metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8:8D6, Q16:7D6, C24.32D4, C12.51D8, C24.29C23, D24.11C22, C4oD8:2S3, (C2xC6).9D8, C3:D16:6C2, C3:C16:4C22, C6.68(C2xD8), (C2xC8).98D6, (C2xD24):22C2, C3:5(C16:C22), C8.6D6:6C2, (C3xD8):8C22, C8.7(C3:D4), C12.C8:6C2, C4.24(D4:S3), (C2xC12).185D4, C12.191(C2xD4), C8.35(C22xS3), (C3xQ16):7C22, C22.5(D4:S3), (C2xC24).104C22, (C3xC4oD8):4C2, C2.23(C2xD4:S3), C4.17(C2xC3:D4), (C2xC4).81(C3:D4), SmallGroup(192,752)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16:D6
G = < a,b,c,d | a8=c6=d2=1, b2=a4, bab-1=dad=a-1, ac=ca, cbc-1=a4b, dbd=a3b, dcd=c-1 >
Subgroups: 360 in 90 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C23, C12, C12, D6, C2xC6, C2xC6, C16, C2xC8, D8, D8, SD16, Q16, C2xD4, C4oD4, C24, D12, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, M5(2), D16, SD32, C2xD8, C4oD8, C3:C16, D24, D24, C2xC24, C3xD8, C3xSD16, C3xQ16, C2xD12, C3xC4oD4, C16:C22, C12.C8, C3:D16, C8.6D6, C2xD24, C3xC4oD8, Q16:D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C3:D4, C22xS3, C2xD8, D4:S3, C2xC3:D4, C16:C22, C2xD4:S3, Q16:D6
Character table of Q16:D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 12A | 12B | 12C | 12D | 12E | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 8 | 24 | 24 | 2 | 2 | 2 | 8 | 2 | 4 | 8 | 8 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 8 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | -1 | -1 | 1 | 1 | 2 | 2 | 2 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | 2 | 0 | 0 | -1 | -2 | 2 | -2 | -1 | 1 | -1 | -1 | 2 | 2 | -2 | 1 | 1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 2 | -1 | 1 | 1 | 1 | 2 | 2 | -2 | 1 | 1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | -√2 | √2 | -√2 | √2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ16 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | √2 | -√2 | √2 | -√2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ17 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | -√2 | -√2 | √2 | √2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ18 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | √2 | √2 | -√2 | -√2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ19 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -2 | 2 | 0 | -1 | 1 | √-3 | -√-3 | -2 | -2 | 2 | 1 | 1 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | complex lifted from C3:D4 |
| ρ20 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | -√-3 | √-3 | -2 | -2 | -2 | -1 | -1 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | complex lifted from C3:D4 |
| ρ21 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -2 | 2 | 0 | -1 | 1 | -√-3 | √-3 | -2 | -2 | 2 | 1 | 1 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | complex lifted from C3:D4 |
| ρ22 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | √-3 | -√-3 | -2 | -2 | -2 | -1 | -1 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | complex lifted from C3:D4 |
| ρ23 | 4 | 4 | 4 | 0 | 0 | 0 | -2 | -4 | -4 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4:S3, Schur index 2 |
| ρ24 | 4 | 4 | -4 | 0 | 0 | 0 | -2 | 4 | -4 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4:S3, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | orthogonal lifted from C16:C22 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | -2√2 | orthogonal lifted from C16:C22 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | -2√3 | 2√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | √2 | -√6 | -√2 | orthogonal faithful |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 2√3 | -2√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | √2 | √6 | -√2 | orthogonal faithful |
| ρ29 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 2√3 | -2√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | -√2 | -√6 | √2 | orthogonal faithful |
| ρ30 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | -2√3 | 2√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | -√2 | √6 | √2 | orthogonal faithful |
►On 48 points
Generators in S48
(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)
(1 13 5 9)(2 12 6 16)(3 11 7 15)(4 10 8 14)(17 28 21 32)(18 27 22 31)(19 26 23 30)(20 25 24 29)(33 47 37 43)(34 46 38 42)(35 45 39 41)(36 44 40 48)
(1 38 25)(2 39 26)(3 40 27)(4 33 28)(5 34 29)(6 35 30)(7 36 31)(8 37 32)(9 42 20 13 46 24)(10 43 21 14 47 17)(11 44 22 15 48 18)(12 45 23 16 41 19)
(1 25)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 17)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(33 35)(36 40)(37 39)(41 48)(42 47)(43 46)(44 45)
G:=sub<Sym(48)| (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), (1,13,5,9)(2,12,6,16)(3,11,7,15)(4,10,8,14)(17,28,21,32)(18,27,22,31)(19,26,23,30)(20,25,24,29)(33,47,37,43)(34,46,38,42)(35,45,39,41)(36,44,40,48), (1,38,25)(2,39,26)(3,40,27)(4,33,28)(5,34,29)(6,35,30)(7,36,31)(8,37,32)(9,42,20,13,46,24)(10,43,21,14,47,17)(11,44,22,15,48,18)(12,45,23,16,41,19), (1,25)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,17)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(33,35)(36,40)(37,39)(41,48)(42,47)(43,46)(44,45)>;
G:=Group( (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), (1,13,5,9)(2,12,6,16)(3,11,7,15)(4,10,8,14)(17,28,21,32)(18,27,22,31)(19,26,23,30)(20,25,24,29)(33,47,37,43)(34,46,38,42)(35,45,39,41)(36,44,40,48), (1,38,25)(2,39,26)(3,40,27)(4,33,28)(5,34,29)(6,35,30)(7,36,31)(8,37,32)(9,42,20,13,46,24)(10,43,21,14,47,17)(11,44,22,15,48,18)(12,45,23,16,41,19), (1,25)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,17)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(33,35)(36,40)(37,39)(41,48)(42,47)(43,46)(44,45) );
G=PermutationGroup([[(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)], [(1,13,5,9),(2,12,6,16),(3,11,7,15),(4,10,8,14),(17,28,21,32),(18,27,22,31),(19,26,23,30),(20,25,24,29),(33,47,37,43),(34,46,38,42),(35,45,39,41),(36,44,40,48)], [(1,38,25),(2,39,26),(3,40,27),(4,33,28),(5,34,29),(6,35,30),(7,36,31),(8,37,32),(9,42,20,13,46,24),(10,43,21,14,47,17),(11,44,22,15,48,18),(12,45,23,16,41,19)], [(1,25),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,17),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(33,35),(36,40),(37,39),(41,48),(42,47),(43,46),(44,45)]])
Matrix representation of Q16:D6 ►in GL4(F97) generated by
| 16 | 18 | 0 | 0 |
| 79 | 95 | 0 | 0 |
| 0 | 0 | 95 | 79 |
| 0 | 0 | 18 | 16 |
| 0 | 0 | 96 | 0 |
| 0 | 0 | 0 | 96 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 96 | 96 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 96 | 0 |
| 96 | 96 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 95 | 16 |
| 0 | 0 | 18 | 2 |
G:=sub<GL(4,GF(97))| [16,79,0,0,18,95,0,0,0,0,95,18,0,0,79,16],[0,0,1,0,0,0,0,1,96,0,0,0,0,96,0,0],[96,1,0,0,96,0,0,0,0,0,1,96,0,0,1,0],[96,0,0,0,96,1,0,0,0,0,95,18,0,0,16,2] >;
Q16:D6 in GAP, Magma, Sage, TeX
Q_{16}\rtimes D_6 % in TeX
G:=Group("Q16:D6"); // GroupNames label
G:=SmallGroup(192,752);
// by ID
G=gap.SmallGroup(192,752);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,387,675,185,192,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^6=d^2=1,b^2=a^4,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations
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