metabelian, supersoluble, monomial
Aliases: D6⋊5Dic6, C62.21C23, (S3×C6)⋊5Q8, C6.20(S3×Q8), Dic3⋊C4⋊2S3, C6.135(S3×D4), C6.8(C2×Dic6), C3⋊7(D6⋊Q8), (C2×C12).258D6, (C2×Dic3).8D6, C2.10(S3×Dic6), C6.19(C4○D12), C32⋊3(C22⋊Q8), D6⋊Dic3.11C2, (C3×Dic3).33D4, (C22×S3).60D6, Dic3⋊Dic3⋊31C2, C3⋊1(C12.48D4), (C6×C12).212C22, C6.Dic6⋊12C2, C2.7(D6.D6), Dic3.14(C3⋊D4), (C6×Dic3).74C22, (C2×C4).40S32, (S3×C2×C4).9S3, (S3×C2×C12).2C2, C22.79(C2×S32), (C3×C6).79(C2×D4), C2.10(S3×C3⋊D4), C6.28(C2×C3⋊D4), (C3×C6).16(C2×Q8), (C3×Dic3⋊C4)⋊4C2, (C2×C32⋊2Q8)⋊1C2, (S3×C2×C6).70C22, (C3×C6).10(C4○D4), (C2×C6).40(C22×S3), (C2×C3⋊Dic3).21C22, SmallGroup(288,499)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊Dic6
G = < a,b,c,d | a6=b2=c12=1, d2=c6, bab=cac-1=dad-1=a-1, cbc-1=a4b, dbd-1=ab, 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, 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, C6.D4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×C12, C32⋊2Q8, S3×C12, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, D6⋊Q8, C12.48D4, D6⋊Dic3, Dic3⋊Dic3, C3×Dic3⋊C4, C6.Dic6, C2×C32⋊2Q8, S3×C2×C12, D6⋊Dic6
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, C4○D12, S3×D4, S3×Q8, C2×C3⋊D4, C2×S32, D6⋊Q8, C12.48D4, S3×Dic6, D6.D6, S3×C3⋊D4, D6⋊Dic6
►On 96 points
(1 96 9 92 5 88)(2 89 6 93 10 85)(3 86 11 94 7 90)(4 91 8 95 12 87)(13 29 21 25 17 33)(14 34 18 26 22 30)(15 31 23 27 19 35)(16 36 20 28 24 32)(37 52 45 60 41 56)(38 57 42 49 46 53)(39 54 47 50 43 58)(40 59 44 51 48 55)(61 75 65 79 69 83)(62 84 70 80 66 76)(63 77 67 81 71 73)(64 74 72 82 68 78)
(1 49)(2 54)(3 51)(4 56)(5 53)(6 58)(7 55)(8 60)(9 57)(10 50)(11 59)(12 52)(13 61)(14 66)(15 63)(16 68)(17 65)(18 70)(19 67)(20 72)(21 69)(22 62)(23 71)(24 64)(25 79)(26 84)(27 81)(28 74)(29 83)(30 76)(31 73)(32 78)(33 75)(34 80)(35 77)(36 82)(37 87)(38 92)(39 89)(40 94)(41 91)(42 96)(43 93)(44 86)(45 95)(46 88)(47 85)(48 90)
(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 23 7 17)(2 22 8 16)(3 21 9 15)(4 20 10 14)(5 19 11 13)(6 18 12 24)(25 96 31 90)(26 95 32 89)(27 94 33 88)(28 93 34 87)(29 92 35 86)(30 91 36 85)(37 64 43 70)(38 63 44 69)(39 62 45 68)(40 61 46 67)(41 72 47 66)(42 71 48 65)(49 81 55 75)(50 80 56 74)(51 79 57 73)(52 78 58 84)(53 77 59 83)(54 76 60 82)
G:=sub<Sym(96)| (1,96,9,92,5,88)(2,89,6,93,10,85)(3,86,11,94,7,90)(4,91,8,95,12,87)(13,29,21,25,17,33)(14,34,18,26,22,30)(15,31,23,27,19,35)(16,36,20,28,24,32)(37,52,45,60,41,56)(38,57,42,49,46,53)(39,54,47,50,43,58)(40,59,44,51,48,55)(61,75,65,79,69,83)(62,84,70,80,66,76)(63,77,67,81,71,73)(64,74,72,82,68,78), (1,49)(2,54)(3,51)(4,56)(5,53)(6,58)(7,55)(8,60)(9,57)(10,50)(11,59)(12,52)(13,61)(14,66)(15,63)(16,68)(17,65)(18,70)(19,67)(20,72)(21,69)(22,62)(23,71)(24,64)(25,79)(26,84)(27,81)(28,74)(29,83)(30,76)(31,73)(32,78)(33,75)(34,80)(35,77)(36,82)(37,87)(38,92)(39,89)(40,94)(41,91)(42,96)(43,93)(44,86)(45,95)(46,88)(47,85)(48,90), (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), (1,23,7,17)(2,22,8,16)(3,21,9,15)(4,20,10,14)(5,19,11,13)(6,18,12,24)(25,96,31,90)(26,95,32,89)(27,94,33,88)(28,93,34,87)(29,92,35,86)(30,91,36,85)(37,64,43,70)(38,63,44,69)(39,62,45,68)(40,61,46,67)(41,72,47,66)(42,71,48,65)(49,81,55,75)(50,80,56,74)(51,79,57,73)(52,78,58,84)(53,77,59,83)(54,76,60,82)>;
G:=Group( (1,96,9,92,5,88)(2,89,6,93,10,85)(3,86,11,94,7,90)(4,91,8,95,12,87)(13,29,21,25,17,33)(14,34,18,26,22,30)(15,31,23,27,19,35)(16,36,20,28,24,32)(37,52,45,60,41,56)(38,57,42,49,46,53)(39,54,47,50,43,58)(40,59,44,51,48,55)(61,75,65,79,69,83)(62,84,70,80,66,76)(63,77,67,81,71,73)(64,74,72,82,68,78), (1,49)(2,54)(3,51)(4,56)(5,53)(6,58)(7,55)(8,60)(9,57)(10,50)(11,59)(12,52)(13,61)(14,66)(15,63)(16,68)(17,65)(18,70)(19,67)(20,72)(21,69)(22,62)(23,71)(24,64)(25,79)(26,84)(27,81)(28,74)(29,83)(30,76)(31,73)(32,78)(33,75)(34,80)(35,77)(36,82)(37,87)(38,92)(39,89)(40,94)(41,91)(42,96)(43,93)(44,86)(45,95)(46,88)(47,85)(48,90), (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), (1,23,7,17)(2,22,8,16)(3,21,9,15)(4,20,10,14)(5,19,11,13)(6,18,12,24)(25,96,31,90)(26,95,32,89)(27,94,33,88)(28,93,34,87)(29,92,35,86)(30,91,36,85)(37,64,43,70)(38,63,44,69)(39,62,45,68)(40,61,46,67)(41,72,47,66)(42,71,48,65)(49,81,55,75)(50,80,56,74)(51,79,57,73)(52,78,58,84)(53,77,59,83)(54,76,60,82) );
G=PermutationGroup([[(1,96,9,92,5,88),(2,89,6,93,10,85),(3,86,11,94,7,90),(4,91,8,95,12,87),(13,29,21,25,17,33),(14,34,18,26,22,30),(15,31,23,27,19,35),(16,36,20,28,24,32),(37,52,45,60,41,56),(38,57,42,49,46,53),(39,54,47,50,43,58),(40,59,44,51,48,55),(61,75,65,79,69,83),(62,84,70,80,66,76),(63,77,67,81,71,73),(64,74,72,82,68,78)], [(1,49),(2,54),(3,51),(4,56),(5,53),(6,58),(7,55),(8,60),(9,57),(10,50),(11,59),(12,52),(13,61),(14,66),(15,63),(16,68),(17,65),(18,70),(19,67),(20,72),(21,69),(22,62),(23,71),(24,64),(25,79),(26,84),(27,81),(28,74),(29,83),(30,76),(31,73),(32,78),(33,75),(34,80),(35,77),(36,82),(37,87),(38,92),(39,89),(40,94),(41,91),(42,96),(43,93),(44,86),(45,95),(46,88),(47,85),(48,90)], [(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)], [(1,23,7,17),(2,22,8,16),(3,21,9,15),(4,20,10,14),(5,19,11,13),(6,18,12,24),(25,96,31,90),(26,95,32,89),(27,94,33,88),(28,93,34,87),(29,92,35,86),(30,91,36,85),(37,64,43,70),(38,63,44,69),(39,62,45,68),(40,61,46,67),(41,72,47,66),(42,71,48,65),(49,81,55,75),(50,80,56,74),(51,79,57,73),(52,78,58,84),(53,77,59,83),(54,76,60,82)]])
48 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 | 6L | 6M | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 12O | 12P | 12Q | 12R |
| 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 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 4 | 2 | 2 | 6 | 6 | 12 | 12 | 36 | 36 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 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 | S3 | D4 | Q8 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | Dic6 | C4○D12 | S32 | S3×D4 | S3×Q8 | C2×S32 | S3×Dic6 | D6.D6 | S3×C3⋊D4 |
| kernel | D6⋊Dic6 | D6⋊Dic3 | Dic3⋊Dic3 | C3×Dic3⋊C4 | C6.Dic6 | C2×C32⋊2Q8 | S3×C2×C12 | Dic3⋊C4 | S3×C2×C4 | C3×Dic3 | S3×C6 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | Dic3 | D6 | C6 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of D6⋊Dic6 ►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 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 2 | 9 | 0 | 0 | 0 | 0 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 7 | 3 | 0 | 0 | 0 | 0 |
| 10 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 3 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 9 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 2 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[2,4,0,0,0,0,9,11,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,0,1],[7,10,0,0,0,0,3,10,0,0,0,0,0,0,10,10,0,0,0,0,3,7,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[11,9,0,0,0,0,11,2,0,0,0,0,0,0,2,2,0,0,0,0,4,11,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
D6⋊Dic6 in GAP, Magma, Sage, TeX
D_6\rtimes {\rm Dic}_6 % in TeX
G:=Group("D6:Dic6"); // GroupNames label
G:=SmallGroup(288,499);
// by ID
G=gap.SmallGroup(288,499);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,219,142,1356,9414]);
// 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=d*a*d^-1=a^-1,c*b*c^-1=a^4*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations