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G = C9⋊S3⋊C32  order 486 = 2·35

2nd semidirect product of C9⋊S3 and C32 acting faithfully

metabelian, supersoluble, monomial

Aliases: C9⋊S3⋊2C32, He3.S3⋊C3, He3.C3⋊1C6, He3.4(C3×S3), (C3×He3).10S3, C33.16(C3×S3), C33.S3⋊2C3, C32.8(C32⋊C6), C32.17(S3×C32), He3.C32⋊1C2, (C3×3- 1+2)⋊3C6, (C3×C9)⋊2(C3×C6), C3.8(C3×C32⋊C6), SmallGroup(486,129)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C9 — C9⋊S3⋊C32
Chief series
C1 — C3 — C32 — C3×C9 — C3×3- 1+2 — He3.C32 — C9⋊S3⋊C32
Lower central
C3×C9 — C9⋊S3⋊C32
Upper central
C1

Generators and relations for C9⋊S3⋊C32
 G = < a,b,c,d,e | a9=b3=c2=d3=e3=1, ab=ba, cac=a-1, dad-1=a4b, eae-1=a7, cbc=b-1, dbd-1=a3b, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 492 in 76 conjugacy classes, 21 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, He3, He3, 3- 1+2, C33, C33, C32⋊C6, C9⋊C6, C9⋊S3, S3×C32, C3×C3⋊S3, He3.C3, He3.C3, C3×He3, C3×3- 1+2, C3×3- 1+2, He3.S3, C3×C32⋊C6, C33.S3, He3.C32, C9⋊S3⋊C32
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C32⋊C6, S3×C32, C3×C32⋊C6, C9⋊S3⋊C32

Permutation representations of C9⋊S3⋊C32
►On 27 points - transitive group 27T148
Generators in S27
(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)

(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)

(2 9)(3 8)(4 7)(5 6)(10 15)(11 14)(12 13)(16 18)(19 21)(22 27)(23 26)(24 25)

(1 20 17)(2 27 18)(3 25 10)(4 23 11)(5 21 12)(6 19 13)(7 26 14)(8 24 15)(9 22 16)

(2 5 8)(3 9 6)(10 16 13)(12 15 18)(19 25 22)(21 24 27)
 
G:=sub<Sym(27)| (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), (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27), (2,9)(3,8)(4,7)(5,6)(10,15)(11,14)(12,13)(16,18)(19,21)(22,27)(23,26)(24,25), (1,20,17)(2,27,18)(3,25,10)(4,23,11)(5,21,12)(6,19,13)(7,26,14)(8,24,15)(9,22,16), (2,5,8)(3,9,6)(10,16,13)(12,15,18)(19,25,22)(21,24,27)>;
 
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), (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27), (2,9)(3,8)(4,7)(5,6)(10,15)(11,14)(12,13)(16,18)(19,21)(22,27)(23,26)(24,25), (1,20,17)(2,27,18)(3,25,10)(4,23,11)(5,21,12)(6,19,13)(7,26,14)(8,24,15)(9,22,16), (2,5,8)(3,9,6)(10,16,13)(12,15,18)(19,25,22)(21,24,27) );
 
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)], [(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27)], [(2,9),(3,8),(4,7),(5,6),(10,15),(11,14),(12,13),(16,18),(19,21),(22,27),(23,26),(24,25)], [(1,20,17),(2,27,18),(3,25,10),(4,23,11),(5,21,12),(6,19,13),(7,26,14),(8,24,15),(9,22,16)], [(2,5,8),(3,9,6),(10,16,13),(12,15,18),(19,25,22),(21,24,27)]])
 
G:=TransitiveGroup(27,148);
 

►On 27 points - transitive group 27T192
Generators in S27
(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)

(1 17 20)(2 18 21)(3 10 22)(4 11 23)(5 12 24)(6 13 25)(7 14 26)(8 15 27)(9 16 19)

(2 9)(3 8)(4 7)(5 6)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)

(2 21 12)(3 10 19)(5 24 15)(6 13 22)(8 27 18)(9 16 25)(11 17 14)(20 23 26)

(2 5 8)(3 9 6)(10 16 13)(12 15 18)(19 25 22)(21 24 27)
 
