Copied to
clipboard

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, (C3xD8):4C2, (S3xD4):2C2, C3:C8:1C22, C8:S3:3C2, C24:C2:3C2, C3:2(C8:C22), D4.S3:1C2, C6.28(C2xD4), C2.16(S3xD4), D4:2S3:1C2, (C3xD4):2C22, C4.2(C22xS3), (C4xS3).1C22, SmallGroup(96,118)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — D8:S3
Chief series
C1 — C3 — C6 — C12 — C4xS3 — S3xD4 — 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, C2xC4, D4, D4, Q8, C23, Dic3, Dic3, C12, D6, D6, C2xC6, M4(2), D8, D8, SD16, C2xD4, C4oD4, C3:C8, C24, Dic6, C4xS3, D12, C2xDic3, C3:D4, C3xD4, C22xS3, C8:C22, C8:S3, C24:C2, D4:S3, D4.S3, C3xD8, S3xD4, D4:2S3, D8:S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C22xS3, C8:C22, S3xD4, 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 S3xD4
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  S3xC8: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:(C4xS3)  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  (C6xD8).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

׿
x
:
Z
F
o
wr
Q
<