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## G = C2×C3.S4order 144 = 24·32

### Direct product of C2 and C3.S4

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 1 2A 2B 2C 2D 2E 3 4A 4B 6A 6B 6C 9A 9B 9C 18A 18B 18C size 1 1 3 3 18 18 2 18 18 2 6 6 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 1 1 1 -1 -1 -1 linear of order 2 ρ3 1 -1 1 -1 -1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 -2 2 -2 0 0 2 0 0 -2 -2 2 -1 -1 -1 1 1 1 orthogonal lifted from D6 ρ6 2 2 2 2 0 0 2 0 0 2 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 2 2 0 0 -1 0 0 -1 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ8 2 -2 2 -2 0 0 -1 0 0 1 1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 orthogonal lifted from D18 ρ9 2 -2 2 -2 0 0 -1 0 0 1 1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 orthogonal lifted from D18 ρ10 2 2 2 2 0 0 -1 0 0 -1 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ11 2 2 2 2 0 0 -1 0 0 -1 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ12 2 -2 2 -2 0 0 -1 0 0 1 1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 orthogonal lifted from D18 ρ13 3 -3 -1 1 -1 1 3 -1 1 -3 1 -1 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ14 3 -3 -1 1 1 -1 3 1 -1 -3 1 -1 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ15 3 3 -1 -1 -1 -1 3 1 1 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from S4 ρ16 3 3 -1 -1 1 1 3 -1 -1 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from S4 ρ17 6 6 -2 -2 0 0 -3 0 0 -3 1 1 0 0 0 0 0 0 orthogonal lifted from C3.S4 ρ18 6 -6 -2 2 0 0 -3 0 0 3 -1 1 0 0 0 0 0 0 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)

 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 35 0 0 0 20 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 3 0 0 0 7 14 0 0 0 0 0 0 0 1 0 0 36 0 0 0 0 0 36 0
,
 14 34 0 0 0 28 23 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 36

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

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