G:=sub<Sym(27)| (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), (1,17,20)(2,18,21)(3,10,22)(4,11,23)(5,12,24)(6,13,25)(7,14,26)(8,15,27)(9,16,19), (2,9)(3,8)(4,7)(5,6)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19), (2,21,12)(3,10,19)(5,24,15)(6,13,22)(8,27,18)(9,16,25)(11,17,14)(20,23,26), (2,5,8)(3,9,6)(10,16,13)(12,15,18)(19,25,22)(21,24,27)>;
 
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), (1,17,20)(2,18,21)(3,10,22)(4,11,23)(5,12,24)(6,13,25)(7,14,26)(8,15,27)(9,16,19), (2,9)(3,8)(4,7)(5,6)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19), (2,21,12)(3,10,19)(5,24,15)(6,13,22)(8,27,18)(9,16,25)(11,17,14)(20,23,26), (2,5,8)(3,9,6)(10,16,13)(12,15,18)(19,25,22)(21,24,27) );
 
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)], [(1,17,20),(2,18,21),(3,10,22),(4,11,23),(5,12,24),(6,13,25),(7,14,26),(8,15,27),(9,16,19)], [(2,9),(3,8),(4,7),(5,6),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19)], [(2,21,12),(3,10,19),(5,24,15),(6,13,22),(8,27,18),(9,16,25),(11,17,14),(20,23,26)], [(2,5,8),(3,9,6),(10,16,13),(12,15,18),(19,25,22),(21,24,27)]])
 
G:=TransitiveGroup(27,192);
 

31 conjugacy classes

class 1  2 3A3B3C3D3E3F3G···3L6A···6H9A···9I
order123333333···36···69···9
size1272336669···927···2718···18

31 irreducible representations

dim1111111822266
type+++++
imageC1C2C3C3C6C6C9⋊S3⋊C32S3C3×S3C3×S3C32⋊C6C3×C32⋊C6
kernelC9⋊S3⋊C32He3.C32He3.S3C33.S3He3.C3C3×3- 1+2C1C3×He3He3C33C32C3
# reps116262116212

Matrix representation of C9⋊S3⋊C32 ►in GL18(ℤ)

001000000000000000
000100000000000000
000010000000000000
000001000000000000
010000000000000000
-1-10000000000000000
000000001000000000
000000000100000000
000000000010000000
000000000001000000
000000010000000000
000000-1-10000000000
000000000000000100
00000000000000-1-100
000000000000000001
0000000000000000-1-1
000000000000-1-10000
000000000000100000
,
100000000000000000
010000000000000000
001000000000000000
000100000000000000
000010000000000000
000001000000000000
000000-1-10000000000
000000100000000000
00000000-1-100000000
000000001000000000
0000000000-1-1000000
000000000010000000
000000000000010000
000000000000-1-10000
000000000000000100
00000000000000-1-100
000000000000000001
0000000000000000-1-1
,
100000000000000000
-1-10000000000000000
0000-1-1000000000000
000001000000000000
00-1-100000000000000
000100000000000000
000000100000000000
000000-1-10000000000
0000000000-1-1000000
000000000001000000
00000000-1-100000000
000000000100000000
000000000000100000
000000000000-1-10000
0000000000000000-1-1
000000000000000001
00000000000000-1-100
000000000000000100
,
000000100000000000
000000010000000000
000000001000000000
000000000100000000
000000000010000000
000000000001000000
000000000000100000
000000000000010000
000000000000001000
000000000000000100
000000000000000010
000000000000000001
100000000000000000
010000000000000000
001000000000000000
000100000000000000
000010000000000000
000001000000000000
,
100000000000000000
010000000000000000
00-1-100000000000000
001000000000000000
000001000000000000
0000-1-1000000000000
000000100000000000
000000010000000000
00000000-1-100000000
000000001000000000
000000000001000000
0000000000-1-1000000
000000000000100000
000000000000010000
00000000000000-1-100
000000000000001000
000000000000000001
0000000000000000-1-1

G:=sub<GL(18,Integers())| [0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1],[1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1] >;
 

C9⋊S3⋊C32 in GAP, Magma, Sage, TeX 

C_9\rtimes S_3\rtimes C_3^2
 
% in TeX
 
G:=Group("C9:S3:C3^2");
 
// GroupNames label
 
G:=SmallGroup(486,129);
 
// by ID
 
G=gap.SmallGroup(486,129);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,8643,873,735,237,3244,3250,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^9=b^3=c^2=d^3=e^3=1,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a^4*b,e*a*e^-1=a^7,c*b*c=b^-1,d*b*d^-1=a^3*b,b*e=e*b,c*d=d*c,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

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