direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×Q16, C8.9D6, Q8.8D6, D6.14D4, Dic12⋊5C2, C12.8C23, C24.7C22, Dic3.5D4, Dic6.4C22, C3⋊2(C2×Q16), (S3×Q8).C2, (S3×C8).1C2, (C3×Q16)⋊2C2, C3⋊Q16⋊3C2, C6.34(C2×D4), C2.22(S3×D4), C3⋊C8.7C22, C4.8(C22×S3), (C3×Q8).3C22, (C4×S3).11C22, SmallGroup(96,124)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×Q16
G = < a,b,c,d | a3=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 130 in 60 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, Q8, Q8, Dic3, Dic3, C12, C12, D6, C2×C8, Q16, Q16, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C3×Q8, C2×Q16, S3×C8, Dic12, C3⋊Q16, C3×Q16, S3×Q8, S3×Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C22×S3, C2×Q16, S3×D4, S3×Q16
Character table of S3×Q16
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6 | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 24A | 24B | |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 4 | 4 | 6 | 12 | 12 | 2 | 2 | 2 | 6 | 6 | 4 | 8 | 8 | 4 | 4 | |
| ρ1 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 0 | 0 | -1 | 2 | -2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 0 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -√2 | √2 | -√2 | √2 | 0 | 0 | 0 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ16 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | √2 | -√2 | √2 | -√2 | 0 | 0 | 0 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | √2 | -√2 | -√2 | √2 | 0 | 0 | 0 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ18 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -√2 | √2 | √2 | -√2 | 0 | 0 | 0 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ19 | 4 | 4 | 0 | 0 | -2 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ20 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | symplectic faithful, Schur index 2 |
►On 48 points
(1 35 46)(2 36 47)(3 37 48)(4 38 41)(5 39 42)(6 40 43)(7 33 44)(8 34 45)(9 30 19)(10 31 20)(11 32 21)(12 25 22)(13 26 23)(14 27 24)(15 28 17)(16 29 18)
(1 5)(2 6)(3 7)(4 8)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 21)(18 22)(19 23)(20 24)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)
(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 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(25 43 29 47)(26 42 30 46)(27 41 31 45)(28 48 32 44)
G:=sub<Sym(48)| (1,35,46)(2,36,47)(3,37,48)(4,38,41)(5,39,42)(6,40,43)(7,33,44)(8,34,45)(9,30,19)(10,31,20)(11,32,21)(12,25,22)(13,26,23)(14,27,24)(15,28,17)(16,29,18), (1,5)(2,6)(3,7)(4,8)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,21)(18,22)(19,23)(20,24)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47), (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)(9,35,13,39)(10,34,14,38)(11,33,15,37)(12,40,16,36)(25,43,29,47)(26,42,30,46)(27,41,31,45)(28,48,32,44)>;
G:=Group( (1,35,46)(2,36,47)(3,37,48)(4,38,41)(5,39,42)(6,40,43)(7,33,44)(8,34,45)(9,30,19)(10,31,20)(11,32,21)(12,25,22)(13,26,23)(14,27,24)(15,28,17)(16,29,18), (1,5)(2,6)(3,7)(4,8)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,21)(18,22)(19,23)(20,24)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47), (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)(9,35,13,39)(10,34,14,38)(11,33,15,37)(12,40,16,36)(25,43,29,47)(26,42,30,46)(27,41,31,45)(28,48,32,44) );
G=PermutationGroup([[(1,35,46),(2,36,47),(3,37,48),(4,38,41),(5,39,42),(6,40,43),(7,33,44),(8,34,45),(9,30,19),(10,31,20),(11,32,21),(12,25,22),(13,26,23),(14,27,24),(15,28,17),(16,29,18)], [(1,5),(2,6),(3,7),(4,8),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25),(17,21),(18,22),(19,23),(20,24),(33,48),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47)], [(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),(9,35,13,39),(10,34,14,38),(11,33,15,37),(12,40,16,36),(25,43,29,47),(26,42,30,46),(27,41,31,45),(28,48,32,44)]])
S3×Q16 is a maximal subgroup of
SD32⋊S3 Q32⋊S3 D12.30D4 D8.10D6 SD16.D6 C24.23D6 Dic6.9D6 Dic10.D6 D15⋊Q16
S3×Q16 is a maximal quotient of
Dic3⋊4Q16 Dic3.1Q16 Q8⋊3Dic6 Dic3⋊Q16 D6⋊Q16 D6.Q16 D6⋊1Q16 Dic3⋊5Q16 C24⋊2Q8 Dic3.Q16 D6.2Q16 D6⋊2Q16 Dic3⋊3Q16 C24.26D4 D6⋊5Q16 D6⋊3Q16 C24.23D6 Dic6.9D6 Dic10.D6 D15⋊Q16
Matrix representation of S3×Q16 ►in GL4(𝔽7) generated by
| 3 | 0 | 3 | 1 |
| 6 | 4 | 1 | 2 |
| 3 | 0 | 4 | 3 |
| 6 | 0 | 4 | 1 |
| 5 | 1 | 1 | 1 |
| 2 | 2 | 0 | 4 |
| 2 | 3 | 6 | 4 |
| 0 | 4 | 3 | 1 |
| 3 | 3 | 0 | 2 |
| 1 | 3 | 1 | 1 |
| 2 | 2 | 6 | 3 |
| 3 | 4 | 2 | 3 |
| 1 | 1 | 0 | 1 |
| 5 | 0 | 4 | 4 |
| 4 | 3 | 4 | 2 |
| 0 | 6 | 3 | 2 |
G:=sub<GL(4,GF(7))| [3,6,3,6,0,4,0,0,3,1,4,4,1,2,3,1],[5,2,2,0,1,2,3,4,1,0,6,3,1,4,4,1],[3,1,2,3,3,3,2,4,0,1,6,2,2,1,3,3],[1,5,4,0,1,0,3,6,0,4,4,3,1,4,2,2] >;
S3×Q16 in GAP, Magma, Sage, TeX
S_3\times Q_{16} % in TeX
G:=Group("S3xQ16"); // GroupNames label
G:=SmallGroup(96,124);
// by ID
G=gap.SmallGroup(96,124);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,116,86,297,159,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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