metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.2C42, C42.242D6, Dic3.2C42, (S3×C8)⋊5C4, (C4×C8)⋊15S3, C8⋊S3⋊5C4, (C4×C24)⋊26C2, C8.42(C4×S3), C24.63(C2×C4), D6⋊C4.19C4, C24⋊C4⋊31C2, C6.5(C8○D4), (C2×C8).340D6, C6.7(C2×C42), C2.8(S3×C42), (C8×Dic3)⋊12C2, C2.3(C8○D12), Dic3⋊C4.19C4, C3⋊1(C8○2M4(2)), C42⋊2S3.13C2, (C2×C12).807C23, (C2×C24).425C22, (C4×C12).340C22, C12.124(C22×C4), C42.S3⋊24C2, (C4×Dic3).265C22, C4.98(S3×C2×C4), C3⋊C8.11(C2×C4), (S3×C2×C8).10C2, (C2×C4).88(C4×S3), C22.38(S3×C2×C4), (C4×S3).31(C2×C4), (C2×C8⋊S3).14C2, (C2×C12).205(C2×C4), (C2×C3⋊C8).292C22, (S3×C2×C4).268C22, (C2×C6).62(C22×C4), (C22×S3).32(C2×C4), (C2×C4).749(C22×S3), (C2×Dic3).47(C2×C4), SmallGroup(192,248)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.C42
G = < a,b,c,d | a6=b2=c4=1, d4=a3, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, cd=dc >
Subgroups: 248 in 130 conjugacy classes, 75 normal (33 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C4×C8, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), S3×C8, C8⋊S3, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C8○2M4(2), C42.S3, C8×Dic3, C24⋊C4, C4×C24, C42⋊2S3, S3×C2×C8, C2×C8⋊S3, D6.C42
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C42, C22×C4, C4×S3, C22×S3, C2×C42, C8○D4, S3×C2×C4, C8○2M4(2), S3×C42, C8○D12, D6.C42
►On 96 points
(1 21 43 5 17 47)(2 22 44 6 18 48)(3 23 45 7 19 41)(4 24 46 8 20 42)(9 88 93 13 84 89)(10 81 94 14 85 90)(11 82 95 15 86 91)(12 83 96 16 87 92)(25 39 69 29 35 65)(26 40 70 30 36 66)(27 33 71 31 37 67)(28 34 72 32 38 68)(49 59 79 53 63 75)(50 60 80 54 64 76)(51 61 73 55 57 77)(52 62 74 56 58 78)
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 93)(10 94)(11 95)(12 96)(13 89)(14 90)(15 91)(16 92)(17 21)(18 22)(19 23)(20 24)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 65)(40 66)(49 53)(50 54)(51 55)(52 56)(57 77)(58 78)(59 79)(60 80)(61 73)(62 74)(63 75)(64 76)
(1 65 75 15)(2 66 76 16)(3 67 77 9)(4 68 78 10)(5 69 79 11)(6 70 80 12)(7 71 73 13)(8 72 74 14)(17 29 53 82)(18 30 54 83)(19 31 55 84)(20 32 56 85)(21 25 49 86)(22 26 50 87)(23 27 51 88)(24 28 52 81)(33 61 93 45)(34 62 94 46)(35 63 95 47)(36 64 96 48)(37 57 89 41)(38 58 90 42)(39 59 91 43)(40 60 92 44)
(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)
G:=sub<Sym(96)| (1,21,43,5,17,47)(2,22,44,6,18,48)(3,23,45,7,19,41)(4,24,46,8,20,42)(9,88,93,13,84,89)(10,81,94,14,85,90)(11,82,95,15,86,91)(12,83,96,16,87,92)(25,39,69,29,35,65)(26,40,70,30,36,66)(27,33,71,31,37,67)(28,34,72,32,38,68)(49,59,79,53,63,75)(50,60,80,54,64,76)(51,61,73,55,57,77)(52,62,74,56,58,78), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,93)(10,94)(11,95)(12,96)(13,89)(14,90)(15,91)(16,92)(17,21)(18,22)(19,23)(20,24)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,65)(40,66)(49,53)(50,54)(51,55)(52,56)(57,77)(58,78)(59,79)(60,80)(61,73)(62,74)(63,75)(64,76), (1,65,75,15)(2,66,76,16)(3,67,77,9)(4,68,78,10)(5,69,79,11)(6,70,80,12)(7,71,73,13)(8,72,74,14)(17,29,53,82)(18,30,54,83)(19,31,55,84)(20,32,56,85)(21,25,49,86)(22,26,50,87)(23,27,51,88)(24,28,52,81)(33,61,93,45)(34,62,94,46)(35,63,95,47)(36,64,96,48)(37,57,89,41)(38,58,90,42)(39,59,91,43)(40,60,92,44), (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)>;
G:=Group( (1,21,43,5,17,47)(2,22,44,6,18,48)(3,23,45,7,19,41)(4,24,46,8,20,42)(9,88,93,13,84,89)(10,81,94,14,85,90)(11,82,95,15,86,91)(12,83,96,16,87,92)(25,39,69,29,35,65)(26,40,70,30,36,66)(27,33,71,31,37,67)(28,34,72,32,38,68)(49,59,79,53,63,75)(50,60,80,54,64,76)(51,61,73,55,57,77)(52,62,74,56,58,78), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,93)(10,94)(11,95)(12,96)(13,89)(14,90)(15,91)(16,92)(17,21)(18,22)(19,23)(20,24)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,65)(40,66)(49,53)(50,54)(51,55)(52,56)(57,77)(58,78)(59,79)(60,80)(61,73)(62,74)(63,75)(64,76), (1,65,75,15)(2,66,76,16)(3,67,77,9)(4,68,78,10)(5,69,79,11)(6,70,80,12)(7,71,73,13)(8,72,74,14)(17,29,53,82)(18,30,54,83)(19,31,55,84)(20,32,56,85)(21,25,49,86)(22,26,50,87)(23,27,51,88)(24,28,52,81)(33,61,93,45)(34,62,94,46)(35,63,95,47)(36,64,96,48)(37,57,89,41)(38,58,90,42)(39,59,91,43)(40,60,92,44), (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) );
G=PermutationGroup([[(1,21,43,5,17,47),(2,22,44,6,18,48),(3,23,45,7,19,41),(4,24,46,8,20,42),(9,88,93,13,84,89),(10,81,94,14,85,90),(11,82,95,15,86,91),(12,83,96,16,87,92),(25,39,69,29,35,65),(26,40,70,30,36,66),(27,33,71,31,37,67),(28,34,72,32,38,68),(49,59,79,53,63,75),(50,60,80,54,64,76),(51,61,73,55,57,77),(52,62,74,56,58,78)], [(1,47),(2,48),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,93),(10,94),(11,95),(12,96),(13,89),(14,90),(15,91),(16,92),(17,21),(18,22),(19,23),(20,24),(33,67),(34,68),(35,69),(36,70),(37,71),(38,72),(39,65),(40,66),(49,53),(50,54),(51,55),(52,56),(57,77),(58,78),(59,79),(60,80),(61,73),(62,74),(63,75),(64,76)], [(1,65,75,15),(2,66,76,16),(3,67,77,9),(4,68,78,10),(5,69,79,11),(6,70,80,12),(7,71,73,13),(8,72,74,14),(17,29,53,82),(18,30,54,83),(19,31,55,84),(20,32,56,85),(21,25,49,86),(22,26,50,87),(23,27,51,88),(24,28,52,81),(33,61,93,45),(34,62,94,46),(35,63,95,47),(36,64,96,48),(37,57,89,41),(38,58,90,42),(39,59,91,43),(40,60,92,44)], [(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)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | 6B | 6C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | ··· | 8T | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | C8○D12 |
| kernel | D6.C42 | C42.S3 | C8×Dic3 | C24⋊C4 | C4×C24 | C42⋊2S3 | S3×C2×C8 | C2×C8⋊S3 | S3×C8 | C8⋊S3 | Dic3⋊C4 | D6⋊C4 | C4×C8 | C42 | C2×C8 | C8 | C2×C4 | C6 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 8 | 4 | 4 | 1 | 1 | 2 | 8 | 4 | 8 | 16 |
Matrix representation of D6.C42 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 14 |
| 0 | 0 | 59 | 66 |
| 63 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 |
| 0 | 0 | 63 | 0 |
| 0 | 0 | 0 | 63 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,72,0,0,1,0],[72,0,0,0,0,1,0,0,0,0,1,0,0,0,1,72],[0,1,0,0,1,0,0,0,0,0,7,59,0,0,14,66],[63,0,0,0,0,63,0,0,0,0,63,0,0,0,0,63] >;
D6.C42 in GAP, Magma, Sage, TeX
D_6.C_4^2
% in TeX
G:=Group("D6.C4^2"); // GroupNames label
G:=SmallGroup(192,248);
// by ID
G=gap.SmallGroup(192,248);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^4=1,d^4=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,c*d=d*c>;
// generators/relations