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G = S3×C4○D4  order 96 = 25·3

Direct product of S3 and C4○D4

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

Aliases: S3×C4○D4, D4⋊7D6, Q8⋊7D6, C6.11C24, D6.6C23, D12⋊10C22, C12.25C23, Dic6⋊10C22, Dic3.10C23, C4○(S3×D4), D4○(C4×S3), C4○(S3×Q8), Q8○(C4×S3), (C2×C4)⋊7D6, (S3×D4)⋊6C2, (S3×Q8)⋊6C2, C4○D12⋊7C2, C4○(D4⋊2S3), D4⋊2S3⋊6C2, (C4×S3)⋊7C22, (C2×C12)⋊4C22, C4○(Q8⋊3S3), Q8⋊3S3⋊6C2, Dic3○(C4○D4), (C3×D4)⋊8C22, C3⋊D4⋊4C22, (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, C3⋊4(C2×C4○D4), (C3×C4○D4)⋊3C2, SmallGroup(96,215)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — S3×C4○D4
Chief series
C1 — C3 — C6 — D6 — C22×S3 — S3×C2×C4 — S3×C4○D4
Lower central
C3 — C6 — S3×C4○D4
Upper central
C1 — C4 — C4○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, C3, C4, C4, C4, C22, C22, S3, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, D6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C2×C4○D4, S3×C2×C4, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C3×C4○D4, S3×C4○D4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, 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 9 5)(2 10 6)(3 11 7)(4 12 8)(13 19 23)(14 20 24)(15 17 21)(16 18 22)

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

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

(1 17)(2 18)(3 19)(4 20)(5 15)(6 16)(7 13)(8 14)(9 21)(10 22)(11 23)(12 24)
 
G:=sub<Sym(24)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,19,23)(14,20,24)(15,17,21)(16,18,22), (1,3)(2,4)(5,11)(6,12)(7,9)(8,10)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,16,15,14)(17,20,19,18)(21,24,23,22), (1,17)(2,18)(3,19)(4,20)(5,15)(6,16)(7,13)(8,14)(9,21)(10,22)(11,23)(12,24)>;
 
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,19,23)(14,20,24)(15,17,21)(16,18,22), (1,3)(2,4)(5,11)(6,12)(7,9)(8,10)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,16,15,14)(17,20,19,18)(21,24,23,22), (1,17)(2,18)(3,19)(4,20)(5,15)(6,16)(7,13)(8,14)(9,21)(10,22)(11,23)(12,24) );
 
G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,19,23),(14,20,24),(15,17,21),(16,18,22)], [(1,3),(2,4),(5,11),(6,12),(7,9),(8,10),(13,21),(14,22),(15,23),(16,24),(17,19),(18,20)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,16,15,14),(17,20,19,18),(21,24,23,22)], [(1,17),(2,18),(3,19),(4,20),(5,15),(6,16),(7,13),(8,14),(9,21),(10,22),(11,23),(12,24)]])
 
G:=TransitiveGroup(24,101);
 

S3×C4○D4 is a maximal subgroup of
 C42⋊3D6  M4(2)⋊28D6  SD16⋊D6  D8⋊4D6  D24⋊C22  C6.C25  D6.C24  D12.39C23  D12⋊23D6  Dic6⋊12D6  D12⋊15D6  D20⋊24D6  D30.C23  D20⋊16D6
S3×C4○D4 is a maximal quotient of
 C42.88D6  C42.188D6  C42⋊10D6  C42⋊12D6  C42.93D6  C42.94D6  C42.95D6  C42.96D6  C42.97D6  C42.98D6  C4×D4⋊2S3  C42.102D6  D4⋊5Dic6  C42.104D6  C4×S3×D4  C42⋊14D6  C42.228D6  D4⋊5D12  C42⋊18D6  C42.229D6  C42.113D6  C42.114D6  C42.122D6  Q8⋊6Dic6  C4×S3×Q8  C4×Q8⋊3S3  Q8⋊6D12  C42.232D6  C42.131D6  C42.132D6  C12⋊(C4○D4)  Dic6⋊20D4  C4⋊C4.178D6  C6.342+ 1+4  C4⋊C4⋊21D6  C6.402+ 1+4  D12⋊20D4  C6.422+ 1+4  C6.432+ 1+4  C6.442+ 1+4  C6.452+ 1+4  (Q8×Dic3)⋊C2  C4⋊C4.187D6  C4⋊C4⋊26D6  D12⋊22D4  Dic6⋊22D4  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⋊C4⋊28D6  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  D12⋊10D4  Dic6⋊10D4  C42⋊22D6  C42⋊23D6  C42.234D6  C42.143D6  Dic6⋊7Q8  C42.236D6  D12⋊7Q8  C42.237D6  C42.150D6  C42.151D6  C42.152D6  C42.153D6  C42.154D6  C42.159D6  C42.160D6  C42⋊25D6  C42⋊26D6  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  D12⋊23D6  Dic6⋊12D6  D12⋊15D6  D20⋊24D6  D30.C23  D20⋊16D6

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

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