metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊20D4, C6.332+ 1+4, C4⋊D4⋊7S3, C4⋊C4.177D6, (C2×D4).90D6, C3⋊3(Q8⋊6D4), C4.109(S3×D4), Dic3⋊D4⋊17C2, C12⋊D4⋊19C2, C12⋊3D4⋊15C2, C12.225(C2×D4), C22⋊C4.46D6, C6.62(C22×D4), Dic3⋊8(C4○D4), Dic3⋊4D4⋊6C2, (C2×C12).35C23, (C2×C6).143C24, D6⋊C4.12C22, Dic3.21(C2×D4), (C22×C4).235D6, Dic6⋊C4⋊20C2, C23.14D6⋊10C2, C2.35(D4⋊6D6), (C6×D4).117C22, C23.20(C22×S3), (C22×C6).14C23, (C2×D12).142C22, Dic3⋊C4.14C22, (C22×S3).62C23, C22.164(S3×C23), (C4×Dic3).90C22, (C22×C12).237C22, (C2×Dic6).293C22, (C2×Dic3).225C23, C6.D4.110C22, (C22×Dic3).104C22, C2.35(C2×S3×D4), (C3×C4⋊D4)⋊8C2, (C4×C3⋊D4)⋊15C2, C2.34(S3×C4○D4), (C2×C4○D12)⋊19C2, C6.148(C2×C4○D4), (S3×C2×C4).82C22, (C2×D4⋊2S3)⋊11C2, (C3×C4⋊C4).139C22, (C2×C4).585(C22×S3), (C2×C3⋊D4).25C22, (C3×C22⋊C4).8C22, SmallGroup(192,1158)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊20D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, cac-1=a7, ad=da, cbc-1=dbd=a6b, dcd=c-1 >
Subgroups: 864 in 312 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C4⋊1D4, C2×C4○D4, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C4○D12, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, Q8⋊6D4, Dic3⋊4D4, Dic3⋊D4, Dic6⋊C4, C12⋊D4, C4×C3⋊D4, C23.14D6, C12⋊3D4, C12⋊3D4, C3×C4⋊D4, C2×C4○D12, C2×D4⋊2S3, Dic6⋊20D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, 2+ 1+4, S3×D4, S3×C23, Q8⋊6D4, C2×S3×D4, D4⋊6D6, S3×C4○D4, Dic6⋊20D4
►On 96 points
(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 52 7 58)(2 51 8 57)(3 50 9 56)(4 49 10 55)(5 60 11 54)(6 59 12 53)(13 85 19 91)(14 96 20 90)(15 95 21 89)(16 94 22 88)(17 93 23 87)(18 92 24 86)(25 80 31 74)(26 79 32 73)(27 78 33 84)(28 77 34 83)(29 76 35 82)(30 75 36 81)(37 70 43 64)(38 69 44 63)(39 68 45 62)(40 67 46 61)(41 66 47 72)(42 65 48 71)
(1 40 76 92)(2 47 77 87)(3 42 78 94)(4 37 79 89)(5 44 80 96)(6 39 81 91)(7 46 82 86)(8 41 83 93)(9 48 84 88)(10 43 73 95)(11 38 74 90)(12 45 75 85)(13 53 68 36)(14 60 69 31)(15 55 70 26)(16 50 71 33)(17 57 72 28)(18 52 61 35)(19 59 62 30)(20 54 63 25)(21 49 64 32)(22 56 65 27)(23 51 66 34)(24 58 67 29)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 37)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 81)(32 82)(33 83)(34 84)(35 73)(36 74)(61 95)(62 96)(63 85)(64 86)(65 87)(66 88)(67 89)(68 90)(69 91)(70 92)(71 93)(72 94)
G:=sub<Sym(96)| (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), (1,52,7,58)(2,51,8,57)(3,50,9,56)(4,49,10,55)(5,60,11,54)(6,59,12,53)(13,85,19,91)(14,96,20,90)(15,95,21,89)(16,94,22,88)(17,93,23,87)(18,92,24,86)(25,80,31,74)(26,79,32,73)(27,78,33,84)(28,77,34,83)(29,76,35,82)(30,75,36,81)(37,70,43,64)(38,69,44,63)(39,68,45,62)(40,67,46,61)(41,66,47,72)(42,65,48,71), (1,40,76,92)(2,47,77,87)(3,42,78,94)(4,37,79,89)(5,44,80,96)(6,39,81,91)(7,46,82,86)(8,41,83,93)(9,48,84,88)(10,43,73,95)(11,38,74,90)(12,45,75,85)(13,53,68,36)(14,60,69,31)(15,55,70,26)(16,50,71,33)(17,57,72,28)(18,52,61,35)(19,59,62,30)(20,54,63,25)(21,49,64,32)(22,56,65,27)(23,51,66,34)(24,58,67,29), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,37)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,81)(32,82)(33,83)(34,84)(35,73)(36,74)(61,95)(62,96)(63,85)(64,86)(65,87)(66,88)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94)>;
