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G = D8⋊S3  order 96 = 25·3

2nd semidirect product of D8 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8⋊2D6, D8⋊2S3, D4⋊2D6, D6.6D4, C24⋊4C22, C12.2C23, Dic3.8D4, Dic6⋊1C22, D12.1C22, D4⋊S3⋊2C2, (C3×D8)⋊4C2, (S3×D4)⋊2C2, C3⋊C8⋊1C22, C8⋊S3⋊3C2, C24⋊C2⋊3C2, C3⋊2(C8⋊C22), D4.S3⋊1C2, C6.28(C2×D4), C2.16(S3×D4), D4⋊2S3⋊1C2, (C3×D4)⋊2C22, C4.2(C22×S3), (C4×S3).1C22, SmallGroup(96,118)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — D8⋊S3
Chief series
C1 — C3 — C6 — C12 — C4×S3 — S3×D4 — D8⋊S3
Lower central
C3 — C6 — C12 — D8⋊S3
Upper central
C1 — C2 — C4 — D8

Generators and relations for D8⋊S3
 G = < a,b,c,d | a8=b2=c3=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >

Subgroups: 194 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, Dic3, Dic3, C12, D6, D6, C2×C6, M4(2), D8, D8, SD16, C2×D4, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C8⋊C22, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D4⋊2S3, D8⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8⋊C22, S3×D4, D8⋊S3

Character table of D8⋊S3

 class 12A2B2C2D2E34A4B4C6A6B6C8A8B1224A24B
 size 114461222612288412444
ρ1111111111111111111    trivial
ρ2111-111111-111-1-1-11-1-1    linear of order 2
ρ3111-1-1-111-1111-1-111-1-1    linear of order 2
ρ411-111-111111-11-1-11-1-1    linear of order 2
ρ511-11-1111-1-11-11-111-1-1    linear of order 2
ρ61111-1-111-1-11111-1111    linear of order 2
ρ711-1-11-1111-11-1-111111    linear of order 2
ρ811-1-1-1111-111-1-11-1111    linear of order 2
ρ92200202-2-2020000-200    orthogonal lifted from D4
ρ10222200-1200-1-1-120-1-1-1    orthogonal lifted from S3
ρ1122-2-200-1200-11120-1-1-1    orthogonal lifted from D6
ρ12222-200-1200-1-11-20-111    orthogonal lifted from D6
ρ1322-2200-1200-11-1-20-111    orthogonal lifted from D6
ρ142200-202-22020000-200    orthogonal lifted from D4
ρ15440000-2-400-20000200    orthogonal lifted from S3×D4
ρ164-400004000-40000000    orthogonal lifted from C8⋊C22
ρ174-40000-2000200000√-6-√-6    complex faithful
ρ184-40000-2000200000-√-6√-6    complex faithful

Permutation representations of D8⋊S3
►On 24 points - transitive group 24T141
Generators in S24
(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 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 21)(18 20)(22 24)

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

(1 5)(3 7)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)
 
G:=sub<Sym(24)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,17,16)(8,18,9), (1,5)(3,7)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21)>;
 
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), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,17,16)(8,18,9), (1,5)(3,7)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21) );
 
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)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15),(17,21),(18,20),(22,24)], [(1,19,10),(2,20,11),(3,21,12),(4,22,13),(5,23,14),(6,24,15),(7,17,16),(8,18,9)], [(1,5),(3,7),(9,18),(10,23),(11,20),(12,17),(13,22),(14,19),(15,24),(16,21)]])
 
G:=TransitiveGroup(24,141);
 

D8⋊S3 is a maximal subgroup of
 D8⋊13D6  SD16⋊D6  D8⋊11D6  S3×C8⋊C22  D8⋊4D6  D8⋊5D6  D8⋊6D6  D8⋊D9  D24⋊S3  C24⋊6D6  Dic6⋊3D6  D12.D6  D12⋊9D6  D12⋊5D6  C24⋊8D6  D40⋊S3  C40⋊8D6  D60.C22  D30.8D4  D20⋊10D6  Dic6⋊D10  D8⋊D15
D8⋊S3 is a maximal quotient of
 D4.S3⋊C4  D4⋊Dic6  Dic6⋊2D4  D4.Dic6  C4⋊C4.D6  C12⋊Q8⋊C2  C4⋊C4⋊19D6  D4⋊(C4×S3)  D6⋊5SD16  D6.SD16  D6⋊C8⋊11C2  C3⋊C8⋊1D4  D4⋊3D12  C3⋊C8⋊D4  D4⋊S3⋊C4  D12.D4  Dic3.Q16  C24⋊4Q8  C8⋊S3⋊C4  D6.2Q16  C2.D8⋊S3  C8⋊3D12  C24⋊C2⋊C4  D12.2Q8  Dic3⋊D8  D8⋊Dic3  (C6×D8).C2  C24⋊11D4  D12⋊D4  Dic6⋊D4  C24⋊12D4  D8⋊D9  D24⋊S3  C24⋊6D6  Dic6⋊3D6  D12.D6  D12⋊9D6  D12⋊5D6  C24⋊8D6  D40⋊S3  C40⋊8D6  D60.C22  D30.8D4  D20⋊10D6  Dic6⋊D10  D8⋊D15

Matrix representation of D8⋊S3 ►in GL4(𝔽5) generated by

0433
1430
0204
0011
,
4200
0100
0301
0210
,
3002
0011
4443
1001
,
4002
0111
0043
0001
G:=sub<GL(4,GF(5))| [0,1,0,0,4,4,2,0,3,3,0,1,3,0,4,1],[4,0,0,0,2,1,3,2,0,0,0,1,0,0,1,0],[3,0,4,1,0,0,4,0,0,1,4,0,2,1,3,1],[4,0,0,0,0,1,0,0,0,1,4,0,2,1,3,1] >;
 

D8⋊S3 in GAP, Magma, Sage, TeX 

D_8\rtimes S_3
 
% in TeX
 
G:=Group("D8:S3");
 
// GroupNames label
 
G:=SmallGroup(96,118);
 
// by ID
 
G=gap.SmallGroup(96,118);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,362,116,297,159,69,2309]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of D8⋊S3 in TeX

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