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G = S3×D8order 96 = 25·3

Direct product of S3 and D8

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

Aliases: S3×D8, C84D6, D41D6, D244C2, C242C22, D6.12D4, D121C22, C12.1C23, Dic3.3D4, C32(C2×D8), (S3×C8)⋊1C2, D4⋊S31C2, (C3×D8)⋊2C2, (S3×D4)⋊1C2, C3⋊C85C22, C2.15(S3×D4), C6.27(C2×D4), (C3×D4)⋊1C22, C4.1(C22×S3), (C4×S3).7C22, SmallGroup(96,117)

Series: Derived Chief Lower central Upper central

C1C12 — S3×D8
C1C3C6C12C4×S3S3×D4 — S3×D8
C3C6C12 — S3×D8
C1C2C4D8

Generators and relations for S3×D8
 G = < a,b,c,d | a3=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 242 in 76 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22 [×9], S3 [×2], S3 [×2], C6, C6 [×2], C8, C8, C2×C4, D4 [×2], D4 [×4], C23 [×2], Dic3, C12, D6, D6 [×6], C2×C6 [×2], C2×C8, D8, D8 [×3], C2×D4 [×2], C3⋊C8, C24, C4×S3, D12 [×2], C3⋊D4 [×2], C3×D4 [×2], C22×S3 [×2], C2×D8, S3×C8, D24, D4⋊S3 [×2], C3×D8, S3×D4 [×2], S3×D8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], D8 [×2], C2×D4, C22×S3, C2×D8, S3×D4, S3×D8

Character table of S3×D8

 class 12A2B2C2D2E2F2G34A4B6A6B6C8A8B8C8D1224A24B
 size 11334412122262882266444
ρ1111111111111111111111    trivial
ρ21111-111-11111-11-1-1-1-11-1-1    linear of order 2
ρ311-1-1-11-1111-11-11-1-1111-1-1    linear of order 2
ρ411-1-11-11-111-111-1-1-1111-1-1    linear of order 2
ρ511111-1-1111111-1-1-1-1-11-1-1    linear of order 2
ρ611-1-111-1-111-111111-1-1111    linear of order 2
ρ711-1-1-1-11111-11-1-111-1-1111    linear of order 2
ρ81111-1-1-1-11111-1-11111111    linear of order 2
ρ9222200002-2-22000000-200    orthogonal lifted from D4
ρ1022-2-200002-222000000-200    orthogonal lifted from D4
ρ112200-2200-120-11-1-2-200-111    orthogonal lifted from D6
ρ122200-2-200-120-1112200-1-1-1    orthogonal lifted from D6
ρ1322002-200-120-1-11-2-200-111    orthogonal lifted from D6
ρ1422002200-120-1-1-12200-1-1-1    orthogonal lifted from S3
ρ152-2-220000200-2002-22-202-2    orthogonal lifted from D8
ρ162-22-20000200-200-222-20-22    orthogonal lifted from D8
ρ172-22-20000200-2002-2-2202-2    orthogonal lifted from D8
ρ182-2-220000200-200-22-220-22    orthogonal lifted from D8
ρ1944000000-2-40-2000000200    orthogonal lifted from S3×D4
ρ204-4000000-200200-22220002-2    orthogonal faithful
ρ214-4000000-20020022-22000-22    orthogonal faithful

Permutation representations of S3×D8
On 24 points - transitive group 24T139
Generators in S24
(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)(2 6)(3 7)(4 8)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(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)

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

G:=Group( (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)(2,6)(3,7)(4,8)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (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) );

G=PermutationGroup([(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),(2,6),(3,7),(4,8),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(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)])

G:=TransitiveGroup(24,139);

S3×D8 is a maximal subgroup of
D8⋊D6  D48⋊C2  D813D6  D815D6  D85D6  C244D6  D12⋊D6  C405D6  D15⋊D8
S3×D8 is a maximal quotient of
Dic34D8  Dic3.D8  Dic3.SD16  D4⋊D12  D6.D8  D6⋊D8  D123D4  Dic35D8  C242Q8  D6.5D8  D62D8  D122Q8  D8⋊D6  D163S3  D48⋊C2  SD32⋊S3  D6.2D8  Q32⋊S3  D485C2  Dic3⋊D8  C245D4  D12⋊D4  D63D8  C244D6  D12⋊D6  C405D6  D15⋊D8

Matrix representation of S3×D8 in GL4(𝔽7) generated by

3046
1265
4024
1035
,
5343
0644
4205
3513
,
6302
1611
2223
3426
,
2314
3603
6222
6464
G:=sub<GL(4,GF(7))| [3,1,4,1,0,2,0,0,4,6,2,3,6,5,4,5],[5,0,4,3,3,6,2,5,4,4,0,1,3,4,5,3],[6,1,2,3,3,6,2,4,0,1,2,2,2,1,3,6],[2,3,6,6,3,6,2,4,1,0,2,6,4,3,2,4] >;

S3×D8 in GAP, Magma, Sage, TeX

S_3\times D_8
% in TeX

G:=Group("S3xD8");
// GroupNames label

G:=SmallGroup(96,117);
// by ID

G=gap.SmallGroup(96,117);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,116,297,159,69,2309]);
// 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^-1>;
// generators/relations

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Character table of S3×D8 in TeX

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