G:=Group( (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), (1,52,7,58)(2,51,8,57)(3,50,9,56)(4,49,10,55)(5,60,11,54)(6,59,12,53)(13,85,19,91)(14,96,20,90)(15,95,21,89)(16,94,22,88)(17,93,23,87)(18,92,24,86)(25,80,31,74)(26,79,32,73)(27,78,33,84)(28,77,34,83)(29,76,35,82)(30,75,36,81)(37,70,43,64)(38,69,44,63)(39,68,45,62)(40,67,46,61)(41,66,47,72)(42,65,48,71), (1,40,76,92)(2,47,77,87)(3,42,78,94)(4,37,79,89)(5,44,80,96)(6,39,81,91)(7,46,82,86)(8,41,83,93)(9,48,84,88)(10,43,73,95)(11,38,74,90)(12,45,75,85)(13,53,68,36)(14,60,69,31)(15,55,70,26)(16,50,71,33)(17,57,72,28)(18,52,61,35)(19,59,62,30)(20,54,63,25)(21,49,64,32)(22,56,65,27)(23,51,66,34)(24,58,67,29), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,37)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,81)(32,82)(33,83)(34,84)(35,73)(36,74)(61,95)(62,96)(63,85)(64,86)(65,87)(66,88)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94) );
G=PermutationGroup([[(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)], [(1,52,7,58),(2,51,8,57),(3,50,9,56),(4,49,10,55),(5,60,11,54),(6,59,12,53),(13,85,19,91),(14,96,20,90),(15,95,21,89),(16,94,22,88),(17,93,23,87),(18,92,24,86),(25,80,31,74),(26,79,32,73),(27,78,33,84),(28,77,34,83),(29,76,35,82),(30,75,36,81),(37,70,43,64),(38,69,44,63),(39,68,45,62),(40,67,46,61),(41,66,47,72),(42,65,48,71)], [(1,40,76,92),(2,47,77,87),(3,42,78,94),(4,37,79,89),(5,44,80,96),(6,39,81,91),(7,46,82,86),(8,41,83,93),(9,48,84,88),(10,43,73,95),(11,38,74,90),(12,45,75,85),(13,53,68,36),(14,60,69,31),(15,55,70,26),(16,50,71,33),(17,57,72,28),(18,52,61,35),(19,59,62,30),(20,54,63,25),(21,49,64,32),(22,56,65,27),(23,51,66,34),(24,58,67,29)], [(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,38),(14,39),(15,40),(16,41),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,37),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,81),(32,82),(33,83),(34,84),(35,73),(36,74),(61,95),(62,96),(63,85),(64,86),(65,87),(66,88),(67,89),(68,90),(69,91),(70,92),(71,93),(72,94)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | C4○D4 | 2+ 1+4 | S3×D4 | D4⋊6D6 | S3×C4○D4 |
| kernel | Dic6⋊20D4 | Dic3⋊4D4 | Dic3⋊D4 | Dic6⋊C4 | C12⋊D4 | C4×C3⋊D4 | C23.14D6 | C12⋊3D4 | C3×C4⋊D4 | C2×C4○D12 | C2×D4⋊2S3 | C4⋊D4 | Dic6 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | Dic3 | C6 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 1 | 3 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of Dic6⋊20D4 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
Dic6⋊20D4 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_{20}D_4 % in TeX
G:=Group("Dic6:20D4"); // GroupNames label
G:=SmallGroup(192,1158);
// by ID
G=gap.SmallGroup(192,1158);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,477,232,184,570,185,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations