metabelian, supersoluble, monomial
Aliases: D6.9D12, C62.61C23, D6⋊C4⋊4S3, (S3×C6).7D4, C6.15(S3×D4), C6.16(C2×D12), C2.19(S3×D12), (C2×C12).21D6, D6⋊Dic3⋊10C2, (C6×C12).4C22, C6.45(C4○D12), C12⋊Dic3⋊3C2, C3⋊3(C23.9D6), (C2×Dic3).70D6, (C22×S3).11D6, Dic3⋊Dic3⋊15C2, C6.23(D4⋊2S3), C2.15(D12⋊5S3), C2.10(D6.4D6), C3⋊2(C23.21D6), (C6×Dic3).63C22, C32⋊7(C22.D4), (C2×C4).26S32, (C3×D6⋊C4)⋊4C2, (C2×S3×Dic3)⋊13C2, (C3×C6).48(C2×D4), C22.107(C2×S32), (S3×C2×C6).20C22, (C2×D6⋊S3).3C2, (C3×C6).65(C4○D4), (C2×C6).80(C22×S3), (C2×C3⋊Dic3).45C22, SmallGroup(288,539)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.9D12
G = < a,b,c,d | a6=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a3b, dcd=a3c-1 >
Subgroups: 650 in 169 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22.D4, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, S3×Dic3, D6⋊S3, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C23.9D6, C23.21D6, D6⋊Dic3, Dic3⋊Dic3, C3×D6⋊C4, C12⋊Dic3, C2×S3×Dic3, C2×D6⋊S3, D6.9D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C22.D4, S32, C2×D12, C4○D12, S3×D4, D4⋊2S3, C2×S32, C23.9D6, C23.21D6, D12⋊5S3, S3×D12, D6.4D6, D6.9D12
►On 96 points
Generators in S96
(1 25 5 29 9 33)(2 26 6 30 10 34)(3 27 7 31 11 35)(4 28 8 32 12 36)(13 79 21 75 17 83)(14 80 22 76 18 84)(15 81 23 77 19 73)(16 82 24 78 20 74)(37 50 45 58 41 54)(38 51 46 59 42 55)(39 52 47 60 43 56)(40 53 48 49 44 57)(61 85 65 89 69 93)(62 86 66 90 70 94)(63 87 67 91 71 95)(64 88 68 92 72 96)
(1 14)(2 77)(3 16)(4 79)(5 18)(6 81)(7 20)(8 83)(9 22)(10 73)(11 24)(12 75)(13 28)(15 30)(17 32)(19 34)(21 36)(23 26)(25 84)(27 74)(29 76)(31 78)(33 80)(35 82)(37 93)(38 66)(39 95)(40 68)(41 85)(42 70)(43 87)(44 72)(45 89)(46 62)(47 91)(48 64)(49 96)(50 69)(51 86)(52 71)(53 88)(54 61)(55 90)(56 63)(57 92)(58 65)(59 94)(60 67)
(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 51)(2 41)(3 49)(4 39)(5 59)(6 37)(7 57)(8 47)(9 55)(10 45)(11 53)(12 43)(13 87)(14 70)(15 85)(16 68)(17 95)(18 66)(19 93)(20 64)(21 91)(22 62)(23 89)(24 72)(25 46)(26 54)(27 44)(28 52)(29 42)(30 50)(31 40)(32 60)(33 38)(34 58)(35 48)(36 56)(61 73)(63 83)(65 81)(67 79)(69 77)(71 75)(74 88)(76 86)(78 96)(80 94)(82 92)(84 90)
G:=sub<Sym(96)| (1,25,5,29,9,33)(2,26,6,30,10,34)(3,27,7,31,11,35)(4,28,8,32,12,36)(13,79,21,75,17,83)(14,80,22,76,18,84)(15,81,23,77,19,73)(16,82,24,78,20,74)(37,50,45,58,41,54)(38,51,46,59,42,55)(39,52,47,60,43,56)(40,53,48,49,44,57)(61,85,65,89,69,93)(62,86,66,90,70,94)(63,87,67,91,71,95)(64,88,68,92,72,96), (1,14)(2,77)(3,16)(4,79)(5,18)(6,81)(7,20)(8,83)(9,22)(10,73)(11,24)(12,75)(13,28)(15,30)(17,32)(19,34)(21,36)(23,26)(25,84)(27,74)(29,76)(31,78)(33,80)(35,82)(37,93)(38,66)(39,95)(40,68)(41,85)(42,70)(43,87)(44,72)(45,89)(46,62)(47,91)(48,64)(49,96)(50,69)(51,86)(52,71)(53,88)(54,61)(55,90)(56,63)(57,92)(58,65)(59,94)(60,67), (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), (1,51)(2,41)(3,49)(4,39)(5,59)(6,37)(7,57)(8,47)(9,55)(10,45)(11,53)(12,43)(13,87)(14,70)(15,85)(16,68)(17,95)(18,66)(19,93)(20,64)(21,91)(22,62)(23,89)(24,72)(25,46)(26,54)(27,44)(28,52)(29,42)(30,50)(31,40)(32,60)(33,38)(34,58)(35,48)(36,56)(61,73)(63,83)(65,81)(67,79)(69,77)(71,75)(74,88)(76,86)(78,96)(80,94)(82,92)(84,90)>;
