metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊3M4(2), C42.29D6, C3⋊C8⋊27D4, C4⋊C8⋊12S3, D6⋊C8⋊28C2, C3⋊4(C8⋊9D4), C6.47(C4×D4), D6⋊C4.13C4, (C2×D12).9C4, (C4×D12).8C2, C4.206(S3×D4), (C2×C8).217D6, C6.12(C8○D4), C12.365(C2×D4), C4⋊Dic3.12C4, C2.14(C8○D12), C42.S3⋊3C2, (C4×C12).62C22, C6.28(C2×M4(2)), C2.17(S3×M4(2)), C12.335(C4○D4), C2.7(Dic3⋊5D4), (C2×C12).833C23, (C2×C24).255C22, C4.55(Q8⋊3S3), (S3×C2×C8)⋊23C2, (C3×C4⋊C8)⋊22C2, (C2×C4).35(C4×S3), (C2×C8⋊S3)⋊21C2, (C2×C12).43(C2×C4), C22.111(S3×C2×C4), (C2×C3⋊C8).195C22, (S3×C2×C4).279C22, (C2×C6).88(C22×C4), (C22×S3).38(C2×C4), (C2×C4).775(C22×S3), (C2×Dic3).22(C2×C4), SmallGroup(192,395)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊3M4(2)
G = < a,b,c,d | a6=b2=c8=d2=1, bab=dad=a-1, ac=ca, cbc-1=a3b, dbd=ab, dcd=c5 >
Subgroups: 312 in 124 conjugacy classes, 51 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C3⋊C8, C3⋊C8, C24, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), S3×C8, C8⋊S3, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C2×D12, C8⋊9D4, C42.S3, D6⋊C8, C3×C4⋊C8, C4×D12, S3×C2×C8, C2×C8⋊S3, D6⋊3M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, M4(2), C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, C2×M4(2), C8○D4, S3×C2×C4, S3×D4, Q8⋊3S3, C8⋊9D4, Dic3⋊5D4, C8○D12, S3×M4(2), D6⋊3M4(2)
►On 96 points
(1 9 83 25 49 78)(2 10 84 26 50 79)(3 11 85 27 51 80)(4 12 86 28 52 73)(5 13 87 29 53 74)(6 14 88 30 54 75)(7 15 81 31 55 76)(8 16 82 32 56 77)(17 67 45 63 33 93)(18 68 46 64 34 94)(19 69 47 57 35 95)(20 70 48 58 36 96)(21 71 41 59 37 89)(22 72 42 60 38 90)(23 65 43 61 39 91)(24 66 44 62 40 92)
(1 92)(2 45)(3 94)(4 47)(5 96)(6 41)(7 90)(8 43)(9 40)(10 67)(11 34)(12 69)(13 36)(14 71)(15 38)(16 65)(17 84)(18 80)(19 86)(20 74)(21 88)(22 76)(23 82)(24 78)(25 44)(26 93)(27 46)(28 95)(29 48)(30 89)(31 42)(32 91)(33 50)(35 52)(37 54)(39 56)(49 66)(51 68)(53 70)(55 72)(57 73)(58 87)(59 75)(60 81)(61 77)(62 83)(63 79)(64 85)
(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 5)(3 7)(9 74)(10 79)(11 76)(12 73)(13 78)(14 75)(15 80)(16 77)(17 63)(18 60)(19 57)(20 62)(21 59)(22 64)(23 61)(24 58)(25 29)(27 31)(33 93)(34 90)(35 95)(36 92)(37 89)(38 94)(39 91)(40 96)(41 71)(42 68)(43 65)(44 70)(45 67)(46 72)(47 69)(48 66)(49 87)(50 84)(51 81)(52 86)(53 83)(54 88)(55 85)(56 82)
G:=sub<Sym(96)| (1,9,83,25,49,78)(2,10,84,26,50,79)(3,11,85,27,51,80)(4,12,86,28,52,73)(5,13,87,29,53,74)(6,14,88,30,54,75)(7,15,81,31,55,76)(8,16,82,32,56,77)(17,67,45,63,33,93)(18,68,46,64,34,94)(19,69,47,57,35,95)(20,70,48,58,36,96)(21,71,41,59,37,89)(22,72,42,60,38,90)(23,65,43,61,39,91)(24,66,44,62,40,92), (1,92)(2,45)(3,94)(4,47)(5,96)(6,41)(7,90)(8,43)(9,40)(10,67)(11,34)(12,69)(13,36)(14,71)(15,38)(16,65)(17,84)(18,80)(19,86)(20,74)(21,88)(22,76)(23,82)(24,78)(25,44)(26,93)(27,46)(28,95)(29,48)(30,89)(31,42)(32,91)(33,50)(35,52)(37,54)(39,56)(49,66)(51,68)(53,70)(55,72)(57,73)(58,87)(59,75)(60,81)(61,77)(62,83)(63,79)(64,85), (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), (1,5)(3,7)(9,74)(10,79)(11,76)(12,73)(13,78)(14,75)(15,80)(16,77)(17,63)(18,60)(19,57)(20,62)(21,59)(22,64)(23,61)(24,58)(25,29)(27,31)(33,93)(34,90)(35,95)(36,92)(37,89)(38,94)(39,91)(40,96)(41,71)(42,68)(43,65)(44,70)(45,67)(46,72)(47,69)(48,66)(49,87)(50,84)(51,81)(52,86)(53,83)(54,88)(55,85)(56,82)>;
G:=Group( (1,9,83,25,49,78)(2,10,84,26,50,79)(3,11,85,27,51,80)(4,12,86,28,52,73)(5,13,87,29,53,74)(6,14,88,30,54,75)(7,15,81,31,55,76)(8,16,82,32,56,77)(17,67,45,63,33,93)(18,68,46,64,34,94)(19,69,47,57,35,95)(20,70,48,58,36,96)(21,71,41,59,37,89)(22,72,42,60,38,90)(23,65,43,61,39,91)(24,66,44,62,40,92), (1,92)(2,45)(3,94)(4,47)(5,96)(6,41)(7,90)(8,43)(9,40)(10,67)(11,34)(12,69)(13,36)(14,71)(15,38)(16,65)(17,84)(18,80)(19,86)(20,74)(21,88)(22,76)(23,82)(24,78)(25,44)(26,93)(27,46)(28,95)(29,48)(30,89)(31,42)(32,91)(33,50)(35,52)(37,54)(39,56)(49,66)(51,68)(53,70)(55,72)(57,73)(58,87)(59,75)(60,81)(61,77)(62,83)(63,79)(64,85), (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), (1,5)(3,7)(9,74)(10,79)(11,76)(12,73)(13,78)(14,75)(15,80)(16,77)(17,63)(18,60)(19,57)(20,62)(21,59)(22,64)(23,61)(24,58)(25,29)(27,31)(33,93)(34,90)(35,95)(36,92)(37,89)(38,94)(39,91)(40,96)(41,71)(42,68)(43,65)(44,70)(45,67)(46,72)(47,69)(48,66)(49,87)(50,84)(51,81)(52,86)(53,83)(54,88)(55,85)(56,82) );
G=PermutationGroup([[(1,9,83,25,49,78),(2,10,84,26,50,79),(3,11,85,27,51,80),(4,12,86,28,52,73),(5,13,87,29,53,74),(6,14,88,30,54,75),(7,15,81,31,55,76),(8,16,82,32,56,77),(17,67,45,63,33,93),(18,68,46,64,34,94),(19,69,47,57,35,95),(20,70,48,58,36,96),(21,71,41,59,37,89),(22,72,42,60,38,90),(23,65,43,61,39,91),(24,66,44,62,40,92)], [(1,92),(2,45),(3,94),(4,47),(5,96),(6,41),(7,90),(8,43),(9,40),(10,67),(11,34),(12,69),(13,36),(14,71),(15,38),(16,65),(17,84),(18,80),(19,86),(20,74),(21,88),(22,76),(23,82),(24,78),(25,44),(26,93),(27,46),(28,95),(29,48),(30,89),(31,42),(32,91),(33,50),(35,52),(37,54),(39,56),(49,66),(51,68),(53,70),(55,72),(57,73),(58,87),(59,75),(60,81),(61,77),(62,83),(63,79),(64,85)], [(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)], [(1,5),(3,7),(9,74),(10,79),(11,76),(12,73),(13,78),(14,75),(15,80),(16,77),(17,63),(18,60),(19,57),(20,62),(21,59),(22,64),(23,61),(24,58),(25,29),(27,31),(33,93),(34,90),(35,95),(36,92),(37,89),(38,94),(39,91),(40,96),(41,71),(42,68),(43,65),(44,70),(45,67),(46,72),(47,69),(48,66),(49,87),(50,84),(51,81),(52,86),(53,83),(54,88),(55,85),(56,82)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | C4○D4 | M4(2) | C4×S3 | C8○D4 | C8○D12 | S3×D4 | Q8⋊3S3 | S3×M4(2) |
| kernel | D6⋊3M4(2) | C42.S3 | D6⋊C8 | C3×C4⋊C8 | C4×D12 | S3×C2×C8 | C2×C8⋊S3 | C4⋊Dic3 | D6⋊C4 | C2×D12 | C4⋊C8 | C3⋊C8 | C42 | C2×C8 | C12 | D6 | C2×C4 | C6 | C2 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of D6⋊3M4(2) ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 7 | 66 |
| 0 | 0 | 59 | 66 |
| 0 | 1 | 0 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 |
| 0 | 0 | 67 | 70 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 72 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,72,0,0,1,0],[72,0,0,0,0,72,0,0,0,0,7,59,0,0,66,66],[0,46,0,0,1,0,0,0,0,0,3,67,0,0,6,70],[72,0,0,0,0,1,0,0,0,0,1,72,0,0,0,72] >;
D6⋊3M4(2) in GAP, Magma, Sage, TeX
D_6\rtimes_3M_4(2)
% in TeX
G:=Group("D6:3M4(2)"); // GroupNames label
G:=SmallGroup(192,395);
// by ID
G=gap.SmallGroup(192,395);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,701,120,219,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^3*b,d*b*d=a*b,d*c*d=c^5>;
// generators/relations