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G = C32:4S4  order 216 = 23·33

2nd semidirect product of C32 and S4 acting via S4/A4=C2

non-abelian, soluble, monomial, rational

Aliases: C32:4S4, C62:9S3, C3:(C3:S4), A4:(C3:S3), (C3xA4):4S3, (C32xA4):3C2, C22:(C33:C2), (C2xC6):2(C3:S3), SmallGroup(216,165)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C32xA4 — C32:4S4
Chief series
C1 — C22 — C2xC6 — C62 — C32xA4 — C32:4S4
Lower central
C32xA4 — C32:4S4
Upper central
C1

Generators and relations for C32:4S4
 G = < a,b,c,d | a6=b6=c3=d2=1, ab=ba, cac-1=a4b3, dad=a2b3, cbc-1=a3b, dbd=a3b2, dcd=c-1 >

Subgroups: 1060 in 130 conjugacy classes, 35 normal (6 characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, C6, D4, C32, C32, Dic3, A4, D6, C2xC6, C3:S3, C3xC6, C3:D4, S4, C33, C3:Dic3, C3xA4, C2xC3:S3, C62, C33:C2, C32:7D4, C3:S4, C32xA4, C32:4S4
Quotients: C1, C2, S3, C3:S3, S4, C33:C2, C3:S4, C32:4S4

Character table of C32:4S4

 class 12A2B3A3B3C3D3E3F3G3H3I3J3K3L3M46A6B6C6D
 size 13542222888888888546666
ρ1111111111111111111111    trivial
ρ211-11111111111111-11111    linear of order 2
ρ3220-12-1-1-1-1-1-122-1-120-1-12-1    orthogonal lifted from S3
ρ42202222-1-1-1-1-1-1-1-1-102222    orthogonal lifted from S3
ρ5220-1-12-1-1-12-1-1-1-12202-1-1-1    orthogonal lifted from S3
ρ6220-1-1-12-1-1-1-1-1222-10-12-1-1    orthogonal lifted from S3
ρ72202-1-1-1-1-1-12-1-12-120-1-1-12    orthogonal lifted from S3
ρ8220-1-12-12-1-1-12-12-1-102-1-1-1    orthogonal lifted from S3
ρ9220-1-1-1222-1-1-1-1-1-120-12-1-1    orthogonal lifted from S3
ρ102202-1-1-12-12-1-12-1-1-10-1-1-12    orthogonal lifted from S3
ρ11220-12-1-1-122-1-1-12-1-10-1-12-1    orthogonal lifted from S3
ρ122202-1-1-1-12-1-12-1-12-10-1-1-12    orthogonal lifted from S3
ρ13220-12-1-12-1-12-1-1-12-10-1-12-1    orthogonal lifted from S3
ρ14220-1-12-1-12-12-12-1-1-102-1-1-1    orthogonal lifted from S3
ρ15220-1-1-12-1-1222-1-1-1-10-12-1-1    orthogonal lifted from S3
ρ163-1-133330000000001-1-1-1-1    orthogonal lifted from S4
ρ173-113333000000000-1-1-1-1-1    orthogonal lifted from S4
ρ186-20-3-36-30000000000-2111    orthogonal lifted from C3:S4
ρ196-20-3-3-3600000000001-211    orthogonal lifted from C3:S4
ρ206-206-3-3-30000000000111-2    orthogonal lifted from C3:S4
ρ216-20-36-3-3000000000011-21    orthogonal lifted from C3:S4

Smallest permutation representation of C32:4S4
►On 36 points
Generators in S36
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)

(1 29 12 19 35 16)(2 30 7 20 36 17)(3 25 8 21 31 18)(4 26 9 22 32 13)(5 27 10 23 33 14)(6 28 11 24 34 15)

(2 20 23)(4 22 19)(6 24 21)(7 17 14)(9 13 16)(11 15 18)(25 34 28)(26 29 32)(27 36 30)

(2 21)(3 5)(4 19)(6 23)(7 25)(8 33)(9 29)(10 31)(11 27)(12 35)(13 26)(14 34)(15 30)(16 32)(17 28)(18 36)(20 24)
 
G:=sub<Sym(36)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,29,12,19,35,16)(2,30,7,20,36,17)(3,25,8,21,31,18)(4,26,9,22,32,13)(5,27,10,23,33,14)(6,28,11,24,34,15), (2,20,23)(4,22,19)(6,24,21)(7,17,14)(9,13,16)(11,15,18)(25,34,28)(26,29,32)(27,36,30), (2,21)(3,5)(4,19)(6,23)(7,25)(8,33)(9,29)(10,31)(11,27)(12,35)(13,26)(14,34)(15,30)(16,32)(17,28)(18,36)(20,24)>;
 
G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,29,12,19,35,16)(2,30,7,20,36,17)(3,25,8,21,31,18)(4,26,9,22,32,13)(5,27,10,23,33,14)(6,28,11,24,34,15), (2,20,23)(4,22,19)(6,24,21)(7,17,14)(9,13,16)(11,15,18)(25,34,28)(26,29,32)(27,36,30), (2,21)(3,5)(4,19)(6,23)(7,25)(8,33)(9,29)(10,31)(11,27)(12,35)(13,26)(14,34)(15,30)(16,32)(17,28)(18,36)(20,24) );
 
G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36)], [(1,29,12,19,35,16),(2,30,7,20,36,17),(3,25,8,21,31,18),(4,26,9,22,32,13),(5,27,10,23,33,14),(6,28,11,24,34,15)], [(2,20,23),(4,22,19),(6,24,21),(7,17,14),(9,13,16),(11,15,18),(25,34,28),(26,29,32),(27,36,30)], [(2,21),(3,5),(4,19),(6,23),(7,25),(8,33),(9,29),(10,31),(11,27),(12,35),(13,26),(14,34),(15,30),(16,32),(17,28),(18,36),(20,24)]])
 

C32:4S4 is a maximal subgroup of   C3:S3xS4  S3xC3:S4
C32:4S4 is a maximal quotient of   C32:4CSU2(F3)  C32:5GL2(F3)  C62:10Dic3

Matrix representation of C32:4S4 ►in GL7(Z)

1000000
0100000
00-1-1000
0010000
00000-11
00000-10
00001-10
,
-1100000
-1000000
0001000
00-1-1000
0000-100
0000-101
0000-110
,
0-100000
1-100000
00-1-1000
0010000
0000001
0000100
0000010
,
-1000000
-1100000
0001000
0010000
0000010
0000100
0000001

G:=sub<GL(7,Integers())| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,-1,0,0,0,0,1,0,0],[-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

C32:4S4 in GAP, Magma, Sage, TeX 

C_3^2\rtimes_4S_4
 
% in TeX
 
G:=Group("C3^2:4S4");
 
// GroupNames label
 
G:=SmallGroup(216,165);
 
// by ID
 
G=gap.SmallGroup(216,165);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-2,2,49,218,867,3244,1630,1949,2927]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=c^3=d^2=1,a*b=b*a,c*a*c^-1=a^4*b^3,d*a*d=a^2*b^3,c*b*c^-1=a^3*b,d*b*d=a^3*b^2,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of C32:4S4 in TeX

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