metabelian, supersoluble, monomial
Aliases: D6⋊1Dic6, C62.57C23, (S3×C6)⋊1Q8, D6⋊C4.8S3, C6.26(S3×Q8), (C2×Dic6)⋊4S3, (C6×Dic6)⋊9C2, (C2×C12).20D6, C6.140(S3×D4), C3⋊3(D6⋊3Q8), (C3×Dic3).6D4, C6.14(C2×Dic6), C2.15(S3×Dic6), C6.9(D4⋊2S3), C32⋊8(C22⋊Q8), D6⋊Dic3.13C2, (C2×Dic3).68D6, (C22×S3).36D6, Dic3⋊Dic3⋊34C2, (C6×C12).182C22, C6.33(Q8⋊3S3), C6.Dic6⋊10C2, Dic3.8(C3⋊D4), C2.16(D12⋊S3), C3⋊6(Dic3.D4), (C6×Dic3).36C22, (C2×C4).23S32, (C3×D6⋊C4).7C2, (C3×C6).95(C2×D4), C6.34(C2×C3⋊D4), C2.14(S3×C3⋊D4), (C3×C6).27(C2×Q8), (C2×S3×Dic3).2C2, C22.104(C2×S32), (S3×C2×C6).18C22, (C3×C6).34(C4○D4), (C2×C6).76(C22×S3), (C2×C3⋊Dic3).41C22, SmallGroup(288,535)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊1Dic6
G = < a,b,c,d | a6=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >
Subgroups: 554 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, C3×Q8, C22×S3, C22×C6, C22⋊Q8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C6×Q8, S3×Dic3, C3×Dic6, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, Dic3.D4, D6⋊3Q8, D6⋊Dic3, Dic3⋊Dic3, C3×D6⋊C4, C6.Dic6, C2×S3×Dic3, C6×Dic6, D6⋊1Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C3⋊D4, C22×S3, C22⋊Q8, S32, C2×Dic6, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C2×C3⋊D4, C2×S32, Dic3.D4, D6⋊3Q8, S3×Dic6, D12⋊S3, S3×C3⋊D4, D6⋊1Dic6
►On 96 points
(1 92 9 88 5 96)(2 93 10 89 6 85)(3 94 11 90 7 86)(4 95 12 91 8 87)(13 60 17 52 21 56)(14 49 18 53 22 57)(15 50 19 54 23 58)(16 51 20 55 24 59)(25 70 33 66 29 62)(26 71 34 67 30 63)(27 72 35 68 31 64)(28 61 36 69 32 65)(37 76 41 80 45 84)(38 77 42 81 46 73)(39 78 43 82 47 74)(40 79 44 83 48 75)
(1 18)(2 58)(3 20)(4 60)(5 22)(6 50)(7 24)(8 52)(9 14)(10 54)(11 16)(12 56)(13 95)(15 85)(17 87)(19 89)(21 91)(23 93)(25 40)(26 84)(27 42)(28 74)(29 44)(30 76)(31 46)(32 78)(33 48)(34 80)(35 38)(36 82)(37 63)(39 65)(41 67)(43 69)(45 71)(47 61)(49 92)(51 94)(53 96)(55 86)(57 88)(59 90)(62 79)(64 81)(66 83)(68 73)(70 75)(72 77)
(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 84 7 78)(2 83 8 77)(3 82 9 76)(4 81 10 75)(5 80 11 74)(6 79 12 73)(13 31 19 25)(14 30 20 36)(15 29 21 35)(16 28 22 34)(17 27 23 33)(18 26 24 32)(37 86 43 92)(38 85 44 91)(39 96 45 90)(40 95 46 89)(41 94 47 88)(42 93 48 87)(49 63 55 69)(50 62 56 68)(51 61 57 67)(52 72 58 66)(53 71 59 65)(54 70 60 64)
G:=sub<Sym(96)| (1,92,9,88,5,96)(2,93,10,89,6,85)(3,94,11,90,7,86)(4,95,12,91,8,87)(13,60,17,52,21,56)(14,49,18,53,22,57)(15,50,19,54,23,58)(16,51,20,55,24,59)(25,70,33,66,29,62)(26,71,34,67,30,63)(27,72,35,68,31,64)(28,61,36,69,32,65)(37,76,41,80,45,84)(38,77,42,81,46,73)(39,78,43,82,47,74)(40,79,44,83,48,75), (1,18)(2,58)(3,20)(4,60)(5,22)(6,50)(7,24)(8,52)(9,14)(10,54)(11,16)(12,56)(13,95)(15,85)(17,87)(19,89)(21,91)(23,93)(25,40)(26,84)(27,42)(28,74)(29,44)(30,76)(31,46)(32,78)(33,48)(34,80)(35,38)(36,82)(37,63)(39,65)(41,67)(43,69)(45,71)(47,61)(49,92)(51,94)(53,96)(55,86)(57,88)(59,90)(62,79)(64,81)(66,83)(68,73)(70,75)(72,77), (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), (1,84,7,78)(2,83,8,77)(3,82,9,76)(4,81,10,75)(5,80,11,74)(6,79,12,73)(13,31,19,25)(14,30,20,36)(15,29,21,35)(16,28,22,34)(17,27,23,33)(18,26,24,32)(37,86,43,92)(38,85,44,91)(39,96,45,90)(40,95,46,89)(41,94,47,88)(42,93,48,87)(49,63,55,69)(50,62,56,68)(51,61,57,67)(52,72,58,66)(53,71,59,65)(54,70,60,64)>;
