Q8, C23, C32, 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, C6×Dic3, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, S3×C2×C6, Dic3.D4, D6⋊Q8, D6⋊Dic3, Dic3⋊Dic3, C62.C22, C3×Dic3⋊C4, C3×D6⋊C4, C2×S3×Dic3, C2×C32⋊4Q8, D6⋊3Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C22×S3, C22⋊Q8, S32, C2×Dic6, C4○D12, S3×D4, D4⋊2S3, S3×Q8, C2×S32, Dic3.D4, D6⋊Q8, S3×Dic6, D12⋊5S3, Dic3⋊D6, D6⋊3Dic6
►On 96 points
(1 80 9 76 5 84)(2 81 10 77 6 73)(3 82 11 78 7 74)(4 83 12 79 8 75)(13 72 17 64 21 68)(14 61 18 65 22 69)(15 62 19 66 23 70)(16 63 20 67 24 71)(25 58 29 50 33 54)(26 59 30 51 34 55)(27 60 31 52 35 56)(28 49 32 53 36 57)(37 89 45 85 41 93)(38 90 46 86 42 94)(39 91 47 87 43 95)(40 92 48 88 44 96)
(1 13)(2 65)(3 15)(4 67)(5 17)(6 69)(7 19)(8 71)(9 21)(10 61)(11 23)(12 63)(14 77)(16 79)(18 81)(20 83)(22 73)(24 75)(25 43)(26 92)(27 45)(28 94)(29 47)(30 96)(31 37)(32 86)(33 39)(34 88)(35 41)(36 90)(38 57)(40 59)(42 49)(44 51)(46 53)(48 55)(50 91)(52 93)(54 95)(56 85)(58 87)(60 89)(62 74)(64 76)(66 78)(68 80)(70 82)(72 84)
(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 43 7 37)(2 42 8 48)(3 41 9 47)(4 40 10 46)(5 39 11 45)(6 38 12 44)(13 29 19 35)(14 28 20 34)(15 27 21 33)(16 26 22 32)(17 25 23 31)(18 36 24 30)(49 63 55 69)(50 62 56 68)(51 61 57 67)(52 72 58 66)(53 71 59 65)(54 70 60 64)(73 94 79 88)(74 93 80 87)(75 92 81 86)(76 91 82 85)(77 90 83 96)(78 89 84 95)
G:=sub<Sym(96)| (1,80,9,76,5,84)(2,81,10,77,6,73)(3,82,11,78,7,74)(4,83,12,79,8,75)(13,72,17,64,21,68)(14,61,18,65,22,69)(15,62,19,66,23,70)(16,63,20,67,24,71)(25,58,29,50,33,54)(26,59,30,51,34,55)(27,60,31,52,35,56)(28,49,32,53,36,57)(37,89,45,85,41,93)(38,90,46,86,42,94)(39,91,47,87,43,95)(40,92,48,88,44,96), (1,13)(2,65)(3,15)(4,67)(5,17)(6,69)(7,19)(8,71)(9,21)(10,61)(11,23)(12,63)(14,77)(16,79)(18,81)(20,83)(22,73)(24,75)(25,43)(26,92)(27,45)(28,94)(29,47)(30,96)(31,37)(32,86)(33,39)(34,88)(35,41)(36,90)(38,57)(40,59)(42,49)(44,51)(46,53)(48,55)(50,91)(52,93)(54,95)(56,85)(58,87)(60,89)(62,74)(64,76)(66,78)(68,80)(70,82)(72,84), (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), (1,43,7,37)(2,42,8,48)(3,41,9,47)(4,40,10,46)(5,39,11,45)(6,38,12,44)(13,29,19,35)(14,28,20,34)(15,27,21,33)(16,26,22,32)(17,25,23,31)(18,36,24,30)(49,63,55,69)(50,62,56,68)(51,61,57,67)(52,72,58,66)(53,71,59,65)(54,70,60,64)(73,94,79,88)(74,93,80,87)(75,92,81,86)(76,91,82,85)(77,90,83,96)(78,89,84,95)>;
G:=Group( (1,80,9,76,5,84)(2,81,10,77,6,73)(3,82,11,78,7,74)(4,83,12,79,8,75)(13,72,17,64,21,68)(14,61,18,65,22,69)(15,62,19,66,23,70)(16,63,20,67,24,71)(25,58,29,50,33,54)(26,59,30,51,34,55)(27,60,31,52,35,56)(28,49,32,53,36,57)(37,89,45,85,41,93)(38,90,46,86,42,94)(39,91,47,87,43,95)(40,92,48,88,44,96), (1,13)(2,65)(3,15)(4,67)(5,17)(6,69)(7,19)(8,71)(9,21)(10,61)(11,23)(12,63)(14,77)(16,79)(18,81)(20,83)(22,73)(24,75)(25,43)(26,92)(27,45)(28,94)(29,47)(30,96)(31,37)(32,86