metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊2M4(2), C3⋊C8⋊25D4, D6⋊C8⋊24C2, C3⋊2(C8⋊9D4), D6⋊C4.5C4, C6.27(C4×D4), C22⋊C8⋊13S3, C24⋊C4⋊13C2, C6.9(C8○D4), C4.197(S3×D4), (C2×C8).195D6, Dic3⋊C8⋊24C2, C12.356(C2×D4), Dic3⋊C4.5C4, C23.18(C4×S3), (C22×C4).95D6, C2.11(C8○D12), C6.D4.5C4, C6.22(C2×M4(2)), C2.13(S3×M4(2)), C12.298(C4○D4), (C2×C12).823C23, (C2×C24).215C22, C4.124(D4⋊2S3), (C22×C12).94C22, C2.11(Dic3⋊4D4), (C4×Dic3).182C22, (S3×C2×C8)⋊15C2, (C2×C4).32(C4×S3), (C2×C3⋊D4).5C4, (C4×C3⋊D4).1C2, (C3×C22⋊C8)⋊22C2, (C2×C12).40(C2×C4), (C2×C4.Dic3)⋊1C2, C22.105(S3×C2×C4), (C2×C3⋊C8).191C22, (S3×C2×C4).275C22, (C2×C6).78(C22×C4), (C22×C6).41(C2×C4), (C22×S3).36(C2×C4), (C2×C4).765(C22×S3), (C2×Dic3).17(C2×C4), SmallGroup(192,287)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊2M4(2)
G = < a,b,c,d | a6=b2=c8=d2=1, bab=cac-1=a-1, ad=da, cbc-1=ab, dbd=a3b, dcd=c5 >
Subgroups: 288 in 124 conjugacy classes, 51 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C3⋊C8, C3⋊C8, C24, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), S3×C8, C2×C3⋊C8, C4.Dic3, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C2×C24, S3×C2×C4, C2×C3⋊D4, C22×C12, C8⋊9D4, Dic3⋊C8, C24⋊C4, D6⋊C8, C3×C22⋊C8, S3×C2×C8, C2×C4.Dic3, C4×C3⋊D4, D6⋊2M4(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, D4⋊2S3, C8⋊9D4, Dic3⋊4D4, C8○D12, S3×M4(2), D6⋊2M4(2)
►On 96 points
(1 84 77 42 52 16)(2 9 53 43 78 85)(3 86 79 44 54 10)(4 11 55 45 80 87)(5 88 73 46 56 12)(6 13 49 47 74 81)(7 82 75 48 50 14)(8 15 51 41 76 83)(17 68 57 38 32 94)(18 95 25 39 58 69)(19 70 59 40 26 96)(20 89 27 33 60 71)(21 72 61 34 28 90)(22 91 29 35 62 65)(23 66 63 36 30 92)(24 93 31 37 64 67)
(1 12)(2 49)(3 14)(4 51)(5 16)(6 53)(7 10)(8 55)(9 13)(11 15)(17 61)(18 65)(19 63)(20 67)(21 57)(22 69)(23 59)(24 71)(25 35)(26 30)(27 37)(28 32)(29 39)(31 33)(34 94)(36 96)(38 90)(40 92)(41 87)(42 73)(43 81)(44 75)(45 83)(46 77)(47 85)(48 79)(50 86)(52 88)(54 82)(56 84)(58 91)(60 93)(62 95)(64 89)(66 70)(68 72)(74 78)(76 80)
(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 17)(2 22)(3 19)(4 24)(5 21)(6 18)(7 23)(8 20)(9 91)(10 96)(11 93)(12 90)(13 95)(14 92)(15 89)(16 94)(25 49)(26 54)(27 51)(28 56)(29 53)(30 50)(31 55)(32 52)(33 41)(34 46)(35 43)(36 48)(37 45)(38 42)(39 47)(40 44)(57 77)(58 74)(59 79)(60 76)(61 73)(62 78)(63 75)(64 80)(65 85)(66 82)(67 87)(68 84)(69 81)(70 86)(71 83)(72 88)
G:=sub<Sym(96)| (1,84,77,42,52,16)(2,9,53,43,78,85)(3,86,79,44,54,10)(4,11,55,45,80,87)(5,88,73,46,56,12)(6,13,49,47,74,81)(7,82,75,48,50,14)(8,15,51,41,76,83)(17,68,57,38,32,94)(18,95,25,39,58,69)(19,70,59,40,26,96)(20,89,27,33,60,71)(21,72,61,34,28,90)(22,91,29,35,62,65)(23,66,63,36,30,92)(24,93,31,37,64,67), (1,12)(2,49)(3,14)(4,51)(5,16)(6,53)(7,10)(8,55)(9,13)(11,15)(17,61)(18,65)(19,63)(20,67)(21,57)(22,69)(23,59)(24,71)(25,35)(26,30)(27,37)(28,32)(29,39)(31,33)(34,94)(36,96)(38,90)(40,92)(41,87)(42,73)(43,81)(44,75)(45,83)(46,77)(47,85)(48,79)(50,86)(52,88)(54,82)(56,84)(58,91)(60,93)(62,95)(64,89)(66,70)(68,72)(74,78)(76,80), (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), (1,17)(2,22)(3,19)(4,24)(5,21)(6,18)(7,23)(8,20)(9,91)(10,96)(11,93)(12,90)(13,95)(14,92)(15,89)(16,94)(25,49)(26,54)(27,51)(28,56)(29,53)(30,50)(31,55)(32,52)(33,41)(34,46)(35,43)(36,48)(37,45)(38,42)(39,47)(40,44)(57,77)(58,74)(59,79)(60,76)(61,73)(62,78)(63,75)(64,80)(65,85)(66,82)(67,87)(68,84)(69,81)(70,86)(71,83)(72,88)>;
