metabelian, supersoluble, monomial
Aliases: D6⋊4Dic6, C62.69C23, (S3×C6)⋊4Q8, D6⋊C4.9S3, C4⋊Dic3⋊8S3, C6.38(S3×D4), C6.30(S3×Q8), (C2×C12).26D6, C3⋊4(D6⋊Q8), D6⋊Dic3.7C2, C3⋊Dic3.17D4, C6.18(C2×Dic6), C2.18(S3×Dic6), C6.56(C4○D12), (C2×Dic3).29D6, C62.C22⋊8C2, (C22×S3).42D6, C2.14(D6⋊D6), C32⋊13(C22⋊Q8), C6.43(D4⋊2S3), (C6×C12).186C22, C6.Dic6⋊11C2, C3⋊3(Dic3.D4), C2.19(D6.3D6), (C6×Dic3).14C22, (C2×C4).31S32, (C3×D6⋊C4).8C2, (C3×C6).54(C2×D4), (C3×C6).33(C2×Q8), (C2×S3×Dic3).9C2, (C3×C4⋊Dic3)⋊16C2, C22.112(C2×S32), (C2×C32⋊2Q8)⋊4C2, (S3×C2×C6).25C22, (C3×C6).68(C4○D4), (C2×C6).88(C22×S3), (C2×C3⋊Dic3).50C22, SmallGroup(288,547)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊4Dic6
G = < a,b,c,d | a6=b2=c12=1, d2=c6, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a4b, dcd-1=c-1 >
Subgroups: 586 in 161 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, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, S3×Dic3, C32⋊2Q8, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, Dic3.D4, D6⋊Q8, D6⋊Dic3, C62.C22, C3×C4⋊Dic3, C3×D6⋊C4, C6.Dic6, C2×S3×Dic3, C2×C32⋊2Q8, D6⋊4Dic6
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, D6⋊D6, D6.3D6, D6⋊4Dic6
►On 96 points
(1 90 9 86 5 94)(2 95 6 87 10 91)(3 92 11 88 7 96)(4 85 8 89 12 93)(13 50 21 58 17 54)(14 55 18 59 22 51)(15 52 23 60 19 56)(16 57 20 49 24 53)(25 46 29 38 33 42)(26 43 34 39 30 47)(27 48 31 40 35 44)(28 45 36 41 32 37)(61 82 69 78 65 74)(62 75 66 79 70 83)(63 84 71 80 67 76)(64 77 68 81 72 73)
(1 20)(2 58)(3 22)(4 60)(5 24)(6 50)(7 14)(8 52)(9 16)(10 54)(11 18)(12 56)(13 87)(15 89)(17 91)(19 93)(21 95)(23 85)(25 61)(26 75)(27 63)(28 77)(29 65)(30 79)(31 67)(32 81)(33 69)(34 83)(35 71)(36 73)(37 68)(38 78)(39 70)(40 80)(41 72)(42 82)(43 62)(44 84)(45 64)(46 74)(47 66)(48 76)(49 94)(51 96)(53 86)(55 88)(57 90)(59 92)
(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 76 7 82)(2 75 8 81)(3 74 9 80)(4 73 10 79)(5 84 11 78)(6 83 12 77)(13 47 19 41)(14 46 20 40)(15 45 21 39)(16 44 22 38)(17 43 23 37)(18 42 24 48)(25 49 31 55)(26 60 32 54)(27 59 33 53)(28 58 34 52)(29 57 35 51)(30 56 36 50)(61 90 67 96)(62 89 68 95)(63 88 69 94)(64 87 70 93)(65 86 71 92)(66 85 72 91)
G:=sub<Sym(96)| (1,90,9,86,5,94)(2,95,6,87,10,91)(3,92,11,88,7,96)(4,85,8,89,12,93)(13,50,21,58,17,54)(14,55,18,59,22,51)(15,52,23,60,19,56)(16,57,20,49,24,53)(25,46,29,38,33,42)(26,43,34,39,30,47)(27,48,31,40,35,44)(28,45,36,41,32,37)(61,82,69,78,65,74)(62,75,66,79,70,83)(63,84,71,80,67,76)(64,77,68,81,72,73), (1,20)(2,58)(3,22)(4,60)(5,24)(6,50)(7,14)(8,52)(9,16)(10,54)(11,18)(12,56)(13,87)(15,89)(17,91)(19,93)(21,95)(23,85)(25,61)(26,75)(27,63)(28,77)(29,65)(30,79)(31,67)(32,81)(33,69)(34,83)(35,71)(36,73)(37,68)(38,78)(39,70)(40,80)(41,72)(42,82)(43,62)(44,84)(45,64)(46,74)(47,66)(48,76)(49,94)(51,96)(53,86)(55,88)(57,90)(59,92), (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), (1,76,7,82)(2,75,8,81)(3,74,9,80)(4,73,10,79)(5,84,11,78)(6,83,12,77)(13,47,19,41)(14,46,20,40)(15,45,21,39)(16,44,22,38)(17,43,23,37)(18,42,24,48)(25,49,31,55)(26,60,32,54)(27,59,33,53)(28,58,34,52)(29,57,35,51)(30,56,36,50)(61,90,67,96)(62,89,68,95)(63,88,69,94)(64,87,70,93)(65,86,71,92)(66,85,72,91)>;
