Copied to
clipboard

G = C2xC3.S4  order 144 = 24·32

Direct product of C2 and C3.S4

direct product, non-abelian, soluble, monomial

Aliases: C2xC3.S4, C23:D9, C6.4S4, C22:D18, C3.(C2xS4), (C2xC6).D6, C3.A4:C22, (C22xC6).3S3, (C2xC3.A4):C2, SmallGroup(144,109)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C3.A4 — C2xC3.S4
Chief series
C1 — C22 — C2xC6 — C3.A4 — C3.S4 — C2xC3.S4
Lower central
C3.A4 — C2xC3.S4
Upper central
C1 — C2

Generators and relations for C2xC3.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, C2xC4, D4, C23, C23, C9, Dic3, D6, C2xC6, C2xC6, C2xD4, D9, C18, C2xDic3, C3:D4, C22xS3, C22xC6, C3.A4, D18, C2xC3:D4, C3.S4, C2xC3.A4, C2xC3.S4
Quotients: C1, C2, C22, S3, D6, D9, S4, D18, C2xS4, C3.S4, C2xC3.S4

Character table of C2xC3.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 C2xS4
ρ143-3-111-131-1-31-1000000    orthogonal lifted from C2xS4
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 C2xC3.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);
 

C2xC3.S4 is a maximal subgroup of   C22:D36  C23.D18
C2xC3.S4 is a maximal quotient of   C12.1S4  C22:D36  Q8.D18  C12.3S4  C12.11S4  C12.4S4  C23.D18

Matrix representation of C2xC3.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] >;
 

C2xC3.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 C2xC3.S4 in TeX

׿
x
:
Z
F
o
wr
Q
<