)(33,39)(34,88)(35,41)(36,90)(38,57)(40,59)(42,49)(44,51)(46,53)(48,55)(50,91)(52,93)(54,95)(56,85)(58,87)(60,89)(62,74)(64,76)(66,78)(68,80)(70,82)(72,84), (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), (1,43,7,37)(2,42,8,48)(3,41,9,47)(4,40,10,46)(5,39,11,45)(6,38,12,44)(13,29,19,35)(14,28,20,34)(15,27,21,33)(16,26,22,32)(17,25,23,31)(18,36,24,30)(49,63,55,69)(50,62,56,68)(51,61,57,67)(52,72,58,66)(53,71,59,65)(54,70,60,64)(73,94,79,88)(74,93,80,87)(75,92,81,86)(76,91,82,85)(77,90,83,96)(78,89,84,95) );
G=PermutationGroup([[(1,80,9,76,5,84),(2,81,10,77,6,73),(3,82,11,78,7,74),(4,83,12,79,8,75),(13,72,17,64,21,68),(14,61,18,65,22,69),(15,62,19,66,23,70),(16,63,20,67,24,71),(25,58,29,50,33,54),(26,59,30,51,34,55),(27,60,31,52,35,56),(28,49,32,53,36,57),(37,89,45,85,41,93),(38,90,46,86,42,94),(39,91,47,87,43,95),(40,92,48,88,44,96)], [(1,13),(2,65),(3,15),(4,67),(5,17),(6,69),(7,19),(8,71),(9,21),(10,61),(11,23),(12,63),(14,77),(16,79),(18,81),(20,83),(22,73),(24,75),(25,43),(26,92),(27,45),(28,94),(29,47),(30,96),(31,37),(32,86),(33,39),(34,88),(35,41),(36,90),(38,57),(40,59),(42,49),(44,51),(46,53),(48,55),(50,91),(52,93),(54,95),(56,85),(58,87),(60,89),(62,74),(64,76),(66,78),(68,80),(70,82),(72,84)], [(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)], [(1,43,7,37),(2,42,8,48),(3,41,9,47),(4,40,10,46),(5,39,11,45),(6,38,12,44),(13,29,19,35),(14,28,20,34),(15,27,21,33),(16,26,22,32),(17,25,23,31),(18,36,24,30),(49,63,55,69),(50,62,56,68),(51,61,57,67),(52,72,58,66),(53,71,59,65),(54,70,60,64),(73,94,79,88),(74,93,80,87),(75,92,81,86),(76,91,82,85),(77,90,83,96),(78,89,84,95)]])
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 | Dic6 | C4○D12 | S32 | S3×D4 | D4⋊2S3 | S3×Q8 | C2×S32 | S3×Dic6 | D12⋊5S3 | Dic3⋊D6 |
| kernel | D6⋊3Dic6 | D6⋊Dic3 | Dic3⋊Dic3 | C62.C22 | C3×Dic3⋊C4 | C3×D6⋊C4 | C2×S3×Dic3 | C2×C32⋊4Q8 | Dic3⋊C4 | D6⋊C4 | C3⋊Dic3 | S3×C6 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | D6 | C6 | 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 | 2 | 1 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of D6⋊3Dic6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 11 | 9 | 0 | 0 | 0 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 |
| 0 | 0 | 0 | 0 | 4 | 11 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[11,4,0,0,0,0,9,2,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,7,10,0,0,0,0,3,6],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,10,7,0,0,0,0,6,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,4,0,0,0,0,2,11] >;
D6⋊3Dic6 in GAP, Magma, Sage, TeX
D_6\rtimes_3{\rm Dic}_6 % in TeX
G:=Group("D6:3Dic6"); // GroupNames label
G:=SmallGroup(288,544);
// by ID
G=gap.SmallGroup(288,544);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,219,58,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^12=1,d^2=c^6,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations