metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊2SD16, C3⋊C8⋊21D4, C3⋊2(C8⋊8D4), C4⋊C4.24D6, Q8⋊C4⋊2S3, D6⋊3Q8⋊1C2, C4.166(S3×D4), (C2×C8).210D6, (C2×Q8).40D6, C12⋊D4.2C2, C6.69(C4○D8), C2.D24⋊29C2, C12.121(C2×D4), C6.32(C2×SD16), C2.18(S3×SD16), C4.32(C4○D12), C12.19(C4○D4), C6.23(C4⋊D4), C12.Q8⋊11C2, (C2×Dic3).94D4, (C22×S3).50D4, C22.197(S3×D4), (C6×Q8).30C22, C2.8(D24⋊C2), C2.26(Dic3⋊D4), (C2×C24).243C22, (C2×C12).247C23, (C2×D12).63C22, C4⋊Dic3.94C22, (S3×C2×C8)⋊21C2, (C2×Q8⋊2S3)⋊3C2, (C2×C6).260(C2×D4), (C3×Q8⋊C4)⋊26C2, (C3×C4⋊C4).48C22, (C2×C3⋊C8).220C22, (S3×C2×C4).228C22, (C2×C4).354(C22×S3), SmallGroup(192,366)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊2SD16
G = < a,b,c,d | a6=b2=c8=d2=1, bab=cac-1=dad=a-1, cbc-1=a4b, dbd=ab, dcd=c3 >
Subgroups: 408 in 124 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, S3×C8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, D6⋊C4, Q8⋊2S3, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, C2×D12, C6×Q8, C8⋊8D4, C12.Q8, C2.D24, C3×Q8⋊C4, C12⋊D4, S3×C2×C8, C2×Q8⋊2S3, D6⋊3Q8, D6⋊2SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C22×S3, C4⋊D4, C2×SD16, C4○D8, C4○D12, S3×D4, C8⋊8D4, Dic3⋊D4, S3×SD16, D24⋊C2, D6⋊2SD16
►On 96 points
(1 85 45 77 23 60)(2 61 24 78 46 86)(3 87 47 79 17 62)(4 63 18 80 48 88)(5 81 41 73 19 64)(6 57 20 74 42 82)(7 83 43 75 21 58)(8 59 22 76 44 84)(9 53 93 67 39 32)(10 25 40 68 94 54)(11 55 95 69 33 26)(12 27 34 70 96 56)(13 49 89 71 35 28)(14 29 36 72 90 50)(15 51 91 65 37 30)(16 31 38 66 92 52)
(1 33)(2 12)(3 35)(4 14)(5 37)(6 16)(7 39)(8 10)(9 21)(11 23)(13 17)(15 19)(18 90)(20 92)(22 94)(24 96)(25 84)(26 60)(27 86)(28 62)(29 88)(30 64)(31 82)(32 58)(34 46)(36 48)(38 42)(40 44)(41 91)(43 93)(45 95)(47 89)(49 79)(50 63)(51 73)(52 57)(53 75)(54 59)(55 77)(56 61)(65 81)(66 74)(67 83)(68 76)(69 85)(70 78)(71 87)(72 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 5)(2 8)(4 6)(9 53)(10 56)(11 51)(12 54)(13 49)(14 52)(15 55)(16 50)(17 47)(18 42)(19 45)(20 48)(21 43)(22 46)(23 41)(24 44)(25 96)(26 91)(27 94)(28 89)(29 92)(30 95)(31 90)(32 93)(33 65)(34 68)(35 71)(36 66)(37 69)(38 72)(39 67)(40 70)(57 88)(58 83)(59 86)(60 81)(61 84)(62 87)(63 82)(64 85)(73 77)(74 80)(76 78)
G:=sub<Sym(96)| (1,85,45,77,23,60)(2,61,24,78,46,86)(3,87,47,79,17,62)(4,63,18,80,48,88)(5,81,41,73,19,64)(6,57,20,74,42,82)(7,83,43,75,21,58)(8,59,22,76,44,84)(9,53,93,67,39,32)(10,25,40,68,94,54)(11,55,95,69,33,26)(12,27,34,70,96,56)(13,49,89,71,35,28)(14,29,36,72,90,50)(15,51,91,65,37,30)(16,31,38,66,92,52), (1,33)(2,12)(3,35)(4,14)(5,37)(6,16)(7,39)(8,10)(9,21)(11,23)(13,17)(15,19)(18,90)(20,92)(22,94)(24,96)(25,84)(26,60)(27,86)(28,62)(29,88)(30,64)(31,82)(32,58)(34,46)(36,48)(38,42)(40,44)(41,91)(43,93)(45,95)(47,89)(49,79)(50,63)(51,73)(52,57)(53,75)(54,59)(55,77)(56,61)(65,81)(66,74)(67,83)(68,76)(69,85)(70,78)(71,87)(72,80), (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), (1,5)(2,8)(4,6)(9,53)(10,56)(11,51)(12,54)(13,49)(14,52)(15,55)(16,50)(17,47)(18,42)(19,45)(20,48)(21,43)(22,46)(23,41)(24,44)(25,96)(26,91)(27,94)(28,89)(29,92)(30,95)(31,90)(32,93)(33,65)(34,68)(35,71)(36,66)(37,69)(38,72)(39,67)(40,70)(57,88)(58,83)(59,86)(60,81)(61,84)(62,87)(63,82)(64,85)(73,77)(74,80)(76,78)>;
