Copied to
clipboard

G = C2×S3×D4  order 96 = 25·3

Direct product of C2, S3 and D4

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

Aliases: C2×S3×D4, C12⋊C23, C23⋊4D6, D6⋊2C23, C6.5C24, D12⋊7C22, Dic3⋊1C23, (C2×C4)⋊6D6, C6⋊2(C2×D4), (C2×C6)⋊C23, (C6×D4)⋊5C2, C3⋊2(C22×D4), C4⋊1(C22×S3), (C2×D12)⋊11C2, (C4×S3)⋊3C22, (S3×C23)⋊4C2, (C2×C12)⋊2C22, (C3×D4)⋊5C22, C3⋊D4⋊1C22, C2.6(S3×C23), C22⋊2(C22×S3), (C22×C6)⋊4C22, (C22×S3)⋊6C22, (C2×Dic3)⋊8C22, (S3×C2×C4)⋊3C2, (C2×C3⋊D4)⋊9C2, SmallGroup(96,209)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C2×S3×D4
Chief series
C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — C2×S3×D4
Lower central
C3 — C6 — C2×S3×D4
Upper central
C1 — C22 — C2×D4

Generators and relations for C2×S3×D4
 G = < a,b,c,d,e | a2=b3=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 562 in 236 conjugacy classes, 97 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, S3, C6, C6, C6, C2×C4, C2×C4, D4, D4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C2×D4, C2×D4, C24, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C22×D4, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C2×S3×D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S3×D4, S3×C23, C2×S3×D4

Character table of C2×S3×D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M2N2O34A4B4C4D6A6B6C6D6E6F6G12A12B
 size 111122223333666622266222444444
ρ1111111111111111111111111111111    trivial
ρ211-1-1-111-11-1-1111-1-11-11-11-1-111-1-111-1    linear of order 2
ρ311-1-11-1-111-1-11-1-1111-11-11-1-11-111-11-1    linear of order 2
ρ41111-1-1-1-11111-1-1-1-111111111-1-1-1-111    linear of order 2
ρ51111-1-1111111-11-111-1-1-1-1111-1-111-1-1    linear of order 2
ρ611-1-11-11-11-1-11-111-111-11-1-1-11-11-11-11    linear of order 2
ρ711-1-1-11-111-1-111-1-1111-11-1-1-111-11-1-11    linear of order 2
ρ8111111-1-111111-11-11-1-1-1-111111-1-1-1-1    linear of order 2
ρ911-1-11-11-1-111-11-1-1111-1-11-1-11-11-11-11    linear of order 2
ρ101111-1-111-1-1-1-11-11-11-1-111111-1-111-1-1    linear of order 2
ρ11111111-1-1-1-1-1-1-11-111-1-11111111-1-1-1-1    linear of order 2
ρ1211-1-1-11-11-111-1-111-111-1-11-1-111-11-1-11    linear of order 2
ρ1311-1-1-111-1-111-1-1-1111-111-1-1-111-1-111-1    linear of order 2
ρ1411111111-1-1-1-1-1-1-1-1111-1-1111111111    linear of order 2
ρ151111-1-1-1-1-1-1-1-11111111-1-1111-1-1-1-111    linear of order 2
ρ1611-1-11-1-11-111-111-1-11-111-1-1-11-111-11-1    linear of order 2
ρ172222-2-2-2-200000000-12200-1-1-11111-1-1    orthogonal lifted from D6
ρ182-2-22000022-2-2000020000-22-2000000    orthogonal lifted from D4
ρ192-22-20000-22-220000200002-2-2000000    orthogonal lifted from D4
ρ20222222-2-200000000-1-2-200-1-1-1-1-11111    orthogonal lifted from D6
ρ2122-2-2-222-200000000-1-220011-1-111-1-11    orthogonal lifted from D6
ρ2222-2-22-22-200000000-12-20011-11-11-11-1    orthogonal lifted from D6
ρ2322-2-22-2-2200000000-1-220011-11-1-11-11    orthogonal lifted from D6
ρ2422-2-2-22-2200000000-12-20011-1-11-111-1    orthogonal lifted from D6
ρ252222222200000000-12200-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ262-22-200002-22-20000200002-2-2000000    orthogonal lifted from D4
ρ272-2-220000-2-222000020000-22-2000000    orthogonal lifted from D4
ρ282222-2-22200000000-1-2-200-1-1-111-1-111    orthogonal lifted from D6
ρ294-44-4000000000000-20000-222000000    orthogonal lifted from S3×D4
ρ304-4-44000000000000-200002-22000000    orthogonal lifted from S3×D4

Permutation representations of C2×S3×D4
►On 24 points - transitive group 24T143
Generators in S24
(1 8)(2 5)(3 6)(4 7)(9 21)(10 22)(11 23)(12 24)(13 20)(14 17)(15 18)(16 19)