G:=Group( (1,25,5,29,9,33)(2,26,6,30,10,34)(3,27,7,31,11,35)(4,28,8,32,12,36)(13,79,21,75,17,83)(14,80,22,76,18,84)(15,81,23,77,19,73)(16,82,24,78,20,74)(37,50,45,58,41,54)(38,51,46,59,42,55)(39,52,47,60,43,56)(40,53,48,49,44,57)(61,85,65,89,69,93)(62,86,66,90,70,94)(63,87,67,91,71,95)(64,88,68,92,72,96), (1,14)(2,77)(3,16)(4,79)(5,18)(6,81)(7,20)(8,83)(9,22)(10,73)(11,24)(12,75)(13,28)(15,30)(17,32)(19,34)(21,36)(23,26)(25,84)(27,74)(29,76)(31,78)(33,80)(35,82)(37,93)(38,66)(39,95)(40,68)(41,85)(42,70)(43,87)(44,72)(45,89)(46,62)(47,91)(48,64)(49,96)(50,69)(51,86)(52,71)(53,88)(54,61)(55,90)(56,63)(57,92)(58,65)(59,94)(60,67), (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), (1,51)(2,41)(3,49)(4,39)(5,59)(6,37)(7,57)(8,47)(9,55)(10,45)(11,53)(12,43)(13,87)(14,70)(15,85)(16,68)(17,95)(18,66)(19,93)(20,64)(21,91)(22,62)(23,89)(24,72)(25,46)(26,54)(27,44)(28,52)(29,42)(30,50)(31,40)(32,60)(33,38)(34,58)(35,48)(36,56)(61,73)(63,83)(65,81)(67,79)(69,77)(71,75)(74,88)(76,86)(78,96)(80,94)(82,92)(84,90) );
G=PermutationGroup([[(1,25,5,29,9,33),(2,26,6,30,10,34),(3,27,7,31,11,35),(4,28,8,32,12,36),(13,79,21,75,17,83),(14,80,22,76,18,84),(15,81,23,77,19,73),(16,82,24,78,20,74),(37,50,45,58,41,54),(38,51,46,59,42,55),(39,52,47,60,43,56),(40,53,48,49,44,57),(61,85,65,89,69,93),(62,86,66,90,70,94),(63,87,67,91,71,95),(64,88,68,92,72,96)], [(1,14),(2,77),(3,16),(4,79),(5,18),(6,81),(7,20),(8,83),(9,22),(10,73),(11,24),(12,75),(13,28),(15,30),(17,32),(19,34),(21,36),(23,26),(25,84),(27,74),(29,76),(31,78),(33,80),(35,82),(37,93),(38,66),(39,95),(40,68),(41,85),(42,70),(43,87),(44,72),(45,89),(46,62),(47,91),(48,64),(49,96),(50,69),(51,86),(52,71),(53,88),(54,61),(55,90),(56,63),(57,92),(58,65),(59,94),(60,67)], [(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)], [(1,51),(2,41),(3,49),(4,39),(5,59),(6,37),(7,57),(8,47),(9,55),(10,45),(11,53),(12,43),(13,87),(14,70),(15,85),(16,68),(17,95),(18,66),(19,93),(20,64),(21,91),(22,62),(23,89),(24,72),(25,46),(26,54),(27,44),(28,52),(29,42),(30,50),(31,40),(32,60),(33,38),(34,58),(35,48),(36,56),(61,73),(63,83),(65,81),(67,79),(69,77),(71,75),(74,88),(76,86),(78,96),(80,94),(82,92),(84,90)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 12A | ··· | 12H | 12I | 12J | 12K | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 18 | 18 | 36 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 12 | 12 | 12 | 12 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | D12 | C4○D12 | S32 | S3×D4 | D4⋊2S3 | C2×S32 | D12⋊5S3 | S3×D12 | D6.4D6 |
| kernel | D6.9D12 | D6⋊Dic3 | Dic3⋊Dic3 | C3×D6⋊C4 | C12⋊Dic3 | C2×S3×Dic3 | C2×D6⋊S3 | D6⋊C4 | S3×C6 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | D6 | C6 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 3 | 1 | 2 | 2 | 2 |
Matrix representation of D6.9D12 ►in GL6(𝔽13)
| 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 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 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 | 3 | 7 |
| 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 9 |
| 0 | 0 | 0 | 0 | 4 | 11 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,10,0,0,0,0,7,10],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,10,7,0,0,0,0,6,3],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,2,4,0,0,0,0,9,11] >;
D6.9D12 in GAP, Magma, Sage, TeX
D_6._9D_{12} % in TeX
G:=Group("D6.9D12"); // GroupNames label
G:=SmallGroup(288,539);
// by ID
G=gap.SmallGroup(288,539);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,254,219,142,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^12=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^3*b,d*c*d=a^3*c^-1>;
// generators/relations