metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6:1Q16, C3:C8.28D4, C4:C4.29D6, Q8:C4:3S3, (C2xC8).211D6, C4.168(S3xD4), (C2xQ8).43D6, C6.19(C2xQ16), C2.11(S3xQ16), C6.49(C4oD8), C6.Q16:11C2, C12.126(C2xD4), C3:1(C8.18D4), C4.D12.2C2, D6:3Q8.3C2, C12.21(C4oD4), C4.34(C4oD12), C6.25(C4:D4), C2.Dic12:29C2, (C2xDic3).95D4, (C22xS3).51D4, C22.203(S3xD4), (C6xQ8).36C22, C2.28(Dic3:D4), (C2xC12).253C23, (C2xC24).245C22, C4:Dic3.97C22, C2.18(Q8.7D6), (C2xDic6).73C22, (S3xC2xC8).12C2, (C2xC3:Q16):5C2, (C2xC6).266(C2xD4), (C3xQ8:C4):28C2, (C3xC4:C4).54C22, (C2xC3:C8).221C22, (S3xC2xC4).229C22, (C2xC4).360(C22xS3), SmallGroup(192,372)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6:1Q16
G = < a,b,c,d | a6=b2=c8=1, d2=c4, bab=cac-1=a-1, ad=da, cbc-1=a4b, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 312 in 114 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2xC4, C2xC4, Q8, C23, Dic3, C12, C12, D6, D6, C2xC6, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, Q16, C22xC4, C2xQ8, C2xQ8, C3:C8, C24, Dic6, C4xS3, C2xDic3, C2xDic3, C2xC12, C2xC12, C3xQ8, C22xS3, Q8:C4, Q8:C4, C2.D8, C22:Q8, C22xC8, C2xQ16, S3xC8, C2xC3:C8, Dic3:C4, C4:Dic3, C4:Dic3, D6:C4, C3:Q16, C3xC4:C4, C2xC24, C2xDic6, S3xC2xC4, C6xQ8, C8.18D4, C6.Q16, C2.Dic12, C3xQ8:C4, C4.D12, S3xC2xC8, C2xC3:Q16, D6:3Q8, D6:1Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2xD4, C4oD4, C22xS3, C4:D4, C2xQ16, C4oD8, C4oD12, S3xD4, C8.18D4, Dic3:D4, Q8.7D6, S3xQ16, D6:1Q16
►On 96 points
(1 93 35 71 21 30)(2 31 22 72 36 94)(3 95 37 65 23 32)(4 25 24 66 38 96)(5 89 39 67 17 26)(6 27 18 68 40 90)(7 91 33 69 19 28)(8 29 20 70 34 92)(9 50 59 83 41 73)(10 74 42 84 60 51)(11 52 61 85 43 75)(12 76 44 86 62 53)(13 54 63 87 45 77)(14 78 46 88 64 55)(15 56 57 81 47 79)(16 80 48 82 58 49)
(1 30)(2 94)(3 32)(4 96)(5 26)(6 90)(7 28)(8 92)(9 41)(11 43)(13 45)(15 47)(17 89)(18 68)(19 91)(20 70)(21 93)(22 72)(23 95)(24 66)(25 38)(27 40)(29 34)(31 36)(33 69)(35 71)(37 65)(39 67)(42 60)(44 62)(46 64)(48 58)(49 80)(50 83)(51 74)(52 85)(53 76)(54 87)(55 78)(56 81)
(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 47 5 43)(2 46 6 42)(3 45 7 41)(4 44 8 48)(9 37 13 33)(10 36 14 40)(11 35 15 39)(12 34 16 38)(17 61 21 57)(18 60 22 64)(19 59 23 63)(20 58 24 62)(25 86 29 82)(26 85 30 81)(27 84 31 88)(28 83 32 87)(49 66 53 70)(50 65 54 69)(51 72 55 68)(52 71 56 67)(73 95 77 91)(74 94 78 90)(75 93 79 89)(76 92 80 96)
G:=sub<Sym(96)| (1,93,35,71,21,30)(2,31,22,72,36,94)(3,95,37,65,23,32)(4,25,24,66,38,96)(5,89,39,67,17,26)(6,27,18,68,40,90)(7,91,33,69,19,28)(8,29,20,70,34,92)(9,50,59,83,41,73)(10,74,42,84,60,51)(11,52,61,85,43,75)(12,76,44,86,62,53)(13,54,63,87,45,77)(14,78,46,88,64,55)(15,56,57,81,47,79)(16,80,48,82,58,49), (1,30)(2,94)(3,32)(4,96)(5,26)(6,90)(7,28)(8,92)(9,41)(11,43)(13,45)(15,47)(17,89)(18,68)(19,91)(20,70)(21,93)(22,72)(23,95)(24,66)(25,38)(27,40)(29,34)(31,36)(33,69)(35,71)(37,65)(39,67)(42,60)(44,62)(46,64)(48,58)(49,80)(50,83)(51,74)(52,85)(53,76)(54,87)(55,78)(56,81), (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), (1,47,5,43)(2,46,6,42)(3,45,7,41)(4,44,8,48)(9,37,13,33)(10,36,14,40)(11,35,15,39)(12,34,16,38)(17,61,21,57)(18,60,22,64)(19,59,23,63)(20,58,24,62)(25,86,29,82)(26,85,30,81)(27,84,31,88)(28,83,32,87)(49,66,53,70)(50,65,54,69)(51,72,55,68)(52,71,56,67)(73,95,77,91)(74,94,78,90)(75,93,79,89)(76,92,80,96)>;
G:=Group( (1,93,35,71,21,30)(2,31,22,72,36,94)(3,95,37,65,23,32)(4,25,24,66,38,96)(5,89,39,67,17,26)(6,27,18,68,40,90)(7,91,33,69,19,28)(8,29,20,70,34,92)(9,50,59,83,41,73)(10,74,42,84,60,51)(11,52,61,85,43,75)(12,76,44,86,62,53)(13,54,63,87,45,77)(14,78,46,88,64,55)(15,56,57,81,47,79)(16,80,48,82,58,49), (1,30)(2,94)(3,32)(4,96)(5,26)(6,90)(7,28)(8,92)(9,41)(11,43)(13,45)(15,47)(17,89)(18,68)(19,91)(20,70)(21,93)(22,72)(23,95)(24,66)(25,38)(27,40)(29,34)(31,36)(33,69)(35,71)(37,65)(39,67)(42,60)(44,62)(46,64)(48,58)(49,80)(50,83)(51,74)(52,85)(53,76)(54,87)(55,78)(56,81), (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), (1,47,5,43)(2,46,6,42)(3,45,7,41)(4,44,8,48)(9,37,13,33)(10,36,14,40)(11,35,15,39)(12,34,16,38)(17,61,21,57)(18,60,22,64)(19,59,23,63)(20,58,24,62)(25,86,29,82)(26,85,30,81)(27,84,31,88)(28,83,32,87)(49,66,53,70)(50,65,54,69)(51,72,55,68)(52,71,56,67)(73,95,77,91)(74,94,78,90)(75,93,79,89)(76,92,80,96) );
G=PermutationGroup([[(1,93,35,71,21,30),(2,31,22,72,36,94),(3,95,37,65,23,32),(4,25,24,66,38,96),(5,89,39,67,17,26),(6,27,18,68,40,90),(7,91,33,69,19,28),(8,29,20,70,34,92),(9,50,59,83,41,73),(10,74,42,84,60,51),(11,52,61,85,43,75),(12,76,44,86,62,53),(13,54,63,87,45,77),(14,78,46,88,64,55),(15,56,57,81,47,79),(16,80,48,82,58,49)], [(1,30),(2,94),(3,32),(4,96),(5,26),(6,90),(7,28),(8,92),(9,41),(11,43),(13,45),(15,47),(17,89),(18,68),(19,91),(20,70),(21,93),(22,72),(23,95),(24,66),(25,38),(27,40),(29,34),(31,36),(33,69),(35,71),(37,65),(39,67),(42,60),(44,62),(46,64),(48,58),(49,80),(50,83),(51,74),(52,85),(53,76),(54,87),(55,78),(56,81)], [(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)], [(1,47,5,43),(2,46,6,42),(3,45,7,41),(4,44,8,48),(9,37,13,33),(10,36,14,40),(11,35,15,39),(12,34,16,38),(17,61,21,57),(18,60,22,64),(19,59,23,63),(20,58,24,62),(25,86,29,82),(26,85,30,81),(27,84,31,88),(28,83,32,87),(49,66,53,70),(50,65,54,69),(51,72,55,68),(52,71,56,67),(73,95,77,91),(74,94,78,90),(75,93,79,89),(76,92,80,96)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 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 | 3 | 4 | 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 | 2 | 2 | 2 | 6 | 6 | 8 | 8 | 24 | 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 | C4oD4 | Q16 | C4oD8 | C4oD12 | S3xD4 | S3xD4 | Q8.7D6 | S3xQ16 |
| kernel | D6:1Q16 | C6.Q16 | C2.Dic12 | C3xQ8:C4 | C4.D12 | S3xC2xC8 | C2xC3:Q16 | D6:3Q8 | Q8:C4 | C3:C8 | C2xDic3 | C22xS3 | C4:C4 | C2xC8 | C2xQ8 | 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 D6:1Q16 ►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 | 72 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 16 | 57 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 57 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 13 | 66 | 0 | 0 | 0 | 0 |
| 66 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 2 | 0 | 0 |
| 0 | 0 | 19 | 67 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,6,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,72,72],[16,16,0,0,0,0,57,16,0,0,0,0,0,0,27,57,0,0,0,0,0,46,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[13,66,0,0,0,0,66,60,0,0,0,0,0,0,6,19,0,0,0,0,2,67,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;
D6:1Q16 in GAP, Magma, Sage, TeX
D_6\rtimes_1Q_{16} % in TeX
G:=Group("D6:1Q16"); // GroupNames label
G:=SmallGroup(192,372);
// by ID
G=gap.SmallGroup(192,372);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,555,184,297,136,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=1,d^2=c^4,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^4*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations