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G = C3.S4order 72 = 23·32

The non-split extension by C3 of S4 acting via S4/A4=C2

non-abelian, soluble, monomial

Aliases: C3.S4, C22⋊D9, C3.A4⋊C2, (C2×C6).S3, SmallGroup(72,15)

Series: Derived Chief Lower central Upper central

C1C22C3.A4 — C3.S4
C1C22C2×C6C3.A4 — C3.S4
C3.A4 — C3.S4
C1

Generators and relations for C3.S4
 G = < a,b,c,d,e | a3=b2=c2=e2=1, d3=a, ab=ba, ac=ca, ad=da, eae=a-1, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=a-1d2 >

3C2
18C2
9C4
9C22
3C6
6S3
4C9
9D4
3D6
3Dic3
4D9
3C3⋊D4

Character table of C3.S4

 class 12A2B3469A9B9C
 size 13182186888
ρ1111111111    trivial
ρ211-11-11111    linear of order 2
ρ3220202-1-1-1    orthogonal lifted from S3
ρ4220-10-1ζ9594ζ9792ζ989    orthogonal lifted from D9
ρ5220-10-1ζ989ζ9594ζ9792    orthogonal lifted from D9
ρ6220-10-1ζ9792ζ989ζ9594    orthogonal lifted from D9
ρ73-1-131-1000    orthogonal lifted from S4
ρ83-113-1-1000    orthogonal lifted from S4
ρ96-20-301000    orthogonal faithful

Permutation representations of C3.S4
On 18 points - transitive group 18T38
Generators in S18
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(1 17)(2 18)(4 11)(5 12)(7 14)(8 15)
(2 18)(3 10)(5 12)(6 13)(8 15)(9 16)
(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 15)(11 14)(12 13)(16 18)

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

G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (1,17)(2,18)(4,11)(5,12)(7,14)(8,15), (2,18)(3,10)(5,12)(6,13)(8,15)(9,16), (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)(10,15)(11,14)(12,13)(16,18) );

G=PermutationGroup([(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18)], [(1,17),(2,18),(4,11),(5,12),(7,14),(8,15)], [(2,18),(3,10),(5,12),(6,13),(8,15),(9,16)], [(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),(10,15),(11,14),(12,13),(16,18)])

G:=TransitiveGroup(18,38);

On 18 points - transitive group 18T39
Generators in S18
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(1 17)(2 18)(4 11)(5 12)(7 14)(8 15)
(2 18)(3 10)(5 12)(6 13)(8 15)(9 16)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(9 18)

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

G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (1,17)(2,18)(4,11)(5,12)(7,14)(8,15), (2,18)(3,10)(5,12)(6,13)(8,15)(9,16), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(9,18) );

G=PermutationGroup([(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18)], [(1,17),(2,18),(4,11),(5,12),(7,14),(8,15)], [(2,18),(3,10),(5,12),(6,13),(8,15),(9,16)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(9,18)])

G:=TransitiveGroup(18,39);

C3.S4 is a maximal subgroup of
C32.S4  C9⋊S4  C32.3S4  C42⋊D9  C24⋊D9  C22⋊D45
C3.S4 is a maximal quotient of
Q8.D9  Q8⋊D9  C6.S4  C9.S4  C32.3S4  C42⋊D9  C24⋊D9  C22⋊D45

Matrix representation of C3.S4 in GL5(𝔽37)

361000
360000
00100
00010
00001
,
10000
01000
003600
000360
00001
,
10000
01000
00100
000360
000036
,
2631000
620000
00001
00100
00010
,
01000
10000
00100
00001
00010

G:=sub<GL(5,GF(37))| [36,36,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36],[26,6,0,0,0,31,20,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C3.S4 in GAP, Magma, Sage, TeX

C_3.S_4
% in TeX

G:=Group("C3.S4");
// GroupNames label

G:=SmallGroup(72,15);
// by ID

G=gap.SmallGroup(72,15);
# by ID

G:=PCGroup([5,-2,-3,-3,-2,2,101,66,182,723,368,454,684]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^2=e^2=1,d^3=a,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=a^-1*d^2>;
// generators/relations

Export

Subgroup lattice of C3.S4 in TeX
Character table of C3.S4 in TeX

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