metabelian, supersoluble, monomial
Aliases: D6⋊7Dic6, C12.76D12, C62.27C23, (S3×C6)⋊7Q8, C4⋊Dic3⋊4S3, C6.22(S3×Q8), (C3×C12).74D4, C6.74(C2×D12), C3⋊5(C4.D12), (C2×C12).277D6, D6⋊Dic3.3C2, C2.12(S3×Dic6), C6.10(C2×Dic6), Dic3⋊Dic3⋊2C2, C6.39(C4○D12), C32⋊5(C22⋊Q8), C12.52(C3⋊D4), (C6×C12).91C22, (C2×Dic3).92D6, (C22×S3).63D6, C2.7(D12⋊5S3), C4.18(C3⋊D12), C6.15(D4⋊2S3), C3⋊2(C12.48D4), (C6×Dic3).6C22, (C2×C4).73S32, (S3×C2×C4).4S3, (S3×C2×C12).7C2, (C3×C4⋊Dic3)⋊8C2, C22.84(C2×S32), (C3×C6).82(C2×D4), C6.10(C2×C3⋊D4), (C3×C6).18(C2×Q8), (S3×C2×C6).73C22, (C3×C6).15(C4○D4), C2.14(C2×C3⋊D12), (C2×C6).46(C22×S3), (C2×C32⋊4Q8)⋊11C2, (C2×C3⋊Dic3).25C22, SmallGroup(288,505)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊7Dic6
G = < a,b,c,d | a6=b2=c12=1, d2=c6, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Subgroups: 578 in 167 conjugacy classes, 56 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, Q8, C23, C32, Dic3, C12, 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×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊Q8, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×C12, S3×C12, C6×Dic3, C6×Dic3, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C4.D12, C12.48D4, D6⋊Dic3, Dic3⋊Dic3, C3×C4⋊Dic3, S3×C2×C12, C2×C32⋊4Q8, D6⋊7Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, D12, C3⋊D4, C22×S3, C22⋊Q8, S32, C2×Dic6, C2×D12, C4○D12, D4⋊2S3, S3×Q8, C2×C3⋊D4, C3⋊D12, C2×S32, C4.D12, C12.48D4, S3×Dic6, D12⋊5S3, C2×C3⋊D12, D6⋊7Dic6
►On 96 points
(1 91 9 87 5 95)(2 92 10 88 6 96)(3 93 11 89 7 85)(4 94 12 90 8 86)(13 67 21 63 17 71)(14 68 22 64 18 72)(15 69 23 65 19 61)(16 70 24 66 20 62)(25 80 29 84 33 76)(26 81 30 73 34 77)(27 82 31 74 35 78)(28 83 32 75 36 79)(37 52 41 56 45 60)(38 53 42 57 46 49)(39 54 43 58 47 50)(40 55 44 59 48 51)
(1 80)(2 81)(3 82)(4 83)(5 84)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 37)(24 38)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 61)(57 62)(58 63)(59 64)(60 65)
(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 69 7 63)(2 68 8 62)(3 67 9 61)(4 66 10 72)(5 65 11 71)(6 64 12 70)(13 87 19 93)(14 86 20 92)(15 85 21 91)(16 96 22 90)(17 95 23 89)(18 94 24 88)(25 52 31 58)(26 51 32 57)(27 50 33 56)(28 49 34 55)(29 60 35 54)(30 59 36 53)(37 74 43 80)(38 73 44 79)(39 84 45 78)(40 83 46 77)(41 82 47 76)(42 81 48 75)
G:=sub<Sym(96)| (1,91,9,87,5,95)(2,92,10,88,6,96)(3,93,11,89,7,85)(4,94,12,90,8,86)(13,67,21,63,17,71)(14,68,22,64,18,72)(15,69,23,65,19,61)(16,70,24,66,20,62)(25,80,29,84,33,76)(26,81,30,73,34,77)(27,82,31,74,35,78)(28,83,32,75,36,79)(37,52,41,56,45,60)(38,53,42,57,46,49)(39,54,43,58,47,50)(40,55,44,59,48,51), (1,80)(2,81)(3,82)(4,83)(5,84)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,37)(24,38)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,61)(57,62)(58,63)(59,64)(60,65), (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), (1,69,7,63)(2,68,8,62)(3,67,9,61)(4,66,10,72)(5,65,11,71)(6,64,12,70)(13,87,19,93)(14,86,20,92)(15,85,21,91)(16,96,22,90)(17,95,23,89)(18,94,24,88)(25,52,31,58)(26,51,32,57)(27,50,33,56)(28,49,34,55)(29,60,35,54)(30,59,36,53)(37,74,43,80)(38,73,44,79)(39,84,45,78)(40,83,46,77)(41,82,47,76)(42,81,48,75)>;
