metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.1Q16, D6⋊C8.3C2, (C2×C8).18D6, C4⋊C4.152D6, Q8⋊C4⋊6S3, C24⋊1C4⋊10C2, C6.Q16⋊9C2, (C2×Q8).41D6, C2.10(S3×Q16), C6.18(C2×Q16), Q8⋊2Dic3⋊8C2, D6⋊3Q8.2C2, C4.56(C4○D12), C2.17(Q8⋊3D6), C6.63(C8⋊C22), (C2×C24).18C22, (C2×Dic3).32D4, (C22×S3).79D4, C22.201(S3×D4), C12.162(C4○D4), (C6×Q8).34C22, C4.87(D4⋊2S3), (C2×C12).251C23, C3⋊2(C23.48D4), C4⋊Dic3.95C22, C2.17(C23.9D6), C6.25(C22.D4), (S3×C4⋊C4).3C2, (C3×Q8⋊C4)⋊6C2, (C2×C6).264(C2×D4), (C2×C3⋊C8).42C22, (S3×C2×C4).24C22, (C3×C4⋊C4).52C22, (C2×C4).358(C22×S3), SmallGroup(192,370)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.Q16
G = < a,b,c,d | a6=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=a3c-1 >
Subgroups: 296 in 104 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22⋊C8, Q8⋊C4, Q8⋊C4, C2.D8, C2×C4⋊C4, C22⋊Q8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, S3×C2×C4, C6×Q8, C23.48D4, C6.Q16, C24⋊1C4, D6⋊C8, Q8⋊2Dic3, C3×Q8⋊C4, S3×C4⋊C4, D6⋊3Q8, D6.Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C4○D4, C22×S3, C22.D4, C2×Q16, C8⋊C22, C4○D12, S3×D4, D4⋊2S3, C23.48D4, C23.9D6, Q8⋊3D6, S3×Q16, D6.Q16
►On 96 points
(1 75 93 23 39 61)(2 76 94 24 40 62)(3 77 95 17 33 63)(4 78 96 18 34 64)(5 79 89 19 35 57)(6 80 90 20 36 58)(7 73 91 21 37 59)(8 74 92 22 38 60)(9 47 55 32 67 83)(10 48 56 25 68 84)(11 41 49 26 69 85)(12 42 50 27 70 86)(13 43 51 28 71 87)(14 44 52 29 72 88)(15 45 53 30 65 81)(16 46 54 31 66 82)
(1 57)(2 90)(3 59)(4 92)(5 61)(6 94)(7 63)(8 96)(9 51)(10 88)(11 53)(12 82)(13 55)(14 84)(15 49)(16 86)(17 91)(18 60)(19 93)(20 62)(21 95)(22 64)(23 89)(24 58)(25 52)(26 81)(27 54)(28 83)(29 56)(30 85)(31 50)(32 87)(33 73)(34 38)(35 75)(36 40)(37 77)(39 79)(41 45)(42 66)(43 47)(44 68)(46 70)(48 72)(65 69)(67 71)(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 32 5 28)(2 16 6 12)(3 30 7 26)(4 14 8 10)(9 19 13 23)(11 17 15 21)(18 29 22 25)(20 27 24 31)(33 45 37 41)(34 72 38 68)(35 43 39 47)(36 70 40 66)(42 76 46 80)(44 74 48 78)(49 63 53 59)(50 94 54 90)(51 61 55 57)(52 92 56 96)(58 86 62 82)(60 84 64 88)(65 73 69 77)(67 79 71 75)(81 91 85 95)(83 89 87 93)
G:=sub<Sym(96)| (1,75,93,23,39,61)(2,76,94,24,40,62)(3,77,95,17,33,63)(4,78,96,18,34,64)(5,79,89,19,35,57)(6,80,90,20,36,58)(7,73,91,21,37,59)(8,74,92,22,38,60)(9,47,55,32,67,83)(10,48,56,25,68,84)(11,41,49,26,69,85)(12,42,50,27,70,86)(13,43,51,28,71,87)(14,44,52,29,72,88)(15,45,53,30,65,81)(16,46,54,31,66,82), (1,57)(2,90)(3,59)(4,92)(5,61)(6,94)(7,63)(8,96)(9,51)(10,88)(11,53)(12,82)(13,55)(14,84)(15,49)(16,86)(17,91)(18,60)(19,93)(20,62)(21,95)(22,64)(23,89)(24,58)(25,52)(26,81)(27,54)(28,83)(29,56)(30,85)(31,50)(32,87)(33,73)(34,38)(35,75)(36,40)(37,77)(39,79)(41,45)(42,66)(43,47)(44,68)(46,70)(48,72)(65,69)(67,71)(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), (1,32,5,28)(2,16,6,12)(3,30,7,26)(4,14,8,10)(9,19,13,23)(11,17,15,21)(18,29,22,25)(20,27,24,31)(33,45,37,41)(34,72,38,68)(35,43,39,47)(36,70,40,66)(42,76,46,80)(44,74,48,78)(49,63,53,59)(50,94,54,90)(51,61,55,57)(52,92,56,96)(58,86,62,82)(60,84,64,88)(65,73,69,77)(67,79,71,75)(81,91,85,95)(83,89,87,93)>;
G:=Group( (1,75,93,23,39,61)(2,76,94,24,40,62)(3,77,95,17,33,63)(4,78,96,18,34,64)(5,79,89,19,35,57)(6,80,90,20,36,58)(7,73,91,21,37,59)(8,74,92,22,38,60)(9,47,55,32,67,83)(10,48,56,25,68,84)(11,41,49,26,69,85)(12,42,50,27,70,86)(13,43,51,28,71,87)(14,44,52,29,72,88)(15,45,53,30,65,81)(16,46,54,31,66,82), (1,57)(2,90)(3,59)(4,92)(5,61)(6,94)(7,63)(8,96)(9,51)(10,88)(11,53)(12,82)(13,55)(14,84)(15,49)(16,86)(17,91)(18,60)(19,93)(20,62)(21,95)(22,64)(23,89)(24,58)(25,52)(26,81)(27,54)(28,83)(29,56)(30,85)(31,50)(32,87)(33,73)(34,38)(35,75)(36,40)(37,77)(39,79)(41,45)(42,66)(43,47)(44,68)(46,70)(48,72)(65,69)(67,71)(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), (1,32,5,28)(2,16,6,12)(3,30,7,26)(4,14,8,10)(9,19,13,23)(11,17,15,21)(18,29,22,25)(20,27,24,31)(33,45,37,41)(34,72,38,68)(35,43,39,47)(36,70,40,66)(42,76,46,80)(44,74,48,78)(49,63,53,59)(50,94,54,90)(51,61,55,57)(52,92,56,96)(58,86,62,82)(60,84,64,88)(65,73,69,77)(67,79,71,75)(81,91,85,95)(83,89,87,93) );
G=PermutationGroup([[(1,75,93,23,39,61),(2,76,94,24,40,62),(3,77,95,17,33,63),(4,78,96,18,34,64),(5,79,89,19,35,57),(6,80,90,20,36,58),(7,73,91,21,37,59),(8,74,92,22,38,60),(9,47,55,32,67,83),(10,48,56,25,68,84),(11,41,49,26,69,85),(12,42,50,27,70,86),(13,43,51,28,71,87),(14,44,52,29,72,88),(15,45,53,30,65,81),(16,46,54,31,66,82)], [(1,57),(2,90),(3,59),(4,92),(5,61),(6,94),(7,63),(8,96),(9,51),(10,88),(11,53),(12,82),(13,55),(14,84),(15,49),(16,86),(17,91),(18,60),(19,93),(20,62),(21,95),(22,64),(23,89),(24,58),(25,52),(26,81),(27,54),(28,83),(29,56),(30,85),(31,50),(32,87),(33,73),(34,38),(35,75),(36,40),(37,77),(39,79),(41,45),(42,66),(43,47),(44,68),(46,70),(48,72),(65,69),(67,71),(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)], [(1,32,5,28),(2,16,6,12),(3,30,7,26),(4,14,8,10),(9,19,13,23),(11,17,15,21),(18,29,22,25),(20,27,24,31),(33,45,37,41),(34,72,38,68),(35,43,39,47),(36,70,40,66),(42,76,46,80),(44,74,48,78),(49,63,53,59),(50,94,54,90),(51,61,55,57),(52,92,56,96),(58,86,62,82),(60,84,64,88),(65,73,69,77),(67,79,71,75),(81,91,85,95),(83,89,87,93)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 8 | 12 | 12 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | Q16 | C4○D12 | C8⋊C22 | D4⋊2S3 | S3×D4 | Q8⋊3D6 | S3×Q16 |
| kernel | D6.Q16 | C6.Q16 | C24⋊1C4 | D6⋊C8 | Q8⋊2Dic3 | C3×Q8⋊C4 | S3×C4⋊C4 | D6⋊3Q8 | Q8⋊C4 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C2×Q8 | C12 | D6 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D6.Q16 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 1 |
| 57 | 16 | 0 | 0 |
| 57 | 57 | 0 | 0 |
| 0 | 0 | 7 | 59 |
| 0 | 0 | 14 | 66 |
| 41 | 17 | 0 | 0 |
| 17 | 32 | 0 | 0 |
| 0 | 0 | 43 | 60 |
| 0 | 0 | 13 | 30 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,1],[72,0,0,0,0,72,0,0,0,0,72,0,0,0,1,1],[57,57,0,0,16,57,0,0,0,0,7,14,0,0,59,66],[41,17,0,0,17,32,0,0,0,0,43,13,0,0,60,30] >;
D6.Q16 in GAP, Magma, Sage, TeX
D_6.Q_{16} % in TeX
G:=Group("D6.Q16"); // GroupNames label
G:=SmallGroup(192,370);
// by ID
G=gap.SmallGroup(192,370);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,926,219,184,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^-1>;
// generators/relations