metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊1SD16, C3⋊C8⋊20D4, C3⋊1(C8⋊8D4), C4⋊C4.10D6, D4⋊C4⋊3S3, C4.D12⋊2C2, (C2×D4).26D6, C4.159(S3×D4), (C2×C8).203D6, D6⋊3D4.3C2, C12.8(C4○D4), C6.23(C4○D8), C12.109(C2×D4), C12.Q8⋊5C2, C2.13(S3×SD16), C6.25(C2×SD16), C4.25(C4○D12), C2.9(D8⋊3S3), C6.16(C4⋊D4), C2.Dic12⋊25C2, (C2×Dic3).90D4, (C6×D4).39C22, (C22×S3).48D4, C22.176(S3×D4), C2.19(Dic3⋊D4), (C2×C24).228C22, (C2×C12).218C23, C4⋊Dic3.72C22, (C2×Dic6).57C22, (S3×C2×C8)⋊19C2, (C2×D4.S3)⋊4C2, (C2×C6).231(C2×D4), (C3×D4⋊C4)⋊28C2, (C3×C4⋊C4).19C22, (C2×C3⋊C8).213C22, (S3×C2×C4).225C22, (C2×C4).325(C22×S3), SmallGroup(192,337)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊SD16
G = < a,b,c,d | a6=b2=c8=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a4b, dbd=a3b, dcd=c3 >
Subgroups: 376 in 124 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, S3×C8, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D4.S3, C6.D4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C6×D4, C8⋊8D4, C12.Q8, C2.Dic12, C3×D4⋊C4, C4.D12, S3×C2×C8, C2×D4.S3, D6⋊3D4, D6⋊SD16
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, D8⋊3S3, S3×SD16, D6⋊SD16
►On 96 points
(1 10 78 53 26 85)(2 86 27 54 79 11)(3 12 80 55 28 87)(4 88 29 56 73 13)(5 14 74 49 30 81)(6 82 31 50 75 15)(7 16 76 51 32 83)(8 84 25 52 77 9)(17 65 39 41 64 95)(18 96 57 42 40 66)(19 67 33 43 58 89)(20 90 59 44 34 68)(21 69 35 45 60 91)(22 92 61 46 36 70)(23 71 37 47 62 93)(24 94 63 48 38 72)
(1 70)(2 93)(3 72)(4 95)(5 66)(6 89)(7 68)(8 91)(9 21)(10 36)(11 23)(12 38)(13 17)(14 40)(15 19)(16 34)(18 81)(20 83)(22 85)(24 87)(25 45)(26 92)(27 47)(28 94)(29 41)(30 96)(31 43)(32 90)(33 50)(35 52)(37 54)(39 56)(42 74)(44 76)(46 78)(48 80)(49 57)(51 59)(53 61)(55 63)(58 82)(60 84)(62 86)(64 88)(65 73)(67 75)(69 77)(71 79)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 47)(18 42)(19 45)(20 48)(21 43)(22 46)(23 41)(24 44)(25 31)(27 29)(28 32)(33 91)(34 94)(35 89)(36 92)(37 95)(38 90)(39 93)(40 96)(50 52)(51 55)(54 56)(57 66)(58 69)(59 72)(60 67)(61 70)(62 65)(63 68)(64 71)(73 79)(75 77)(76 80)(82 84)(83 87)(86 88)
G:=sub<Sym(96)| (1,10,78,53,26,85)(2,86,27,54,79,11)(3,12,80,55,28,87)(4,88,29,56,73,13)(5,14,74,49,30,81)(6,82,31,50,75,15)(7,16,76,51,32,83)(8,84,25,52,77,9)(17,65,39,41,64,95)(18,96,57,42,40,66)(19,67,33,43,58,89)(20,90,59,44,34,68)(21,69,35,45,60,91)(22,92,61,46,36,70)(23,71,37,47,62,93)(24,94,63,48,38,72), (1,70)(2,93)(3,72)(4,95)(5,66)(6,89)(7,68)(8,91)(9,21)(10,36)(11,23)(12,38)(13,17)(14,40)(15,19)(16,34)(18,81)(20,83)(22,85)(24,87)(25,45)(26,92)(27,47)(28,94)(29,41)(30,96)(31,43)(32,90)(33,50)(35,52)(37,54)(39,56)(42,74)(44,76)(46,78)(48,80)(49,57)(51,59)(53,61)(55,63)(58,82)(60,84)(62,86)(64,88)(65,73)(67,75)(69,77)(71,79), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,47)(18,42)(19,45)(20,48)(21,43)(22,46)(23,41)(24,44)(25,31)(27,29)(28,32)(33,91)(34,94)(35,89)(36,92)(37,95)(38,90)(39,93)(40,96)(50,52)(51,55)(54,56)(57,66)(58,69)(59,72)(60,67)(61,70)(62,65)(63,68)(64,71)(73,79)(75,77)(76,80)(82,84)(83,87)(86,88)>;
