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G = C32xS4order 216 = 23·33

Direct product of C32 and S4

direct product, non-abelian, soluble, monomial

Aliases: C32xS4, C62:1S3, A4:(C3xC6), (C3xA4):3C6, C22:(S3xC32), (C32xA4):1C2, (C2xC6):1(C3xS3), SmallGroup(216,163)

Series: Derived Chief Lower central Upper central

C1C22A4 — C32xS4
C1C22A4C3xA4C32xA4 — C32xS4
A4 — C32xS4
C1C32

Generators and relations for C32xS4
 G = < a,b,c,d,e,f | a3=b3=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 260 in 82 conjugacy classes, 24 normal (8 characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, C6, D4, C32, C32, C12, A4, A4, C2xC6, C2xC6, C3xS3, C3xC6, C3xD4, S4, C33, C3xC12, C3xA4, C3xA4, C62, C62, S3xC32, D4xC32, C3xS4, C32xA4, C32xS4
Quotients: C1, C2, C3, S3, C6, C32, C3xS3, C3xC6, S4, S3xC32, C3xS4, C32xS4

Smallest permutation representation of C32xS4
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 23 28)(2 24 29)(3 22 30)(4 31 26)(5 32 27)(6 33 25)(7 15 35)(8 13 36)(9 14 34)(10 21 18)(11 19 16)(12 20 17)
(1 16)(2 17)(3 18)(4 9)(5 7)(6 8)(10 22)(11 23)(12 24)(13 33)(14 31)(15 32)(19 28)(20 29)(21 30)(25 36)(26 34)(27 35)
(1 32)(2 33)(3 31)(4 30)(5 28)(6 29)(7 19)(8 20)(9 21)(10 34)(11 35)(12 36)(13 17)(14 18)(15 16)(22 26)(23 27)(24 25)
(4 9 21)(5 7 19)(6 8 20)(10 26 34)(11 27 35)(12 25 36)(13 17 33)(14 18 31)(15 16 32)
(7 19)(8 20)(9 21)(10 34)(11 35)(12 36)(13 17)(14 18)(15 16)

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

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

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

45 conjugacy classes

class 1 2A2B3A···3H3I···3Q 4 6A···6H6I···6P12A···12H
order1223···33···346···66···612···12
size1361···18···863···36···66···6

45 irreducible representations

dim11112233
type++++
imageC1C2C3C6S3C3xS3S4C3xS4
kernelC32xS4C32xA4C3xS4C3xA4C62C2xC6C32C3
# reps118818216

Matrix representation of C32xS4 in GL7(F13)

9000000
0900000
0010000
0001000
0000100
0000010
0000001
,
1000000
0100000
0090000
0009000
0000100
0000010
0000001
,
1000000
0100000
0010000
0001000
0000001
0000121212
0000100
,
1000000
0100000
0010000
0001000
0000121212
0000001
0000010
,
121200000
1000000
00012000
00112000
0000100
0000121212
0000010
,
121200000
0100000
00120000
00121000
0000100
0000001
0000010

G:=sub<GL(7,GF(13))| [9,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0,0,0,0,0,1,12,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,1,0,0,0,0,12,1,0],[12,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0] >;

C32xS4 in GAP, Magma, Sage, TeX

C_3^2\times S_4
% in TeX

G:=Group("C3^2xS4");
// GroupNames label

G:=SmallGroup(216,163);
// by ID

G=gap.SmallGroup(216,163);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,867,3244,202,1949,347]);
// Polycyclic

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

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