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G = C32.S4order 216 = 23·33

The non-split extension by C32 of S4 acting via S4/C22=S3

non-abelian, soluble, monomial

Aliases: C32.S4, C62.5S3, C3.S4⋊C3, C3.A4⋊C6, C22⋊(C9⋊C6), C3.1(C3×S4), C32.A4⋊C2, (C2×C6).2(C3×S3), SmallGroup(216,90)

Series: Derived Chief Lower central Upper central

C1C22C3.A4 — C32.S4
C1C22C2×C6C3.A4C32.A4 — C32.S4
C3.A4 — C32.S4
C1

Generators and relations for C32.S4
 G = < a,b,c,d,e,f | a3=b3=c2=d2=f2=1, e3=b, ab=ba, ac=ca, ad=da, eae-1=ab-1, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=b-1e2 >

3C2
18C2
3C3
9C22
9C4
3C6
3C6
6S3
6C6
18C6
4C9
8C9
9D4
3D6
3Dic3
3C2×C6
9C2×C6
9C12
3C3×C6
4D9
6C3×S3
43- 1+2
3C3⋊D4
9C3×D4
2C3.A4
3C3×Dic3
3S3×C6
4C9⋊C6
3C3×C3⋊D4

Character table of C32.S4

 class 12A2B3A3B3C46A6B6C6D6E6F6G9A9B9C12A12B
 size 1318233183366618182424241818
ρ11111111111111111111    trivial
ρ211-1111-111111-1-1111-1-1    linear of order 2
ρ311-11ζ3ζ32-1ζ3ζ32ζ3ζ321ζ65ζ6ζ31ζ32ζ6ζ65    linear of order 6
ρ41111ζ3ζ321ζ3ζ32ζ3ζ321ζ3ζ32ζ31ζ32ζ32ζ3    linear of order 3
ρ511-11ζ32ζ3-1ζ32ζ3ζ32ζ31ζ6ζ65ζ321ζ3ζ65ζ6    linear of order 6
ρ61111ζ32ζ31ζ32ζ3ζ32ζ31ζ32ζ3ζ321ζ3ζ3ζ32    linear of order 3
ρ722022202222200-1-1-100    orthogonal lifted from S3
ρ82202-1+-3-1--30-1+-3-1--3-1+-3-1--3200ζ65-1ζ600    complex lifted from C3×S3
ρ92202-1--3-1+-30-1--3-1+-3-1--3-1+-3200ζ6-1ζ6500    complex lifted from C3×S3
ρ103-11333-1-1-1-1-1-111000-1-1    orthogonal lifted from S4
ρ113-1-13331-1-1-1-1-1-1-100011    orthogonal lifted from S4
ρ123-113-3-3-3/2-3+3-3/2-1ζ6ζ65ζ6ζ65-1ζ32ζ3000ζ65ζ6    complex lifted from C3×S4
ρ133-1-13-3+3-3/2-3-3-3/21ζ65ζ6ζ65ζ6-1ζ65ζ6000ζ32ζ3    complex lifted from C3×S4
ρ143-113-3+3-3/2-3-3-3/2-1ζ65ζ6ζ65ζ6-1ζ3ζ32000ζ6ζ65    complex lifted from C3×S4
ρ153-1-13-3-3-3/2-3+3-3/21ζ6ζ65ζ6ζ65-1ζ6ζ65000ζ3ζ32    complex lifted from C3×S4
ρ166-20-300044-2-210000000    orthogonal faithful
ρ17660-30000000-30000000    orthogonal lifted from C9⋊C6
ρ186-20-3000-2-2-3-2+2-31+-31--310000000    complex faithful
ρ196-20-3000-2+2-3-2-2-31--31+-310000000    complex faithful

Permutation representations of C32.S4
On 18 points - transitive group 18T98
Generators in S18
(2 8 5)(3 6 9)(10 16 13)(11 14 17)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(1 15)(2 16)(4 18)(5 10)(7 12)(8 13)
(2 16)(3 17)(5 10)(6 11)(8 13)(9 14)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 18)(8 17)(9 16)

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

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

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

G:=TransitiveGroup(18,98);

On 18 points - transitive group 18T101
Generators in S18
(2 8 5)(3 6 9)(10 13 16)(12 18 15)
(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)| (2,8,5)(3,6,9)(10,13,16)(12,18,15), (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( (2,8,5)(3,6,9)(10,13,16)(12,18,15), (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([(2,8,5),(3,6,9),(10,13,16),(12,18,15)], [(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(18,101);

C32.S4 is a maximal quotient of   C32.CSU2(𝔽3)  C32.GL2(𝔽3)  C62.Dic3

Matrix representation of C32.S4 in GL6(ℤ)

100000
010000
000-100
001-100
0000-11
0000-10
,
-110000
-100000
00-1100
00-1000
0000-11
0000-10
,
-100000
0-10000
00-1000
000-100
000010
000001
,
100000
010000
00-1000
000-100
0000-10
00000-1
,
0000-11
0000-10
100000
010000
001000
000100
,
010000
100000
0000-10
0000-11
00-1000
00-1100

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,-1,-1,0,0,0,0,1,0,0,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,0,-1,-1,0,0,0,0,0,1,0,0] >;

C32.S4 in GAP, Magma, Sage, TeX

C_3^2.S_4
% in TeX

G:=Group("C3^2.S4");
// GroupNames label

G:=SmallGroup(216,90);
// by ID

G=gap.SmallGroup(216,90);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,542,224,122,867,3244,556,1949,989]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^3=c^2=d^2=f^2=1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^-1,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

Subgroup lattice of C32.S4 in TeX
Character table of C32.S4 in TeX

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