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## G = C9×S4order 216 = 23·33

### Direct product of C9 and S4

Aliases: C9×S4, A4⋊C18, (C3×S4).C3, C22⋊(S3×C9), (C9×A4)⋊1C2, (C2×C18)⋊1S3, C3.4(C3×S4), (C3×A4).2C6, (C2×C6).1(C3×S3), SmallGroup(216,89)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C9×S4
 Chief series C1 — C22 — A4 — C3×A4 — C9×A4 — C9×S4
 Lower central A4 — C9×S4
 Upper central C1 — C9

Generators and relations for C9×S4
G = < a,b,c,d,e | a9=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

Smallest permutation representation of C9×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)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 34)(11 35)(12 36)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 28)(7 29)(8 30)(9 31)(10 25)(11 26)(12 27)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(10 25 34)(11 26 35)(12 27 36)(13 19 28)(14 20 29)(15 21 30)(16 22 31)(17 23 32)(18 24 33)
(10 25)(11 26)(12 27)(13 19)(14 20)(15 21)(16 22)(17 23)(18 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), (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33), (1,32)(2,33)(3,34)(4,35)(5,36)(6,28)(7,29)(8,30)(9,31)(10,25)(11,26)(12,27)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (10,25,34)(11,26,35)(12,27,36)(13,19,28)(14,20,29)(15,21,30)(16,22,31)(17,23,32)(18,24,33), (10,25)(11,26)(12,27)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)>;

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

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

45 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4 6A 6B 6C 6D 9A ··· 9F 9G ··· 9L 12A 12B 18A ··· 18F 18G ··· 18L 36A ··· 36F order 1 2 2 3 3 3 3 3 4 6 6 6 6 9 ··· 9 9 ··· 9 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 3 6 1 1 8 8 8 6 3 3 6 6 1 ··· 1 8 ··· 8 6 6 3 ··· 3 6 ··· 6 6 ··· 6

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 3 type + + + + image C1 C2 C3 C6 C9 C18 S3 C3×S3 S3×C9 S4 C3×S4 C9×S4 kernel C9×S4 C9×A4 C3×S4 C3×A4 S4 A4 C2×C18 C2×C6 C22 C9 C3 C1 # reps 1 1 2 2 6 6 1 2 6 2 4 12

Matrix representation of C9×S4 in GL5(𝔽37)

 34 0 0 0 0 0 34 0 0 0 0 0 26 0 0 0 0 0 26 0 0 0 0 0 26
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 36 1 0 0 0 36 0 0 0 1 36 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 36 0 0 1 0 36 0 0 0 0 36
,
 36 36 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 36 36 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(5,GF(37))| [34,0,0,0,0,0,34,0,0,0,0,0,26,0,0,0,0,0,26,0,0,0,0,0,26],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,36,36,36,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,36,36,36],[36,1,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[36,0,0,0,0,36,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

C9×S4 in GAP, Magma, Sage, TeX

C_9\times S_4
% in TeX

G:=Group("C9xS4");
// GroupNames label

G:=SmallGroup(216,89);
// by ID

G=gap.SmallGroup(216,89);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,43,867,3244,202,1949,347]);
// Polycyclic

G:=Group<a,b,c,d,e|a^9=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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