G:=Group( (1,90,9,86,5,94)(2,95,6,87,10,91)(3,92,11,88,7,96)(4,85,8,89,12,93)(13,50,21,58,17,54)(14,55,18,59,22,51)(15,52,23,60,19,56)(16,57,20,49,24,53)(25,46,29,38,33,42)(26,43,34,39,30,47)(27,48,31,40,35,44)(28,45,36,41,32,37)(61,82,69,78,65,74)(62,75,66,79,70,83)(63,84,71,80,67,76)(64,77,68,81,72,73), (1,20)(2,58)(3,22)(4,60)(5,24)(6,50)(7,14)(8,52)(9,16)(10,54)(11,18)(12,56)(13,87)(15,89)(17,91)(19,93)(21,95)(23,85)(25,61)(26,75)(27,63)(28,77)(29,65)(30,79)(31,67)(32,81)(33,69)(34,83)(35,71)(36,73)(37,68)(38,78)(39,70)(40,80)(41,72)(42,82)(43,62)(44,84)(45,64)(46,74)(47,66)(48,76)(49,94)(51,96)(53,86)(55,88)(57,90)(59,92), (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), (1,76,7,82)(2,75,8,81)(3,74,9,80)(4,73,10,79)(5,84,11,78)(6,83,12,77)(13,47,19,41)(14,46,20,40)(15,45,21,39)(16,44,22,38)(17,43,23,37)(18,42,24,48)(25,49,31,55)(26,60,32,54)(27,59,33,53)(28,58,34,52)(29,57,35,51)(30,56,36,50)(61,90,67,96)(62,89,68,95)(63,88,69,94)(64,87,70,93)(65,86,71,92)(66,85,72,91) );
G=PermutationGroup([[(1,90,9,86,5,94),(2,95,6,87,10,91),(3,92,11,88,7,96),(4,85,8,89,12,93),(13,50,21,58,17,54),(14,55,18,59,22,51),(15,52,23,60,19,56),(16,57,20,49,24,53),(25,46,29,38,33,42),(26,43,34,39,30,47),(27,48,31,40,35,44),(28,45,36,41,32,37),(61,82,69,78,65,74),(62,75,66,79,70,83),(63,84,71,80,67,76),(64,77,68,81,72,73)], [(1,20),(2,58),(3,22),(4,60),(5,24),(6,50),(7,14),(8,52),(9,16),(10,54),(11,18),(12,56),(13,87),(15,89),(17,91),(19,93),(21,95),(23,85),(25,61),(26,75),(27,63),(28,77),(29,65),(30,79),(31,67),(32,81),(33,69),(34,83),(35,71),(36,73),(37,68),(38,78),(39,70),(40,80),(41,72),(42,82),(43,62),(44,84),(45,64),(46,74),(47,66),(48,76),(49,94),(51,96),(53,86),(55,88),(57,90),(59,92)], [(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)], [(1,76,7,82),(2,75,8,81),(3,74,9,80),(4,73,10,79),(5,84,11,78),(6,83,12,77),(13,47,19,41),(14,46,20,40),(15,45,21,39),(16,44,22,38),(17,43,23,37),(18,42,24,48),(25,49,31,55),(26,60,32,54),(27,59,33,53),(28,58,34,52),(29,57,35,51),(30,56,36,50),(61,90,67,96),(62,89,68,95),(63,88,69,94),(64,87,70,93),(65,86,71,92),(66,85,72,91)]])
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 | D6⋊D6 | D6.3D6 |
| kernel | D6⋊4Dic6 | D6⋊Dic3 | C62.C22 | C3×C4⋊Dic3 | C3×D6⋊C4 | C6.Dic6 | C2×S3×Dic3 | C2×C32⋊2Q8 | C4⋊Dic3 | 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⋊4Dic6 ►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 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 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 |
| 8 | 3 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 0 | 0 | 0 | 0 | 2 | 4 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,7,0,0,0,0,10,10],[8,5,0,0,0,0,3,5,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,9,2,0,0,0,0,11,4] >;
D6⋊4Dic6 in GAP, Magma, Sage, TeX
D_6\rtimes_4{\rm Dic}_6 % in TeX
G:=Group("D6:4Dic6"); // GroupNames label
G:=SmallGroup(288,547);
// by ID
G=gap.SmallGroup(288,547);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,219,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^12=1,d^2=c^6,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations