Copied to
clipboard

G = S3×C4○D4order 96 = 25·3

Direct product of S3 and C4○D4

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

Aliases: S3×C4○D4, D47D6, Q87D6, C6.11C24, D6.6C23, D1210C22, C12.25C23, Dic610C22, Dic3.10C23, C4(S3×D4), D4(C4×S3), C4(S3×Q8), Q8(C4×S3), (C2×C4)⋊7D6, (S3×D4)⋊6C2, (S3×Q8)⋊6C2, C4○D127C2, C4(D42S3), D42S36C2, (C4×S3)⋊7C22, (C2×C12)⋊4C22, C4(Q83S3), Q83S36C2, Dic3(C4○D4), (C3×D4)⋊8C22, C3⋊D44C22, (C2×C6).3C23, (C3×Q8)⋊7C22, C4.25(C22×S3), C2.12(S3×C23), C22.2(C22×S3), (C2×Dic3)⋊10C22, (C22×S3).31C22, (S3×C2×C4)⋊6C2, C34(C2×C4○D4), (C3×C4○D4)⋊3C2, SmallGroup(96,215)

Series: Derived Chief Lower central Upper central

C1C6 — S3×C4○D4
C1C3C6D6C22×S3S3×C2×C4 — S3×C4○D4
C3C6 — S3×C4○D4
C1C4C4○D4

Generators and relations for S3×C4○D4
 G = < a,b,c,d,e | a3=b2=c4=e2=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 330 in 164 conjugacy classes, 87 normal (16 characteristic)
C1, C2, C2 [×8], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×10], S3 [×2], S3 [×3], C6, C6 [×3], C2×C4 [×3], C2×C4 [×13], D4 [×3], D4 [×9], Q8, Q8 [×3], C23 [×3], Dic3, Dic3 [×3], C12, C12 [×3], D6, D6 [×3], D6 [×6], C2×C6 [×3], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4, C4○D4 [×7], Dic6 [×3], C4×S3, C4×S3 [×9], D12 [×3], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×3], C2×C4○D4, S3×C2×C4 [×3], C4○D12 [×3], S3×D4 [×3], D42S3 [×3], S3×Q8, Q83S3, C3×C4○D4, S3×C4○D4
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, S3×C23, S3×C4○D4

Character table of S3×C4○D4

 class 12A2B2C2D2E2F2G2H2I34A4B4C4D4E4F4G4H4I4J6A6B6C6D12A12B12C12D12E
 size 112223366621122233666244422444
ρ1111111111111111111111111111111    trivial
ρ211-1-1-1-1-11111-1-111111-1-1-11-1-1-1-1-1111    linear of order 2
ρ3111-111111-11-1-11-1-1-1-1-11-11-111-1-11-1-1    linear of order 2
ρ411-11-1-1-111-11111-1-1-1-11-1111-1-1111-1-1    linear of order 2
ρ511-111111-111-1-1-11-1-1-11-1-111-11-1-1-11-1    linear of order 2
ρ6111-1-1-1-11-11111-11-1-1-1-1111-11-111-11-1    linear of order 2
ρ711-1-11111-1-1111-1-1111-1-111-1-1111-1-11    linear of order 2
ρ81111-1-1-11-1-11-1-1-1-111111-1111-1-1-1-1-11    linear of order 2
ρ911-1-11-1-1-111111-1-11-1-111-11-1-1111-1-11    linear of order 2
ρ101111-111-1111-1-1-1-11-1-1-1-11111-1-1-1-1-11    linear of order 2
ρ1111-111-1-1-11-11-1-1-11-111-11111-11-1-1-11-1    linear of order 2
ρ12111-1-111-11-1111-11-1111-1-11-11-111-11-1    linear of order 2
ρ13111-11-1-1-1-111-1-11-1-1111-111-111-1-11-1-1    linear of order 2
ρ1411-11-111-1-111111-1-111-11-111-1-1111-1-1    linear of order 2
ρ1511111-1-1-1-1-1111111-1-1-1-1-1111111111    linear of order 2
ρ1611-1-1-111-1-1-11-1-1111-1-11111-1-1-1-1-1111    linear of order 2
ρ17222-2200000-1-2-22-2-200000-11-1-111-111    orthogonal lifted from D6
ρ18222-2-200000-122-22-200000-11-11-1-11-11    orthogonal lifted from D6
ρ192222-200000-1-2-2-2-2200000-1-1-111111-1    orthogonal lifted from D6
ρ202222200000-12222200000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ2122-2-2-200000-1-2-222200000-111111-1-1-1    orthogonal lifted from D6
ρ2222-2-2200000-122-2-2200000-111-1-1-111-1    orthogonal lifted from D6
ρ2322-22200000-1-2-2-22-200000-1-11-1111-11    orthogonal lifted from D6
ρ2422-22-200000-1222-2-200000-1-111-1-1-111    orthogonal lifted from D6
ρ252-20002-200022i-2i0002i-2i000-20002i-2i000    complex lifted from C4○D4
ρ262-20002-20002-2i2i000-2i2i000-2000-2i2i000    complex lifted from C4○D4
ρ272-2000-2200022i-2i000-2i2i000-20002i-2i000    complex lifted from C4○D4
ρ282-2000-220002-2i2i0002i-2i000-2000-2i2i000    complex lifted from C4○D4
ρ294-400000000-2-4i4i0000000020002i-2i000    complex faithful
ρ304-400000000-24i-4i000000002000-2i2i000    complex faithful

Permutation representations of S3×C4○D4
On 24 points - transitive group 24T101
Generators in S24
(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 23 13)(10 24 14)(11 21 15)(12 22 16)
(1 3)(2 4)(5 20)(6 17)(7 18)(8 19)(9 11)(10 12)(13 21)(14 22)(15 23)(16 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)
(1 2 3 4)(5 6 7 8)(9 12 11 10)(13 16 15 14)(17 18 19 20)(21 24 23 22)
(1 11)(2 12)(3 9)(4 10)(5 24)(6 21)(7 22)(8 23)(13 17)(14 18)(15 19)(16 20)

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

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

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

G:=TransitiveGroup(24,101);

S3×C4○D4 is a maximal subgroup of
C423D6  M4(2)⋊28D6  SD16⋊D6  D84D6  D24⋊C22  C6.C25  D6.C24  D12.39C23  D1223D6  Dic612D6  D1215D6  D2024D6  D30.C23  D2016D6
S3×C4○D4 is a maximal quotient of
C42.88D6  C42.188D6  C4210D6  C4212D6  C42.93D6  C42.94D6  C42.95D6  C42.96D6  C42.97D6  C42.98D6  C4×D42S3  C42.102D6  D45Dic6  C42.104D6  C4×S3×D4  C4214D6  C42.228D6  D45D12  C4218D6  C42.229D6  C42.113D6  C42.114D6  C42.122D6  Q86Dic6  C4×S3×Q8  C4×Q83S3  Q86D12  C42.232D6  C42.131D6  C42.132D6  C12⋊(C4○D4)  Dic620D4  C4⋊C4.178D6  C6.342+ 1+4  C4⋊C421D6  C6.402+ 1+4  D1220D4  C6.422+ 1+4  C6.432+ 1+4  C6.442+ 1+4  C6.452+ 1+4  (Q8×Dic3)⋊C2  C4⋊C4.187D6  C4⋊C426D6  D1222D4  Dic622D4  C6.522+ 1+4  C6.532+ 1+4  C6.202- 1+4  C6.212- 1+4  C6.222- 1+4  C6.232- 1+4  C4⋊C4.197D6  C6.802- 1+4  C6.1212+ 1+4  C6.822- 1+4  C4⋊C428D6  C6.612+ 1+4  C6.1222+ 1+4  C6.622+ 1+4  C6.632+ 1+4  C6.642+ 1+4  C6.652+ 1+4  C6.662+ 1+4  C6.672+ 1+4  C42.233D6  C42.137D6  C42.138D6  C42.139D6  D1210D4  Dic610D4  C4222D6  C4223D6  C42.234D6  C42.143D6  Dic67Q8  C42.236D6  D127Q8  C42.237D6  C42.150D6  C42.151D6  C42.152D6  C42.153D6  C42.154D6  C42.159D6  C42.160D6  C4225D6  C4226D6  C42.189D6  C42.161D6  C42.162D6  C42.163D6  C42.164D6  C6.1042- 1+4  (C2×D4)⋊43D6  C6.1452+ 1+4  C6.1072- 1+4  (C2×C12)⋊17D4  C6.1482+ 1+4  D1223D6  Dic612D6  D1215D6  D2024D6  D30.C23  D2016D6

Matrix representation of S3×C4○D4 in GL4(𝔽5) generated by

4030
0402
3000
0200
,
1000
0003
2040
0200
,
3000
0300
0030
0003
,
2000
0300
0020
0003
,
0201
0040
0400
1020
G:=sub<GL(4,GF(5))| [4,0,3,0,0,4,0,2,3,0,0,0,0,2,0,0],[1,0,2,0,0,0,0,2,0,0,4,0,0,3,0,0],[3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[2,0,0,0,0,3,0,0,0,0,2,0,0,0,0,3],[0,0,0,1,2,0,4,0,0,4,0,2,1,0,0,0] >;

S3×C4○D4 in GAP, Magma, Sage, TeX

S_3\times C_4\circ D_4
% in TeX

G:=Group("S3xC4oD4");
// GroupNames label

G:=SmallGroup(96,215);
// by ID

G=gap.SmallGroup(96,215);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,86,297,2309]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^4=e^2=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d>;
// generators/relations

Export

Character table of S3×C4○D4 in TeX

׿
×
𝔽