G:=Group( (1,10,78,53,26,85)(2,86,27,54,79,11)(3,12,80,55,28,87)(4,88,29,56,73,13)(5,14,74,49,30,81)(6,82,31,50,75,15)(7,16,76,51,32,83)(8,84,25,52,77,9)(17,65,39,41,64,95)(18,96,57,42,40,66)(19,67,33,43,58,89)(20,90,59,44,34,68)(21,69,35,45,60,91)(22,92,61,46,36,70)(23,71,37,47,62,93)(24,94,63,48,38,72), (1,70)(2,93)(3,72)(4,95)(5,66)(6,89)(7,68)(8,91)(9,21)(10,36)(11,23)(12,38)(13,17)(14,40)(15,19)(16,34)(18,81)(20,83)(22,85)(24,87)(25,45)(26,92)(27,47)(28,94)(29,41)(30,96)(31,43)(32,90)(33,50)(35,52)(37,54)(39,56)(42,74)(44,76)(46,78)(48,80)(49,57)(51,59)(53,61)(55,63)(58,82)(60,84)(62,86)(64,88)(65,73)(67,75)(69,77)(71,79), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,47)(18,42)(19,45)(20,48)(21,43)(22,46)(23,41)(24,44)(25,31)(27,29)(28,32)(33,91)(34,94)(35,89)(36,92)(37,95)(38,90)(39,93)(40,96)(50,52)(51,55)(54,56)(57,66)(58,69)(59,72)(60,67)(61,70)(62,65)(63,68)(64,71)(73,79)(75,77)(76,80)(82,84)(83,87)(86,88) );
G=PermutationGroup([[(1,10,78,53,26,85),(2,86,27,54,79,11),(3,12,80,55,28,87),(4,88,29,56,73,13),(5,14,74,49,30,81),(6,82,31,50,75,15),(7,16,76,51,32,83),(8,84,25,52,77,9),(17,65,39,41,64,95),(18,96,57,42,40,66),(19,67,33,43,58,89),(20,90,59,44,34,68),(21,69,35,45,60,91),(22,92,61,46,36,70),(23,71,37,47,62,93),(24,94,63,48,38,72)], [(1,70),(2,93),(3,72),(4,95),(5,66),(6,89),(7,68),(8,91),(9,21),(10,36),(11,23),(12,38),(13,17),(14,40),(15,19),(16,34),(18,81),(20,83),(22,85),(24,87),(25,45),(26,92),(27,47),(28,94),(29,41),(30,96),(31,43),(32,90),(33,50),(35,52),(37,54),(39,56),(42,74),(44,76),(46,78),(48,80),(49,57),(51,59),(53,61),(55,63),(58,82),(60,84),(62,86),(64,88),(65,73),(67,75),(69,77),(71,79)], [(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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,47),(18,42),(19,45),(20,48),(21,43),(22,46),(23,41),(24,44),(25,31),(27,29),(28,32),(33,91),(34,94),(35,89),(36,92),(37,95),(38,90),(39,93),(40,96),(50,52),(51,55),(54,56),(57,66),(58,69),(59,72),(60,67),(61,70),(62,65),(63,68),(64,71),(73,79),(75,77),(76,80),(82,84),(83,87),(86,88)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 8 | 2 | 2 | 2 | 6 | 6 | 8 | 24 | 24 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 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 | D8⋊3S3 | S3×SD16 |
| kernel | D6⋊SD16 | C12.Q8 | C2.Dic12 | C3×D4⋊C4 | C4.D12 | S3×C2×C8 | C2×D4.S3 | D6⋊3D4 | D4⋊C4 | C3⋊C8 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C2×D4 | 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⋊SD16 ►in GL4(𝔽73) generated by
| 65 | 0 | 0 | 0 |
| 66 | 9 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 57 | 18 | 0 | 0 |
| 71 | 16 | 0 | 0 |
| 0 | 0 | 46 | 34 |
| 0 | 0 | 43 | 27 |
| 48 | 19 | 0 | 0 |
| 17 | 25 | 0 | 0 |
| 0 | 0 | 0 | 33 |
| 0 | 0 | 31 | 61 |
| 1 | 0 | 0 | 0 |
| 18 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 66 | 72 |
G:=sub<GL(4,GF(73))| [65,66,0,0,0,9,0,0,0,0,72,0,0,0,0,72],[57,71,0,0,18,16,0,0,0,0,46,43,0,0,34,27],[48,17,0,0,19,25,0,0,0,0,0,31,0,0,33,61],[1,18,0,0,0,72,0,0,0,0,1,66,0,0,0,72] >;
D6⋊SD16 in GAP, Magma, Sage, TeX
D_6\rtimes {\rm SD}_{16} % in TeX
G:=Group("D6:SD16"); // GroupNames label
G:=SmallGroup(192,337);
// by ID
G=gap.SmallGroup(192,337);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,219,297,136,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^4*b,d*b*d=a^3*b,d*c*d=c^3>;
// generators/relations