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G = S3×SD16order 96 = 25·3

Direct product of S3 and SD16

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

Aliases: S3×SD16, C85D6, Q82D6, D4.2D6, C245C22, D6.13D4, C12.4C23, Dic3.4D4, Dic62C22, D12.2C22, (S3×D4).C2, (S3×C8)⋊4C2, C3⋊C86C22, (S3×Q8)⋊1C2, C32(C2×SD16), C24⋊C25C2, D4.S33C2, C6.30(C2×D4), C2.18(S3×D4), Q82S31C2, (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

C1C12 — S3×SD16
C1C3C6C12C4×S3S3×D4 — S3×SD16
C3C6C12 — S3×SD16
C1C2C4SD16

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 [×4], C3, C4, C4 [×3], C22 [×5], S3 [×2], S3, C6, C6, C8, C8, C2×C4 [×2], D4, D4 [×2], Q8, Q8 [×2], C23, Dic3, Dic3, C12, C12, D6, D6 [×3], C2×C6, C2×C8, SD16, SD16 [×3], 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, Q82S3, C3×SD16, S3×D4, S3×Q8, S3×SD16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], SD16 [×2], 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 9 21)(2 10 22)(3 11 23)(4 12 24)(5 13 17)(6 14 18)(7 15 19)(8 16 20)
(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)
(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)(10 12)(11 15)(14 16)(18 20)(19 23)(22 24)

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

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

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

G:=TransitiveGroup(24,142);

S3×SD16 is a maximal subgroup of
SD1613D6  D811D6  D86D6  C24.C23  C249D6  Dic6⋊D6  D12.9D6  C4014D6  Dic10⋊D6  D15⋊SD16
S3×SD16 is a maximal quotient of
Dic36SD16  Dic3.SD16  D4⋊Dic6  Dic62D4  D65SD16  D6.SD16  D6⋊SD16  Dic37SD16  Q82Dic6  Dic3.1Q16  D6.1SD16  Q83D12  D62SD16  Dic3⋊SD16  Dic38SD16  Dic6⋊Q8  C245Q8  D6.2SD16  D6.4SD16  C88D12  D12⋊Q8  Dic33SD16  Dic35SD16  D66SD16  D68SD16  C2414D4  C2415D4  C249D6  Dic6⋊D6  D12.9D6  C4014D6  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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