G:=Group( (1,92,9,88,5,96)(2,93,10,89,6,85)(3,94,11,90,7,86)(4,95,12,91,8,87)(13,60,17,52,21,56)(14,49,18,53,22,57)(15,50,19,54,23,58)(16,51,20,55,24,59)(25,70,33,66,29,62)(26,71,34,67,30,63)(27,72,35,68,31,64)(28,61,36,69,32,65)(37,76,41,80,45,84)(38,77,42,81,46,73)(39,78,43,82,47,74)(40,79,44,83,48,75), (1,18)(2,58)(3,20)(4,60)(5,22)(6,50)(7,24)(8,52)(9,14)(10,54)(11,16)(12,56)(13,95)(15,85)(17,87)(19,89)(21,91)(23,93)(25,40)(26,84)(27,42)(28,74)(29,44)(30,76)(31,46)(32,78)(33,48)(34,80)(35,38)(36,82)(37,63)(39,65)(41,67)(43,69)(45,71)(47,61)(49,92)(51,94)(53,96)(55,86)(57,88)(59,90)(62,79)(64,81)(66,83)(68,73)(70,75)(72,77), (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), (1,84,7,78)(2,83,8,77)(3,82,9,76)(4,81,10,75)(5,80,11,74)(6,79,12,73)(13,31,19,25)(14,30,20,36)(15,29,21,35)(16,28,22,34)(17,27,23,33)(18,26,24,32)(37,86,43,92)(38,85,44,91)(39,96,45,90)(40,95,46,89)(41,94,47,88)(42,93,48,87)(49,63,55,69)(50,62,56,68)(51,61,57,67)(52,72,58,66)(53,71,59,65)(54,70,60,64) );
G=PermutationGroup([[(1,92,9,88,5,96),(2,93,10,89,6,85),(3,94,11,90,7,86),(4,95,12,91,8,87),(13,60,17,52,21,56),(14,49,18,53,22,57),(15,50,19,54,23,58),(16,51,20,55,24,59),(25,70,33,66,29,62),(26,71,34,67,30,63),(27,72,35,68,31,64),(28,61,36,69,32,65),(37,76,41,80,45,84),(38,77,42,81,46,73),(39,78,43,82,47,74),(40,79,44,83,48,75)], [(1,18),(2,58),(3,20),(4,60),(5,22),(6,50),(7,24),(8,52),(9,14),(10,54),(11,16),(12,56),(13,95),(15,85),(17,87),(19,89),(21,91),(23,93),(25,40),(26,84),(27,42),(28,74),(29,44),(30,76),(31,46),(32,78),(33,48),(34,80),(35,38),(36,82),(37,63),(39,65),(41,67),(43,69),(45,71),(47,61),(49,92),(51,94),(53,96),(55,86),(57,88),(59,90),(62,79),(64,81),(66,83),(68,73),(70,75),(72,77)], [(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)], [(1,84,7,78),(2,83,8,77),(3,82,9,76),(4,81,10,75),(5,80,11,74),(6,79,12,73),(13,31,19,25),(14,30,20,36),(15,29,21,35),(16,28,22,34),(17,27,23,33),(18,26,24,32),(37,86,43,92),(38,85,44,91),(39,96,45,90),(40,95,46,89),(41,94,47,88),(42,93,48,87),(49,63,55,69),(50,62,56,68),(51,61,57,67),(52,72,58,66),(53,71,59,65),(54,70,60,64)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | + | - | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | Q8 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | Dic6 | S32 | S3×D4 | D4⋊2S3 | S3×Q8 | Q8⋊3S3 | C2×S32 | S3×Dic6 | D12⋊S3 | S3×C3⋊D4 |
| kernel | D6⋊1Dic6 | D6⋊Dic3 | Dic3⋊Dic3 | C3×D6⋊C4 | C6.Dic6 | C2×S3×Dic3 | C6×Dic6 | D6⋊C4 | C2×Dic6 | C3×Dic3 | S3×C6 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | Dic3 | D6 | C2×C4 | C6 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of D6⋊1Dic6 ►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 |
| 2 | 9 | 0 | 0 | 0 | 0 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 10 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 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 | 5 | 0 |
| 0 | 0 | 0 | 0 | 5 | 8 |
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],[2,4,0,0,0,0,9,11,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,6,3,0,0,0,0,10,3],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,5,0,0,0,0,0,8] >;
D6⋊1Dic6 in GAP, Magma, Sage, TeX
D_6\rtimes_1{\rm Dic}_6 % in TeX
G:=Group("D6:1Dic6"); // GroupNames label
G:=SmallGroup(288,535);
// by ID
G=gap.SmallGroup(288,535);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,422,135,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^12=1,d^2=c^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations