metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6:5Q16, Dic6.17D4, D6:C8.9C2, (C2xQ16):3S3, (C2xC8).38D6, C4.67(S3xD4), (C3xQ8).9D4, (C6xQ16):13C2, C12.52(C2xD4), (C2xQ8).86D6, C6.29(C2xQ16), C2.18(S3xQ16), C6.61C22wrC2, D6:3Q8.7C2, C3:4(C22:Q16), Q8:2Dic3:33C2, C2.Dic12:18C2, (C2xDic3).76D4, Q8.15(C3:D4), (C22xS3).93D4, C22.277(S3xD4), (C6xQ8).89C22, C2.29(C23:2D6), (C2xC12).460C23, (C2xC24).180C22, C2.28(Q16:S3), C6.78(C8.C22), C4:Dic3.183C22, (C2xDic6).132C22, (C2xS3xQ8).5C2, C4.48(C2xC3:D4), (C2xC3:Q16):20C2, (C2xC6).371(C2xD4), (S3xC2xC4).52C22, (C2xC3:C8).165C22, (C2xC4).548(C22xS3), SmallGroup(192,745)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6:5Q16
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=c-1 >
Subgroups: 408 in 148 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2xC4, C2xC4, Q8, Q8, C23, Dic3, C12, C12, D6, D6, C2xC6, C22:C4, C4:C4, C2xC8, C2xC8, Q16, C22xC4, C2xQ8, C2xQ8, C3:C8, C24, Dic6, Dic6, C4xS3, C2xDic3, C2xDic3, C2xC12, C2xC12, C3xQ8, C3xQ8, C22xS3, C22:C8, Q8:C4, C22:Q8, C2xQ16, C2xQ16, C22xQ8, C2xC3:C8, Dic3:C4, C4:Dic3, D6:C4, C3:Q16, C2xC24, C3xQ16, C2xDic6, C2xDic6, S3xC2xC4, S3xC2xC4, S3xQ8, C6xQ8, C22:Q16, C2.Dic12, D6:C8, Q8:2Dic3, C2xC3:Q16, D6:3Q8, C6xQ16, C2xS3xQ8, D6:5Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2xD4, C3:D4, C22xS3, C22wrC2, C2xQ16, C8.C22, S3xD4, C2xC3:D4, C22:Q16, S3xQ16, Q16:S3, C23:2D6, D6:5Q16
►On 96 points
(1 49 31 87 9 74)(2 50 32 88 10 75)(3 51 25 81 11 76)(4 52 26 82 12 77)(5 53 27 83 13 78)(6 54 28 84 14 79)(7 55 29 85 15 80)(8 56 30 86 16 73)(17 71 91 60 47 36)(18 72 92 61 48 37)(19 65 93 62 41 38)(20 66 94 63 42 39)(21 67 95 64 43 40)(22 68 96 57 44 33)(23 69 89 58 45 34)(24 70 90 59 46 35)
(1 78)(2 28)(3 80)(4 30)(5 74)(6 32)(7 76)(8 26)(9 53)(10 14)(11 55)(12 16)(13 49)(15 51)(17 95)(18 33)(19 89)(20 35)(21 91)(22 37)(23 93)(24 39)(25 85)(27 87)(29 81)(31 83)(34 62)(36 64)(38 58)(40 60)(41 45)(42 70)(43 47)(44 72)(46 66)(48 68)(50 54)(52 56)(57 92)(59 94)(61 96)(63 90)(65 69)(67 71)(73 82)(75 84)(77 86)(79 88)
(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 64 5 60)(2 63 6 59)(3 62 7 58)(4 61 8 57)(9 67 13 71)(10 66 14 70)(11 65 15 69)(12 72 16 68)(17 87 21 83)(18 86 22 82)(19 85 23 81)(20 84 24 88)(25 38 29 34)(26 37 30 33)(27 36 31 40)(28 35 32 39)(41 55 45 51)(42 54 46 50)(43 53 47 49)(44 52 48 56)(73 96 77 92)(74 95 78 91)(75 94 79 90)(76 93 80 89)
G:=sub<Sym(96)| (1,49,31,87,9,74)(2,50,32,88,10,75)(3,51,25,81,11,76)(4,52,26,82,12,77)(5,53,27,83,13,78)(6,54,28,84,14,79)(7,55,29,85,15,80)(8,56,30,86,16,73)(17,71,91,60,47,36)(18,72,92,61,48,37)(19,65,93,62,41,38)(20,66,94,63,42,39)(21,67,95,64,43,40)(22,68,96,57,44,33)(23,69,89,58,45,34)(24,70,90,59,46,35), (1,78)(2,28)(3,80)(4,30)(5,74)(6,32)(7,76)(8,26)(9,53)(10,14)(11,55)(12,16)(13,49)(15,51)(17,95)(18,33)(19,89)(20,35)(21,91)(22,37)(23,93)(24,39)(25,85)(27,87)(29,81)(31,83)(34,62)(36,64)(38,58)(40,60)(41,45)(42,70)(43,47)(44,72)(46,66)(48,68)(50,54)(52,56)(57,92)(59,94)(61,96)(63,90)(65,69)(67,71)(73,82)(75,84)(77,86)(79,88), (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), (1,64,5,60)(2,63,6,59)(3,62,7,58)(4,61,8,57)(9,67,13,71)(10,66,14,70)(11,65,15,69)(12,72,16,68)(17,87,21,83)(18,86,22,82)(19,85,23,81)(20,84,24,88)(25,38,29,34)(26,37,30,33)(27,36,31,40)(28,35,32,39)(41,55,45,51)(42,54,46,50)(43,53,47,49)(44,52,48,56)(73,96,77,92)(74,95,78,91)(75,94,79,90)(76,93,80,89)>;
G:=Group( (1,49,31,87,9,74)(2,50,32,88,10,75)(3,51,25,81,11,76)(4,52,26,82,12,77)(5,53,27,83,13,78)(6,54,28,84,14,79)(7,55,29,85,15,80)(8,56,30,86,16,73)(17,71,91,60,47,36)(18,72,92,61,48,37)(19,65,93,62,41,38)(20,66,94,63,42,39)(21,67,95,64,43,40)(22,68,96,57,44,33)(23,69,89,58,45,34)(24,70,90,59,46,35), (1,78)(2,28)(3,80)(4,30)(5,74)(6,32)(7,76)(8,26)(9,53)(10,14)(11,55)(12,16)(13,49)(15,51)(17,95)(18,33)(19,89)(20,35)(21,91)(22,37)(23,93)(24,39)(25,85)(27,87)(29,81)(31,83)(34,62)(36,64)(38,58)(40,60)(41,45)(42,70)(43,47)(44,72)(46,66)(48,68)(50,54)(52,56)(57,92)(59,94)(61,96)(63,90)(65,69)(67,71)(73,82)(75,84)(77,86)(79,88), (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), (1,64,5,60)(2,63,6,59)(3,62,7,58)(4,61,8,57)(9,67,13,71)(10,66,14,70)(11,65,15,69)(12,72,16,68)(17,87,21,83)(18,86,22,82)(19,85,23,81)(20,84,24,88)(25,38,29,34)(26,37,30,33)(27,36,31,40)(28,35,32,39)(41,55,45,51)(42,54,46,50)(43,53,47,49)(44,52,48,56)(73,96,77,92)(74,95,78,91)(75,94,79,90)(76,93,80,89) );
G=PermutationGroup([[(1,49,31,87,9,74),(2,50,32,88,10,75),(3,51,25,81,11,76),(4,52,26,82,12,77),(5,53,27,83,13,78),(6,54,28,84,14,79),(7,55,29,85,15,80),(8,56,30,86,16,73),(17,71,91,60,47,36),(18,72,92,61,48,37),(19,65,93,62,41,38),(20,66,94,63,42,39),(21,67,95,64,43,40),(22,68,96,57,44,33),(23,69,89,58,45,34),(24,70,90,59,46,35)], [(1,78),(2,28),(3,80),(4,30),(5,74),(6,32),(7,76),(8,26),(9,53),(10,14),(11,55),(12,16),(13,49),(15,51),(17,95),(18,33),(19,89),(20,35),(21,91),(22,37),(23,93),(24,39),(25,85),(27,87),(29,81),(31,83),(34,62),(36,64),(38,58),(40,60),(41,45),(42,70),(43,47),(44,72),(46,66),(48,68),(50,54),(52,56),(57,92),(59,94),(61,96),(63,90),(65,69),(67,71),(73,82),(75,84),(77,86),(79,88)], [(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)], [(1,64,5,60),(2,63,6,59),(3,62,7,58),(4,61,8,57),(9,67,13,71),(10,66,14,70),(11,65,15,69),(12,72,16,68),(17,87,21,83),(18,86,22,82),(19,85,23,81),(20,84,24,88),(25,38,29,34),(26,37,30,33),(27,36,31,40),(28,35,32,39),(41,55,45,51),(42,54,46,50),(43,53,47,49),(44,52,48,56),(73,96,77,92),(74,95,78,91),(75,94,79,90),(76,93,80,89)]])
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 | D4 | D4 | D6 | D6 | Q16 | C3:D4 | C8.C22 | S3xD4 | S3xD4 | S3xQ16 | Q16:S3 |
| kernel | D6:5Q16 | C2.Dic12 | D6:C8 | Q8:2Dic3 | C2xC3:Q16 | D6:3Q8 | C6xQ16 | C2xS3xQ8 | C2xQ16 | Dic6 | C2xDic3 | C3xQ8 | C22xS3 | C2xC8 | C2xQ8 | D6 | Q8 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D6:5Q16 ►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 |
| 0 | 41 | 0 | 0 |
| 16 | 41 | 0 | 0 |
| 0 | 0 | 30 | 13 |
| 0 | 0 | 60 | 43 |
| 6 | 14 | 0 | 0 |
| 13 | 67 | 0 | 0 |
| 0 | 0 | 30 | 13 |
| 0 | 0 | 60 | 43 |
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],[0,16,0,0,41,41,0,0,0,0,30,60,0,0,13,43],[6,13,0,0,14,67,0,0,0,0,30,60,0,0,13,43] >;
D6:5Q16 in GAP, Magma, Sage, TeX
D_6\rtimes_5Q_{16} % in TeX
G:=Group("D6:5Q16"); // GroupNames label
G:=SmallGroup(192,745);
// by ID
G=gap.SmallGroup(192,745);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,184,851,438,102,6278]);
// 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=c^-1>;
// generators/relations