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## G = C32×S4order 216 = 23·33

### Direct product of C32 and S4

Aliases: C32×S4, C621S3, A4⋊(C3×C6), (C3×A4)⋊3C6, C22⋊(S3×C32), (C32×A4)⋊1C2, (C2×C6)⋊1(C3×S3), SmallGroup(216,163)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C32×S4
 Chief series C1 — C22 — A4 — C3×A4 — C32×A4 — C32×S4
 Lower central A4 — C32×S4
 Upper central C1 — C32

Generators and relations for C32×S4
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, C2×C6, C2×C6, C3×S3, C3×C6, C3×D4, S4, C33, C3×C12, C3×A4, C3×A4, C62, C62, S3×C32, D4×C32, C3×S4, C32×A4, C32×S4
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, S4, S3×C32, C3×S4, C32×S4

Smallest permutation representation of C32×S4
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 2A 2B 3A ··· 3H 3I ··· 3Q 4 6A ··· 6H 6I ··· 6P 12A ··· 12H order 1 2 2 3 ··· 3 3 ··· 3 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 3 6 1 ··· 1 8 ··· 8 6 3 ··· 3 6 ··· 6 6 ··· 6

45 irreducible representations

 dim 1 1 1 1 2 2 3 3 type + + + + image C1 C2 C3 C6 S3 C3×S3 S4 C3×S4 kernel C32×S4 C32×A4 C3×S4 C3×A4 C62 C2×C6 C32 C3 # reps 1 1 8 8 1 8 2 16

Matrix representation of C32×S4 in GL7(𝔽13)

 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 0 1 0 0 0 0 12 12 12 0 0 0 0 1 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 12 12 12 0 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 12 12 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 1 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 12 0 0 0 0 0 1 0
,
 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 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

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] >;

C32×S4 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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