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G = C92⋊6S3  order 486 = 2·35

6th semidirect product of C92 and S3 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C92⋊6S3, C9⋊C9⋊3S3, C92⋊7C3⋊2C2, C32⋊C9.8C6, C32⋊C9.18S3, C33.22(C3×S3), C32⋊2D9.7C3, C3.4(He3.4C6), C3.12(He3.4S3), (C3×C9).6(C3×S3), (C3×C9).6(C3⋊S3), C32.43(C3×C3⋊S3), SmallGroup(486,153)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C32⋊C9 — C92⋊6S3
Chief series
C1 — C3 — C32 — C33 — C32⋊C9 — C92⋊7C3 — C92⋊6S3
Lower central
C32⋊C9 — C92⋊6S3
Upper central
C1 — C3

Generators and relations for C92⋊6S3
 G = < a,b,c,d | a9=b9=c3=d2=1, ab=ba, cac-1=ab3, ad=da, cbc-1=a3b, dbd=b-1, dcd=c-1 >

Subgroups: 308 in 63 conjugacy classes, 17 normal (13 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, D9, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, 3- 1+2, C33, C3×D9, S3×C9, C3×C3⋊S3, C92, C32⋊C9, C9⋊C9, C9⋊C9, C3×3- 1+2, C9×D9, C32⋊C18, C9⋊C18, C32⋊2D9, C92⋊7C3, C92⋊6S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C3×C3⋊S3, He3.4S3, He3.4C6, C92⋊6S3

Permutation representations of C92⋊6S3
►On 18 points - transitive group 18T166
Generators in S18
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)

(1 9 8 7 6 5 4 3 2)(10 11 12 13 14 15 16 17 18)

(2 5 8)(3 9 6)(11 17 14)(12 15 18)

(1 16)(2 17)(3 18)(4 10)(5 11)(6 12)(7 13)(8 14)(9 15)
 
G:=sub<Sym(18)| (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,9,8,7,6,5,4,3,2)(10,11,12,13,14,15,16,17,18), (2,5,8)(3,9,6)(11,17,14)(12,15,18), (1,16)(2,17)(3,18)(4,10)(5,11)(6,12)(7,13)(8,14)(9,15)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,9,8,7,6,5,4,3,2)(10,11,12,13,14,15,16,17,18), (2,5,8)(3,9,6)(11,17,14)(12,15,18), (1,16)(2,17)(3,18)(4,10)(5,11)(6,12)(7,13)(8,14)(9,15) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,9,8,7,6,5,4,3,2),(10,11,12,13,14,15,16,17,18)], [(2,5,8),(3,9,6),(11,17,14),(12,15,18)], [(1,16),(2,17),(3,18),(4,10),(5,11),(6,12),(7,13),(8,14),(9,15)]])
 
G:=TransitiveGroup(18,166);
 

39 conjugacy classes

class 1  2 3A3B3C3D3E3F6A6B9A···9F9G···9O9P···9W18A···18F
order12333333669···99···99···918···18
size127112221827273···36···618···1827···27

39 irreducible representations

dim111122222366
type++++++
imageC1C2C3C6S3S3S3C3×S3C3×S3He3.4C6He3.4S3C92⋊6S3
kernelC92⋊6S3C92⋊7C3C32⋊2D9C32⋊C9C92C32⋊C9C9⋊C9C3×C9C33C3C3C1
# reps1122112621236

Matrix representation of C92⋊6S3 ►in GL6(𝔽19)

001000
1100000
0110000
000001
0001100
0000110
,
070000
007000
100000
000001
0001100
0000110
,
100000
070000
0011000
000100
0000110
000007
,
000100
000010
000001
100000
010000
001000

G:=sub<GL(6,GF(19))| [0,11,0,0,0,0,0,0,11,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[0,0,1,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;
 

C92⋊6S3 in GAP, Magma, Sage, TeX 

C_9^2\rtimes_6S_3
 
% in TeX
 
G:=Group("C9^2:6S3");
 
// GroupNames label
 
G:=SmallGroup(486,153);
 
// by ID
 
G=gap.SmallGroup(486,153);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1190,338,867,873,1383,3244]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^9=b^9=c^3=d^2=1,a*b=b*a,c*a*c^-1=a*b^3,a*d=d*a,c*b*c^-1=a^3*b,d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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