G:=Group( (1,85,45,77,23,60)(2,61,24,78,46,86)(3,87,47,79,17,62)(4,63,18,80,48,88)(5,81,41,73,19,64)(6,57,20,74,42,82)(7,83,43,75,21,58)(8,59,22,76,44,84)(9,53,93,67,39,32)(10,25,40,68,94,54)(11,55,95,69,33,26)(12,27,34,70,96,56)(13,49,89,71,35,28)(14,29,36,72,90,50)(15,51,91,65,37,30)(16,31,38,66,92,52), (1,33)(2,12)(3,35)(4,14)(5,37)(6,16)(7,39)(8,10)(9,21)(11,23)(13,17)(15,19)(18,90)(20,92)(22,94)(24,96)(25,84)(26,60)(27,86)(28,62)(29,88)(30,64)(31,82)(32,58)(34,46)(36,48)(38,42)(40,44)(41,91)(43,93)(45,95)(47,89)(49,79)(50,63)(51,73)(52,57)(53,75)(54,59)(55,77)(56,61)(65,81)(66,74)(67,83)(68,76)(69,85)(70,78)(71,87)(72,80), (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), (1,5)(2,8)(4,6)(9,53)(10,56)(11,51)(12,54)(13,49)(14,52)(15,55)(16,50)(17,47)(18,42)(19,45)(20,48)(21,43)(22,46)(23,41)(24,44)(25,96)(26,91)(27,94)(28,89)(29,92)(30,95)(31,90)(32,93)(33,65)(34,68)(35,71)(36,66)(37,69)(38,72)(39,67)(40,70)(57,88)(58,83)(59,86)(60,81)(61,84)(62,87)(63,82)(64,85)(73,77)(74,80)(76,78) );
G=PermutationGroup([[(1,85,45,77,23,60),(2,61,24,78,46,86),(3,87,47,79,17,62),(4,63,18,80,48,88),(5,81,41,73,19,64),(6,57,20,74,42,82),(7,83,43,75,21,58),(8,59,22,76,44,84),(9,53,93,67,39,32),(10,25,40,68,94,54),(11,55,95,69,33,26),(12,27,34,70,96,56),(13,49,89,71,35,28),(14,29,36,72,90,50),(15,51,91,65,37,30),(16,31,38,66,92,52)], [(1,33),(2,12),(3,35),(4,14),(5,37),(6,16),(7,39),(8,10),(9,21),(11,23),(13,17),(15,19),(18,90),(20,92),(22,94),(24,96),(25,84),(26,60),(27,86),(28,62),(29,88),(30,64),(31,82),(32,58),(34,46),(36,48),(38,42),(40,44),(41,91),(43,93),(45,95),(47,89),(49,79),(50,63),(51,73),(52,57),(53,75),(54,59),(55,77),(56,61),(65,81),(66,74),(67,83),(68,76),(69,85),(70,78),(71,87),(72,80)], [(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)], [(1,5),(2,8),(4,6),(9,53),(10,56),(11,51),(12,54),(13,49),(14,52),(15,55),(16,50),(17,47),(18,42),(19,45),(20,48),(21,43),(22,46),(23,41),(24,44),(25,96),(26,91),(27,94),(28,89),(29,92),(30,95),(31,90),(32,93),(33,65),(34,68),(35,71),(36,66),(37,69),(38,72),(39,67),(40,70),(57,88),(58,83),(59,86),(60,81),(61,84),(62,87),(63,82),(64,85),(73,77),(74,80),(76,78)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 24 | 2 | 2 | 2 | 6 | 6 | 8 | 8 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C4○D4 | SD16 | C4○D8 | C4○D12 | S3×D4 | S3×D4 | S3×SD16 | D24⋊C2 |
| kernel | D6⋊2SD16 | C12.Q8 | C2.D24 | C3×Q8⋊C4 | C12⋊D4 | S3×C2×C8 | C2×Q8⋊2S3 | D6⋊3Q8 | Q8⋊C4 | C3⋊C8 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C2×Q8 | C12 | D6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of D6⋊2SD16 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 61 | 18 | 0 | 0 | 0 | 0 |
| 69 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 67 | 67 | 0 | 0 |
| 0 | 0 | 6 | 67 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 48 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,0,72,72],[61,69,0,0,0,0,18,0,0,0,0,0,0,0,67,6,0,0,0,0,67,67,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,48,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
D6⋊2SD16 in GAP, Magma, Sage, TeX
D_6\rtimes_2{\rm SD}_{16} % in TeX
G:=Group("D6:2SD16"); // GroupNames label
G:=SmallGroup(192,366);
// by ID
G=gap.SmallGroup(192,366);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,219,184,297,136,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^4*b,d*b*d=a*b,d*c*d=c^3>;
// generators/relations