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G = S3×Q16order 96 = 25·3

Direct product of S3 and Q16

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: S3×Q16, C8.9D6, Q8.8D6, D6.14D4, Dic125C2, C12.8C23, C24.7C22, Dic3.5D4, Dic6.4C22, C32(C2×Q16), (S3×Q8).C2, (S3×C8).1C2, (C3×Q16)⋊2C2, C3⋊Q163C2, 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

C1C12 — S3×Q16
C1C3C6C12C4×S3S3×Q8 — S3×Q16
C3C6C12 — S3×Q16
C1C2C4Q16

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 [×2], C3, C4, C4 [×5], C22, S3 [×2], C6, C8, C8, C2×C4 [×3], Q8 [×2], Q8 [×4], Dic3, Dic3 [×2], C12, C12 [×2], D6, C2×C8, Q16, Q16 [×3], C2×Q8 [×2], C3⋊C8, C24, Dic6 [×2], Dic6 [×2], C4×S3, C4×S3 [×2], C3×Q8 [×2], C2×Q16, S3×C8, Dic12, C3⋊Q16 [×2], C3×Q16, S3×Q8 [×2], S3×Q16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], Q16 [×2], C2×D4, C22×S3, C2×Q16, S3×D4, S3×Q16

Character table of S3×Q16

 class 12A2B2C34A4B4C4D4E4F68A8B8C8D12A12B12C24A24B
 size 11332244612122226648844
ρ1111111111111111111111    trivial
ρ2111111-1-11-1-1111111-1-111    linear of order 2
ρ311-1-111-1-1-111111-1-11-1-111    linear of order 2
ρ411-1-1111-1-11-11-1-1111-11-1-1    linear of order 2
ρ51111111-11-111-1-1-1-11-11-1-1    linear of order 2
ρ611-1-11111-1-1-1111-1-111111    linear of order 2
ρ711-1-111-11-1-111-1-11111-1-1-1    linear of order 2
ρ8111111-1111-11-1-1-1-111-1-1-1    linear of order 2
ρ922222-200-20020000-20000    orthogonal lifted from D4
ρ102200-1222000-12200-1-1-1-1-1    orthogonal lifted from S3
ρ1122-2-22-20020020000-20000    orthogonal lifted from D4
ρ122200-12-22000-1-2-200-1-1111    orthogonal lifted from D6
ρ132200-12-2-2000-12200-111-1-1    orthogonal lifted from D6
ρ142200-122-2000-1-2-200-11-111    orthogonal lifted from D6
ρ152-22-22000000-2-22-220002-2    symplectic lifted from Q16, Schur index 2
ρ162-22-22000000-22-22-2000-22    symplectic lifted from Q16, Schur index 2
ρ172-2-222000000-22-2-22000-22    symplectic lifted from Q16, Schur index 2
ρ182-2-222000000-2-222-20002-2    symplectic lifted from Q16, Schur index 2
ρ194400-2-400000-2000020000    orthogonal lifted from S3×D4
ρ204-400-2000000222-22000002-2    symplectic faithful, Schur index 2
ρ214-400-20000002-222200000-22    symplectic faithful, Schur index 2

Smallest permutation representation of S3×Q16
On 48 points
Generators in S48
(1 31 46)(2 32 47)(3 25 48)(4 26 41)(5 27 42)(6 28 43)(7 29 44)(8 30 45)(9 33 24)(10 34 17)(11 35 18)(12 36 19)(13 37 20)(14 38 21)(15 39 22)(16 40 23)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 38)(18 39)(19 40)(20 33)(21 34)(22 35)(23 36)(24 37)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)
(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 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 47 21 43)(18 46 22 42)(19 45 23 41)(20 44 24 48)(25 37 29 33)(26 36 30 40)(27 35 31 39)(28 34 32 38)

G:=sub<Sym(48)| (1,31,46)(2,32,47)(3,25,48)(4,26,41)(5,27,42)(6,28,43)(7,29,44)(8,30,45)(9,33,24)(10,34,17)(11,35,18)(12,36,19)(13,37,20)(14,38,21)(15,39,22)(16,40,23), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,38)(18,39)(19,40)(20,33)(21,34)(22,35)(23,36)(24,37)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43), (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), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,47,21,43)(18,46,22,42)(19,45,23,41)(20,44,24,48)(25,37,29,33)(26,36,30,40)(27,35,31,39)(28,34,32,38)>;

G:=Group( (1,31,46)(2,32,47)(3,25,48)(4,26,41)(5,27,42)(6,28,43)(7,29,44)(8,30,45)(9,33,24)(10,34,17)(11,35,18)(12,36,19)(13,37,20)(14,38,21)(15,39,22)(16,40,23), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,38)(18,39)(19,40)(20,33)(21,34)(22,35)(23,36)(24,37)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43), (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), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,47,21,43)(18,46,22,42)(19,45,23,41)(20,44,24,48)(25,37,29,33)(26,36,30,40)(27,35,31,39)(28,34,32,38) );

G=PermutationGroup([(1,31,46),(2,32,47),(3,25,48),(4,26,41),(5,27,42),(6,28,43),(7,29,44),(8,30,45),(9,33,24),(10,34,17),(11,35,18),(12,36,19),(13,37,20),(14,38,21),(15,39,22),(16,40,23)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,38),(18,39),(19,40),(20,33),(21,34),(22,35),(23,36),(24,37),(25,44),(26,45),(27,46),(28,47),(29,48),(30,41),(31,42),(32,43)], [(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)], [(1,15,5,11),(2,14,6,10),(3,13,7,9),(4,12,8,16),(17,47,21,43),(18,46,22,42),(19,45,23,41),(20,44,24,48),(25,37,29,33),(26,36,30,40),(27,35,31,39),(28,34,32,38)])

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
Dic34Q16  Dic3.1Q16  Q83Dic6  Dic3⋊Q16  D6⋊Q16  D6.Q16  D61Q16  Dic35Q16  C242Q8  Dic3.Q16  D6.2Q16  D62Q16  Dic33Q16  C24.26D4  D65Q16  D63Q16  C24.23D6  Dic6.9D6  Dic10.D6  D15⋊Q16

Matrix representation of S3×Q16 in GL4(𝔽7) generated by

3031
6412
3043
6041
,
5111
2204
2364
0431
,
3302
1311
2263
3423
,
1101
5044
4342
0632
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

Export

Character table of S3×Q16 in TeX

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