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

(1 8)(2 5)(3 6)(4 7)(9 20)(10 17)(11 18)(12 19)(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 8)(2 7)(3 6)(4 5)(9 21)(10 24)(11 23)(12 22)(13 20)(14 19)(15 18)(16 17)
 
G:=sub<Sym(24)| (1,8)(2,5)(3,6)(4,7)(9,21)(10,22)(11,23)(12,24)(13,20)(14,17)(15,18)(16,19), (1,21,20)(2,22,17)(3,23,18)(4,24,19)(5,10,14)(6,11,15)(7,12,16)(8,9,13), (1,8)(2,5)(3,6)(4,7)(9,20)(10,17)(11,18)(12,19)(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), (1,8)(2,7)(3,6)(4,5)(9,21)(10,24)(11,23)(12,22)(13,20)(14,19)(15,18)(16,17)>;
 
G:=Group( (1,8)(2,5)(3,6)(4,7)(9,21)(10,22)(11,23)(12,24)(13,20)(14,17)(15,18)(16,19), (1,21,20)(2,22,17)(3,23,18)(4,24,19)(5,10,14)(6,11,15)(7,12,16)(8,9,13), (1,8)(2,5)(3,6)(4,7)(9,20)(10,17)(11,18)(12,19)(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), (1,8)(2,7)(3,6)(4,5)(9,21)(10,24)(11,23)(12,22)(13,20)(14,19)(15,18)(16,17) );
 
G=PermutationGroup([[(1,8),(2,5),(3,6),(4,7),(9,21),(10,22),(11,23),(12,24),(13,20),(14,17),(15,18),(16,19)], [(1,21,20),(2,22,17),(3,23,18),(4,24,19),(5,10,14),(6,11,15),(7,12,16),(8,9,13)], [(1,8),(2,5),(3,6),(4,7),(9,20),(10,17),(11,18),(12,19),(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)], [(1,8),(2,7),(3,6),(4,5),(9,21),(10,24),(11,23),(12,22),(13,20),(14,19),(15,18),(16,17)]])
 
G:=TransitiveGroup(24,143);
 

C2×S3×D4 is a maximal subgroup of
 C4⋊C4⋊19D6  D4⋊D12  D6⋊5SD16  D12⋊D4  D6⋊6SD16  C42⋊13D6  D4⋊5D12  C24⋊7D6  C24⋊8D6  C6.372+ 1+4  C6.382+ 1+4  D12⋊19D4  C6.402+ 1+4  D12⋊20D4  C6.1202+ 1+4  C6.1212+ 1+4  C42⋊20D6  D12⋊10D4  C42⋊28D6  D12⋊11D4  C6.1452+ 1+4
C2×S3×D4 is a maximal quotient of
 C24.38D6  C6.2- 1+4  C42⋊14D6  C42.228D6  D12⋊23D4  D12⋊24D4  Dic6⋊23D4  Dic6⋊24D4  C24.67D6  C24⋊7D6  C24⋊8D6  C24.44D6  C24.45D6  C12⋊(C4○D4)  C6.322+ 1+4  Dic6⋊19D4  Dic6⋊20D4  C6.372+ 1+4  C4⋊C4⋊21D6  C6.382+ 1+4  C6.722- 1+4  D12⋊19D4  C6.402+ 1+4  C6.732- 1+4  D12⋊20D4  C4⋊C4⋊26D6  C6.162- 1+4  C6.172- 1+4  D12⋊21D4  D12⋊22D4  Dic6⋊21D4  Dic6⋊22D4  C6.792- 1+4  C6.1202+ 1+4  C6.1212+ 1+4  C6.822- 1+4  C4⋊C4⋊28D6  C42.233D6  C42⋊20D6  C42.141D6  D12⋊10D4  Dic6⋊10D4  C42⋊28D6  C42.238D6  D12⋊11D4  Dic6⋊11D4  C42.171D6  C42.240D6  D12⋊12D4  D12⋊8Q8  D8⋊13D6  SD16⋊13D6  D12.30D4  SD16⋊D6  D8⋊15D6  D8⋊11D6  D8.10D6  D8⋊4D6  D8⋊5D6  D8⋊6D6  D24⋊C22  C24.C23  SD16.D6

Matrix representation of C2×S3×D4 ►in GL4(𝔽13) generated by

12000
01200
0010
0001
,
01200
11200
0010
0001
,
01200
12000
00120
00012
,
12000
01200
00110
00512
,
1000
0100
0010
00512
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,12,12,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,12,0,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,5,0,0,10,12],[1,0,0,0,0,1,0,0,0,0,1,5,0,0,0,12] >;
 

C2×S3×D4 in GAP, Magma, Sage, TeX 

C_2\times S_3\times D_4
 
% in TeX
 
G:=Group("C2xS3xD4");
 
// GroupNames label
 
G:=SmallGroup(96,209);
 
// by ID
 
G=gap.SmallGroup(96,209);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,159,2309]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

Export

Character table of C2×S3×D4 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