G:=Group( (1,91,9,87,5,95)(2,92,10,88,6,96)(3,93,11,89,7,85)(4,94,12,90,8,86)(13,67,21,63,17,71)(14,68,22,64,18,72)(15,69,23,65,19,61)(16,70,24,66,20,62)(25,80,29,84,33,76)(26,81,30,73,34,77)(27,82,31,74,35,78)(28,83,32,75,36,79)(37,52,41,56,45,60)(38,53,42,57,46,49)(39,54,43,58,47,50)(40,55,44,59,48,51), (1,80)(2,81)(3,82)(4,83)(5,84)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,37)(24,38)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,61)(57,62)(58,63)(59,64)(60,65), (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), (1,69,7,63)(2,68,8,62)(3,67,9,61)(4,66,10,72)(5,65,11,71)(6,64,12,70)(13,87,19,93)(14,86,20,92)(15,85,21,91)(16,96,22,90)(17,95,23,89)(18,94,24,88)(25,52,31,58)(26,51,32,57)(27,50,33,56)(28,49,34,55)(29,60,35,54)(30,59,36,53)(37,74,43,80)(38,73,44,79)(39,84,45,78)(40,83,46,77)(41,82,47,76)(42,81,48,75) );
G=PermutationGroup([[(1,91,9,87,5,95),(2,92,10,88,6,96),(3,93,11,89,7,85),(4,94,12,90,8,86),(13,67,21,63,17,71),(14,68,22,64,18,72),(15,69,23,65,19,61),(16,70,24,66,20,62),(25,80,29,84,33,76),(26,81,30,73,34,77),(27,82,31,74,35,78),(28,83,32,75,36,79),(37,52,41,56,45,60),(38,53,42,57,46,49),(39,54,43,58,47,50),(40,55,44,59,48,51)], [(1,80),(2,81),(3,82),(4,83),(5,84),(6,73),(7,74),(8,75),(9,76),(10,77),(11,78),(12,79),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,37),(24,38),(25,91),(26,92),(27,93),(28,94),(29,95),(30,96),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(49,66),(50,67),(51,68),(52,69),(53,70),(54,71),(55,72),(56,61),(57,62),(58,63),(59,64),(60,65)], [(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)], [(1,69,7,63),(2,68,8,62),(3,67,9,61),(4,66,10,72),(5,65,11,71),(6,64,12,70),(13,87,19,93),(14,86,20,92),(15,85,21,91),(16,96,22,90),(17,95,23,89),(18,94,24,88),(25,52,31,58),(26,51,32,57),(27,50,33,56),(28,49,34,55),(29,60,35,54),(30,59,36,53),(37,74,43,80),(38,73,44,79),(39,84,45,78),(40,83,46,77),(41,82,47,76),(42,81,48,75)]])
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 | 2 | 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 | S3 | S3 | D4 | Q8 | D6 | D6 | D6 | C4○D4 | D12 | C3⋊D4 | Dic6 | C4○D12 | S32 | D4⋊2S3 | S3×Q8 | C3⋊D12 | C2×S32 | S3×Dic6 | D12⋊5S3 |
| kernel | D6⋊7Dic6 | D6⋊Dic3 | Dic3⋊Dic3 | C3×C4⋊Dic3 | S3×C2×C12 | C2×C32⋊4Q8 | C4⋊Dic3 | S3×C2×C4 | C3×C12 | S3×C6 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | C12 | C12 | D6 | C6 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 |
Matrix representation of D6⋊7Dic6 ►in GL6(𝔽13)
| 12 | 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 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 2 | 9 | 0 | 0 | 0 | 0 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 3 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[2,4,0,0,0,0,9,11,0,0,0,0,0,0,2,9,0,0,0,0,4,11,0,0,0,0,0,0,12,12,0,0,0,0,0,1],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,10,10,0,0,0,0,3,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,8,5,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
D6⋊7Dic6 in GAP, Magma, Sage, TeX
D_6\rtimes_7{\rm Dic}_6 % in TeX
G:=Group("D6:7Dic6"); // GroupNames label
G:=SmallGroup(288,505);
// by ID
G=gap.SmallGroup(288,505);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,176,219,100,1356,9414]);
// 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,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations