metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.3D6, Q16⋊2S3, D6.9D4, Q8.9D6, C12.9C23, C24.10C22, Dic3.11D4, D12.4C22, Dic6.5C22, (S3×Q8)⋊3C2, C24⋊C2⋊4C2, C8⋊S3⋊4C2, (C3×Q16)⋊4C2, C3⋊Q16⋊4C2, C2.23(S3×D4), C6.35(C2×D4), C3⋊C8.2C22, Q8⋊3S3.C2, Q8⋊2S3⋊3C2, C3⋊3(C8.C22), C4.9(C22×S3), (C4×S3).4C22, (C3×Q8).4C22, SmallGroup(96,125)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16⋊S3
G = < a,b,c,d | a8=c3=d2=1, b2=a4, bab-1=a-1, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 146 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, D4, Q8, Q8, Dic3, Dic3, C12, C12, D6, D6, M4(2), SD16, Q16, Q16, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C3×Q8, C8.C22, C8⋊S3, C24⋊C2, Q8⋊2S3, C3⋊Q16, C3×Q16, S3×Q8, Q8⋊3S3, Q16⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8.C22, S3×D4, Q16⋊S3
Character table of Q16⋊S3
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6 | 8A | 8B | 12A | 12B | 12C | 24A | 24B | |
| size | 1 | 1 | 6 | 12 | 2 | 2 | 4 | 4 | 6 | 12 | 2 | 4 | 12 | 4 | 8 | 8 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 0 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 0 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 0 | 0 | -1 | 2 | -2 | 2 | 0 | 0 | -1 | -2 | 0 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 0 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | 0 | -1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | 0 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 4 | 4 | 0 | 0 | -2 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ16 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ17 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -√-6 | √-6 | complex faithful |
| ρ18 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | √-6 | -√-6 | complex faithful |
►On 48 points
(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 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 39 13 35)(10 38 14 34)(11 37 15 33)(12 36 16 40)(25 43 29 47)(26 42 30 46)(27 41 31 45)(28 48 32 44)
(1 13 46)(2 14 47)(3 15 48)(4 16 41)(5 9 42)(6 10 43)(7 11 44)(8 12 45)(17 37 28)(18 38 29)(19 39 30)(20 40 31)(21 33 32)(22 34 25)(23 35 26)(24 36 27)
(1 19)(2 24)(3 21)(4 18)(5 23)(6 20)(7 17)(8 22)(9 26)(10 31)(11 28)(12 25)(13 30)(14 27)(15 32)(16 29)(33 48)(34 45)(35 42)(36 47)(37 44)(38 41)(39 46)(40 43)
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), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,39,13,35)(10,38,14,34)(11,37,15,33)(12,36,16,40)(25,43,29,47)(26,42,30,46)(27,41,31,45)(28,48,32,44), (1,13,46)(2,14,47)(3,15,48)(4,16,41)(5,9,42)(6,10,43)(7,11,44)(8,12,45)(17,37,28)(18,38,29)(19,39,30)(20,40,31)(21,33,32)(22,34,25)(23,35,26)(24,36,27), (1,19)(2,24)(3,21)(4,18)(5,23)(6,20)(7,17)(8,22)(9,26)(10,31)(11,28)(12,25)(13,30)(14,27)(15,32)(16,29)(33,48)(34,45)(35,42)(36,47)(37,44)(38,41)(39,46)(40,43)>;
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), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,39,13,35)(10,38,14,34)(11,37,15,33)(12,36,16,40)(25,43,29,47)(26,42,30,46)(27,41,31,45)(28,48,32,44), (1,13,46)(2,14,47)(3,15,48)(4,16,41)(5,9,42)(6,10,43)(7,11,44)(8,12,45)(17,37,28)(18,38,29)(19,39,30)(20,40,31)(21,33,32)(22,34,25)(23,35,26)(24,36,27), (1,19)(2,24)(3,21)(4,18)(5,23)(6,20)(7,17)(8,22)(9,26)(10,31)(11,28)(12,25)(13,30)(14,27)(15,32)(16,29)(33,48)(34,45)(35,42)(36,47)(37,44)(38,41)(39,46)(40,43) );
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)], [(1,23,5,19),(2,22,6,18),(3,21,7,17),(4,20,8,24),(9,39,13,35),(10,38,14,34),(11,37,15,33),(12,36,16,40),(25,43,29,47),(26,42,30,46),(27,41,31,45),(28,48,32,44)], [(1,13,46),(2,14,47),(3,15,48),(4,16,41),(5,9,42),(6,10,43),(7,11,44),(8,12,45),(17,37,28),(18,38,29),(19,39,30),(20,40,31),(21,33,32),(22,34,25),(23,35,26),(24,36,27)], [(1,19),(2,24),(3,21),(4,18),(5,23),(6,20),(7,17),(8,22),(9,26),(10,31),(11,28),(12,25),(13,30),(14,27),(15,32),(16,29),(33,48),(34,45),(35,42),(36,47),(37,44),(38,41),(39,46),(40,43)]])
Q16⋊S3 is a maximal subgroup of
D12.30D4 SD16⋊D6 D8⋊11D6 S3×C8.C22 D24⋊C22 C24.C23 SD16.D6 Q16⋊D9 Dic12⋊S3 D12.4D6 D12.11D6 Dic6.10D6 Dic6.22D6 D12.15D6 C24.35D6 CSU2(𝔽3)⋊S3 D6.2S4 Dic20⋊S3 D30.4D4 Dic10.26D6 C60.C23 C60.44C23 D30.44D4 Q16⋊D15
Q16⋊S3 is a maximal quotient of
C3⋊Q16⋊C4 Q8⋊2Dic6 (C2×C8).D6 (C2×Q8).36D6 Dic6.11D4 Q8.4Dic6 (S3×Q8)⋊C4 Q8⋊7(C4×S3) D6.1SD16 Q8⋊3D12 Q8.11D12 C3⋊(C8⋊D4) C8⋊Dic3⋊C2 C3⋊C8.D4 Q8⋊3(C4×S3) Dic3⋊SD16 C24⋊4Q8 Dic6.2Q8 C8⋊S3⋊C4 D6.5D8 C8⋊3D12 C2.D8⋊7S3 C24⋊C2⋊C4 D12⋊2Q8 Dic3⋊3Q16 Q16⋊Dic3 (C2×Q16)⋊S3 D6⋊5Q16 D12.17D4 C24.36D4 C24.37D4 Q16⋊D9 Dic12⋊S3 D12.4D6 D12.11D6 Dic6.10D6 Dic6.22D6 D12.15D6 C24.35D6 Dic20⋊S3 D30.4D4 Dic10.26D6 C60.C23 C60.44C23 D30.44D4 Q16⋊D15
Matrix representation of Q16⋊S3 ►in GL4(𝔽5) generated by
| 0 | 1 | 1 | 1 |
| 0 | 4 | 1 | 3 |
| 1 | 4 | 2 | 4 |
| 0 | 0 | 3 | 4 |
| 0 | 3 | 2 | 2 |
| 0 | 2 | 0 | 1 |
| 2 | 2 | 0 | 3 |
| 0 | 0 | 0 | 3 |
| 4 | 2 | 1 | 4 |
| 0 | 1 | 3 | 0 |
| 0 | 4 | 3 | 0 |
| 1 | 1 | 4 | 0 |
| 1 | 2 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 4 | 4 | 0 |
| 0 | 1 | 0 | 4 |
G:=sub<GL(4,GF(5))| [0,0,1,0,1,4,4,0,1,1,2,3,1,3,4,4],[0,0,2,0,3,2,2,0,2,0,0,0,2,1,3,3],[4,0,0,1,2,1,4,1,1,3,3,4,4,0,0,0],[1,0,0,0,2,1,4,1,0,0,4,0,1,0,0,4] >;
Q16⋊S3 in GAP, Magma, Sage, TeX
Q_{16}\rtimes S_3 % in TeX
G:=Group("Q16:S3"); // GroupNames label
G:=SmallGroup(96,125);
// by ID
G=gap.SmallGroup(96,125);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,362,116,86,297,159,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^3=d^2=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations
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