G:=Group( (1,84,77,42,52,16)(2,9,53,43,78,85)(3,86,79,44,54,10)(4,11,55,45,80,87)(5,88,73,46,56,12)(6,13,49,47,74,81)(7,82,75,48,50,14)(8,15,51,41,76,83)(17,68,57,38,32,94)(18,95,25,39,58,69)(19,70,59,40,26,96)(20,89,27,33,60,71)(21,72,61,34,28,90)(22,91,29,35,62,65)(23,66,63,36,30,92)(24,93,31,37,64,67), (1,12)(2,49)(3,14)(4,51)(5,16)(6,53)(7,10)(8,55)(9,13)(11,15)(17,61)(18,65)(19,63)(20,67)(21,57)(22,69)(23,59)(24,71)(25,35)(26,30)(27,37)(28,32)(29,39)(31,33)(34,94)(36,96)(38,90)(40,92)(41,87)(42,73)(43,81)(44,75)(45,83)(46,77)(47,85)(48,79)(50,86)(52,88)(54,82)(56,84)(58,91)(60,93)(62,95)(64,89)(66,70)(68,72)(74,78)(76,80), (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), (1,17)(2,22)(3,19)(4,24)(5,21)(6,18)(7,23)(8,20)(9,91)(10,96)(11,93)(12,90)(13,95)(14,92)(15,89)(16,94)(25,49)(26,54)(27,51)(28,56)(29,53)(30,50)(31,55)(32,52)(33,41)(34,46)(35,43)(36,48)(37,45)(38,42)(39,47)(40,44)(57,77)(58,74)(59,79)(60,76)(61,73)(62,78)(63,75)(64,80)(65,85)(66,82)(67,87)(68,84)(69,81)(70,86)(71,83)(72,88) );
G=PermutationGroup([[(1,84,77,42,52,16),(2,9,53,43,78,85),(3,86,79,44,54,10),(4,11,55,45,80,87),(5,88,73,46,56,12),(6,13,49,47,74,81),(7,82,75,48,50,14),(8,15,51,41,76,83),(17,68,57,38,32,94),(18,95,25,39,58,69),(19,70,59,40,26,96),(20,89,27,33,60,71),(21,72,61,34,28,90),(22,91,29,35,62,65),(23,66,63,36,30,92),(24,93,31,37,64,67)], [(1,12),(2,49),(3,14),(4,51),(5,16),(6,53),(7,10),(8,55),(9,13),(11,15),(17,61),(18,65),(19,63),(20,67),(21,57),(22,69),(23,59),(24,71),(25,35),(26,30),(27,37),(28,32),(29,39),(31,33),(34,94),(36,96),(38,90),(40,92),(41,87),(42,73),(43,81),(44,75),(45,83),(46,77),(47,85),(48,79),(50,86),(52,88),(54,82),(56,84),(58,91),(60,93),(62,95),(64,89),(66,70),(68,72),(74,78),(76,80)], [(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)], [(1,17),(2,22),(3,19),(4,24),(5,21),(6,18),(7,23),(8,20),(9,91),(10,96),(11,93),(12,90),(13,95),(14,92),(15,89),(16,94),(25,49),(26,54),(27,51),(28,56),(29,53),(30,50),(31,55),(32,52),(33,41),(34,46),(35,43),(36,48),(37,45),(38,42),(39,47),(40,44),(57,77),(58,74),(59,79),(60,76),(61,73),(62,78),(63,75),(64,80),(65,85),(66,82),(67,87),(68,84),(69,81),(70,86),(71,83),(72,88)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | C4○D4 | M4(2) | C4×S3 | C4×S3 | C8○D4 | C8○D12 | S3×D4 | D4⋊2S3 | S3×M4(2) |
| kernel | D6⋊2M4(2) | Dic3⋊C8 | C24⋊C4 | D6⋊C8 | C3×C22⋊C8 | S3×C2×C8 | C2×C4.Dic3 | C4×C3⋊D4 | Dic3⋊C4 | D6⋊C4 | C6.D4 | C2×C3⋊D4 | C22⋊C8 | C3⋊C8 | C2×C8 | C22×C4 | C12 | D6 | C2×C4 | C23 | C6 | C2 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 2 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of D6⋊2M4(2) ►in GL6(𝔽73)
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 56 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 51 | 1 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 46 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 7 | 0 | 0 |
| 0 | 0 | 43 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 2 |
| 0 | 0 | 0 | 0 | 14 | 22 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 62 | 3 | 0 | 0 |
| 0 | 0 | 33 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 2 |
| 0 | 0 | 0 | 0 | 14 | 22 |
G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,72,1,0,0,0,0,0,0,1,56,0,0,0,0,0,72,0,0,0,0,0,0,72,51,0,0,0,0,0,1],[27,46,0,0,0,0,0,46,0,0,0,0,0,0,23,43,0,0,0,0,7,50,0,0,0,0,0,0,51,14,0,0,0,0,2,22],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,62,33,0,0,0,0,3,11,0,0,0,0,0,0,51,14,0,0,0,0,2,22] >;
D6⋊2M4(2) in GAP, Magma, Sage, TeX
D_6\rtimes_2M_4(2)
% in TeX
G:=Group("D6:2M4(2)"); // GroupNames label
G:=SmallGroup(192,287);
// by ID
G=gap.SmallGroup(192,287);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,701,219,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a*b,d*b*d=a^3*b,d*c*d=c^5>;
// generators/relations