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G = S3×SD16  order 96 = 25·3

Direct product of S3 and SD16

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

Aliases: S3×SD16, C8⋊5D6, Q8⋊2D6, D4.2D6, C24⋊5C22, D6.13D4, C12.4C23, Dic3.4D4, Dic6⋊2C22, D12.2C22, (S3×D4).C2, (S3×C8)⋊4C2, C3⋊C8⋊6C22, (S3×Q8)⋊1C2, C3⋊2(C2×SD16), C24⋊C2⋊5C2, D4.S3⋊3C2, C6.30(C2×D4), C2.18(S3×D4), Q8⋊2S3⋊1C2, (C3×SD16)⋊3C2, C4.4(C22×S3), (C3×Q8)⋊1C22, (C4×S3).9C22, (C3×D4).2C22, SmallGroup(96,120)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — S3×SD16
Chief series
C1 — C3 — C6 — C12 — C4×S3 — S3×D4 — S3×SD16
Lower central
C3 — C6 — C12 — S3×SD16
Upper central
C1 — C2 — C4 — SD16

Generators and relations for S3×SD16
 G = < a,b,c,d | a3=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 186 in 68 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, SD16, SD16, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C2×SD16, S3×C8, C24⋊C2, D4.S3, Q8⋊2S3, C3×SD16, S3×D4, S3×Q8, S3×SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C22×S3, C2×SD16, S3×D4, S3×SD16

Character table of S3×SD16

 class 12A2B2C2D2E34A4B4C4D6A6B8A8B8C8D12A12B24A24B
 size 11334122246122822664844
ρ1111111111111111111111    trivial
ρ211-1-11-1111-1-11111-1-11111    linear of order 2
ρ311-1-1-11111-1-11-1-1-11111-1-1    linear of order 2
ρ411-1-1-1111-1-111-111-1-11-111    linear of order 2
ρ511-1-11-111-1-1111-1-1111-1-1-1    linear of order 2
ρ61111-1-1111111-1-1-1-1-111-1-1    linear of order 2
ρ71111-1-111-11-11-111111-111    linear of order 2
ρ811111111-11-111-1-1-1-11-1-1-1    linear of order 2
ρ92222002-20-20200000-2000    orthogonal lifted from D4
ρ102200-20-12200-11-2-200-1-111    orthogonal lifted from D6
ρ11220020-12200-1-12200-1-1-1-1    orthogonal lifted from S3
ρ122200-20-12-200-112200-11-1-1    orthogonal lifted from D6
ρ13220020-12-200-1-1-2-200-1111    orthogonal lifted from D6
ρ1422-2-2002-2020200000-2000    orthogonal lifted from D4
ρ152-22-20020000-20√-2-√-2√-2-√-200√-2-√-2    complex lifted from SD16
ρ162-2-220020000-20√-2-√-2-√-2√-200√-2-√-2    complex lifted from SD16
ρ172-2-220020000-20-√-2√-2√-2-√-200-√-2√-2    complex lifted from SD16
ρ182-22-20020000-20-√-2√-2-√-2√-200-√-2√-2    complex lifted from SD16
ρ19440000-2-4000-2000002000    orthogonal lifted from S3×D4
ρ204-40000-20000202√-2-2√-20000-√-2√-2    complex faithful
ρ214-40000-2000020-2√-22√-20000√-2-√-2    complex faithful

Permutation representations of S3×SD16
►On 24 points - transitive group 24T142
Generators in S24
(1 10 21)(2 11 22)(3 12 23)(4 13 24)(5 14 17)(6 15 18)(7 16 19)(8 9 20)

(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)

(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(18 20)(19 23)(22 24)
 
G:=sub<Sym(24)| (1,10,21)(2,11,22)(3,12,23)(4,13,24)(5,14,17)(6,15,18)(7,16,19)(8,9,20), (9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(18,20)(19,23)(22,24)>;
 
G:=Group( (1,10,21)(2,11,22)(3,12,23)(4,13,24)(5,14,17)(6,15,18)(7,16,19)(8,9,20), (9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(18,20)(19,23)(22,24) );
 
G=PermutationGroup([[(1,10,21),(2,11,22),(3,12,23),(4,13,24),(5,14,17),(6,15,18),(7,16,19),(8,9,20)], [(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(18,20),(19,23),(22,24)]])
 
G:=TransitiveGroup(24,142);
 

S3×SD16 is a maximal subgroup of
 SD16⋊13D6  D8⋊11D6  D8⋊6D6  C24.C23  C24⋊9D6  Dic6⋊D6  D12.9D6  C40⋊14D6  Dic10⋊D6  D15⋊SD16
S3×SD16 is a maximal quotient of
 Dic3⋊6SD16  Dic3.SD16  D4⋊Dic6  Dic6⋊2D4  D6⋊5SD16  D6.SD16  D6⋊SD16  Dic3⋊7SD16  Q8⋊2Dic6  Dic3.1Q16  D6.1SD16  Q8⋊3D12  D6⋊2SD16  Dic3⋊SD16  Dic3⋊8SD16  Dic6⋊Q8  C24⋊5Q8  D6.2SD16  D6.4SD16  C8⋊8D12  D12⋊Q8  Dic3⋊3SD16  Dic3⋊5SD16  D6⋊6SD16  D6⋊8SD16  C24⋊14D4  C24⋊15D4  C24⋊9D6  Dic6⋊D6  D12.9D6  C40⋊14D6  Dic10⋊D6  D15⋊SD16

Matrix representation of S3×SD16 ►in GL4(𝔽73) generated by

1000
0100
007272
0010
,
72000
07200
0010
007272
,
0400
551200
0010
0001
,
14800
07200
0010
0001
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,1,0,0,72,0],[72,0,0,0,0,72,0,0,0,0,1,72,0,0,0,72],[0,55,0,0,4,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,48,72,0,0,0,0,1,0,0,0,0,1] >;
 

S3×SD16 in GAP, Magma, Sage, TeX 

S_3\times {\rm SD}_{16}
 
% in TeX
 
G:=Group("S3xSD16");
 
// GroupNames label
 
G:=SmallGroup(96,120);
 
// by ID
 
G=gap.SmallGroup(96,120);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,116,86,297,159,69,2309]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
 
// generators/relations
 

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Character table of S3×SD16 in TeX

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