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G = C2×S3×D4order 96 = 25·3

Direct product of C2, S3 and D4

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

Aliases: C2×S3×D4, C12⋊C23, C234D6, D62C23, C6.5C24, D127C22, Dic31C23, (C2×C4)⋊6D6, C62(C2×D4), (C2×C6)⋊C23, (C6×D4)⋊5C2, C32(C22×D4), C41(C22×S3), (C2×D12)⋊11C2, (C4×S3)⋊3C22, (S3×C23)⋊4C2, (C2×C12)⋊2C22, (C3×D4)⋊5C22, C3⋊D41C22, C2.6(S3×C23), C222(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

C1C6 — C2×S3×D4
C1C3C6D6C22×S3S3×C23 — C2×S3×D4
C3C6 — C2×S3×D4
C1C22C2×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 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22, C22 [×4], C22 [×34], S3 [×4], S3 [×4], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×5], D4 [×4], D4 [×12], C23 [×2], C23 [×19], Dic3 [×2], C12 [×2], D6 [×10], D6 [×20], C2×C6, C2×C6 [×4], C2×C6 [×4], C22×C4, C2×D4, C2×D4 [×11], C24 [×2], C4×S3 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×S3 [×10], C22×S3 [×8], C22×C6 [×2], C22×D4, S3×C2×C4, C2×D12, S3×D4 [×8], C2×C3⋊D4 [×2], C6×D4, S3×C23 [×2], C2×S3×D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, S3×D4 [×2], 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⋊C419D6  D4⋊D12  D65SD16  D12⋊D4  D66SD16  C4213D6  D45D12  C247D6  C248D6  C6.372+ 1+4  C6.382+ 1+4  D1219D4  C6.402+ 1+4  D1220D4  C6.1202+ 1+4  C6.1212+ 1+4  C4220D6  D1210D4  C4228D6  D1211D4  C6.1452+ 1+4
C2×S3×D4 is a maximal quotient of
C24.38D6  C6.2- 1+4  C4214D6  C42.228D6  D1223D4  D1224D4  Dic623D4  Dic624D4  C24.67D6  C247D6  C248D6  C24.44D6  C24.45D6  C12⋊(C4○D4)  C6.322+ 1+4  Dic619D4  Dic620D4  C6.372+ 1+4  C4⋊C421D6  C6.382+ 1+4  C6.722- 1+4  D1219D4  C6.402+ 1+4  C6.732- 1+4  D1220D4  C4⋊C426D6  C6.162- 1+4  C6.172- 1+4  D1221D4  D1222D4  Dic621D4  Dic622D4  C6.792- 1+4  C6.1202+ 1+4  C6.1212+ 1+4  C6.822- 1+4  C4⋊C428D6  C42.233D6  C4220D6  C42.141D6  D1210D4  Dic610D4  C4228D6  C42.238D6  D1211D4  Dic611D4  C42.171D6  C42.240D6  D1212D4  D128Q8  D813D6  SD1613D6  D12.30D4  SD16⋊D6  D815D6  D811D6  D8.10D6  D84D6  D85D6  D86D6  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

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Character table of C2×S3×D4 in TeX

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