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G = C2×C3.S4  order 144 = 24·32

Direct product of C2 and C3.S4

direct product, non-abelian, soluble, monomial

Aliases: C2×C3.S4, C23⋊D9, C6.4S4, C22⋊D18, C3.(C2×S4), (C2×C6).D6, C3.A4⋊C22, (C22×C6).3S3, (C2×C3.A4)⋊C2, SmallGroup(144,109)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C3.A4 — C2×C3.S4
Chief series
C1 — C22 — C2×C6 — C3.A4 — C3.S4 — C2×C3.S4
Lower central
C3.A4 — C2×C3.S4
Upper central
C1 — C2

Generators and relations for C2×C3.S4
 G = < a,b,c,d,e,f | a2=b3=c2=d2=f2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=b-1e2 >

Subgroups: 285 in 56 conjugacy classes, 13 normal (11 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C9, Dic3, D6, C2×C6, C2×C6, C2×D4, D9, C18, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C3.A4, D18, C2×C3⋊D4, C3.S4, C2×C3.A4, C2×C3.S4
Quotients: C1, C2, C22, S3, D6, D9, S4, D18, C2×S4, C3.S4, C2×C3.S4

Character table of C2×C3.S4

 class 12A2B2C2D2E34A4B6A6B6C9A9B9C18A18B18C
 size 1133181821818266888888
ρ1111111111111111111    trivial
ρ21-11-11-11-11-1-11111-1-1-1    linear of order 2
ρ31-11-1-1111-1-1-11111-1-1-1    linear of order 2
ρ41111-1-11-1-1111111111    linear of order 2
ρ52-22-200200-2-22-1-1-1111    orthogonal lifted from D6
ρ6222200200222-1-1-1-1-1-1    orthogonal lifted from S3
ρ7222200-100-1-1-1ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ97+ζ92ζ95+ζ94ζ98+ζ9    orthogonal lifted from D9
ρ82-22-200-10011-1ζ95+ζ94ζ98+ζ9ζ97+ζ92-ζ98-ζ9-ζ97-ζ92-ζ95-ζ94    orthogonal lifted from D18
ρ92-22-200-10011-1ζ98+ζ9ζ97+ζ92ζ95+ζ94-ζ97-ζ92-ζ95-ζ94-ζ98-ζ9    orthogonal lifted from D18
ρ10222200-100-1-1-1ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ95+ζ94ζ98+ζ9ζ97+ζ92    orthogonal lifted from D9
ρ11222200-100-1-1-1ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ98+ζ9ζ97+ζ92ζ95+ζ94    orthogonal lifted from D9
ρ122-22-200-10011-1ζ97+ζ92ζ95+ζ94ζ98+ζ9-ζ95-ζ94-ζ98-ζ9-ζ97-ζ92    orthogonal lifted from D18
ρ133-3-11-113-11-31-1000000    orthogonal lifted from C2×S4
ρ143-3-111-131-1-31-1000000    orthogonal lifted from C2×S4
ρ1533-1-1-1-13113-1-1000000    orthogonal lifted from S4
ρ1633-1-1113-1-13-1-1000000    orthogonal lifted from S4
ρ1766-2-200-300-311000000    orthogonal lifted from C3.S4
ρ186-6-2200-3003-11000000    orthogonal faithful

Permutation representations of C2×C3.S4
►On 18 points - transitive group 18T67
Generators in S18
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 10)

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

(1 11)(2 12)(4 14)(5 15)(7 17)(8 18)

(2 12)(3 13)(5 15)(6 16)(8 18)(9 10)

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

(2 9)(3 8)(4 7)(5 6)(10 12)(13 18)(14 17)(15 16)
 
G:=sub<Sym(18)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,10), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (1,11)(2,12)(4,14)(5,15)(7,17)(8,18), (2,12)(3,13)(5,15)(6,16)(8,18)(9,10), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (2,9)(3,8)(4,7)(5,6)(10,12)(13,18)(14,17)(15,16)>;
 
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,10), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (1,11)(2,12)(4,14)(5,15)(7,17)(8,18), (2,12)(3,13)(5,15)(6,16)(8,18)(9,10), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (2,9)(3,8)(4,7)(5,6)(10,12)(13,18)(14,17)(15,16) );
 
G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,10)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18)], [(1,11),(2,12),(4,14),(5,15),(7,17),(8,18)], [(2,12),(3,13),(5,15),(6,16),(8,18),(9,10)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(2,9),(3,8),(4,7),(5,6),(10,12),(13,18),(14,17),(15,16)]])
 
G:=TransitiveGroup(18,67);
 

C2×C3.S4 is a maximal subgroup of   C22⋊D36  C23.D18
C2×C3.S4 is a maximal quotient of   C12.1S4  C22⋊D36  Q8.D18  C12.3S4  C12.11S4  C12.4S4  C23.D18

Matrix representation of C2×C3.S4 ►in GL5(𝔽37)

360000
036000
00100
00010
00001
,
135000
2035000
00100
00010
00001
,
10000
01000
003600
00010
000036
,
10000
01000
003600
000360
00001
,
283000
714000
00001
003600
000360
,
1434000
2823000
00010
00100
000036

G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,20,0,0,0,35,35,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,1],[28,7,0,0,0,3,14,0,0,0,0,0,0,36,0,0,0,0,0,36,0,0,1,0,0],[14,28,0,0,0,34,23,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,36] >;
 

C2×C3.S4 in GAP, Magma, Sage, TeX 

C_2\times C_3.S_4
 
% in TeX
 
G:=Group("C2xC3.S4");
 
// GroupNames label
 
G:=SmallGroup(144,109);
 
// by ID
 
G=gap.SmallGroup(144,109);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-2,2,362,122,579,2164,556,1301,989]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^3=c^2=d^2=f^2=1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=b^-1*e^2>;
 
// generators/relations
 

Export

Character table of C2×C3.S4 in TeX

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