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G = C33⋊1C18  order 486 = 2·35

1st semidirect product of C33 and C18 acting via C18/C3=C6

metabelian, supersoluble, monomial

Aliases: C33⋊1C18, C34.1C6, C32⋊C9⋊3S3, C33⋊C2⋊1C9, C33⋊C9⋊1C2, C32.6(S3×C9), C33.49(C3×S3), C3.2(C32⋊C18), C3.2(C33⋊C6), C32.34(C32⋊C6), (C3×C33⋊C2).1C3, SmallGroup(486,18)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C33⋊1C18
Chief series
C1 — C3 — C32 — C33 — C34 — C33⋊C9 — C33⋊1C18
Lower central
C33 — C33⋊1C18
Upper central
C1 — C3

Generators and relations for C33⋊1C18
 G = < a,b,c,d | a3=b3=c3=d18=1, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, dbd-1=b-1c-1, dcd-1=c-1 >

Subgroups: 660 in 87 conjugacy classes, 13 normal (all characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C18, C3×S3, C3⋊S3, C3×C9, C33, C33, S3×C9, C3×C3⋊S3, C33⋊C2, C32⋊C9, C32⋊C9, C34, C32⋊C18, C3×C33⋊C2, C33⋊C9, C33⋊1C18
Quotients: C1, C2, C3, S3, C6, C9, C18, C3×S3, S3×C9, C32⋊C6, C32⋊C18, C33⋊C6, C33⋊1C18

Permutation representations of C33⋊1C18
►On 18 points - transitive group 18T159
Generators in S18
(1 7 13)(2 14 8)(4 16 10)(5 11 17)

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

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

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)
 
G:=sub<Sym(18)| (1,7,13)(2,14,8)(4,16,10)(5,11,17), (2,8,14)(3,9,15)(5,17,11)(6,18,12), (1,13,7)(2,8,14)(3,15,9)(4,10,16)(5,17,11)(6,12,18), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)>;
 
G:=Group( (1,7,13)(2,14,8)(4,16,10)(5,11,17), (2,8,14)(3,9,15)(5,17,11)(6,18,12), (1,13,7)(2,8,14)(3,15,9)(4,10,16)(5,17,11)(6,12,18), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18) );
 
G=PermutationGroup([[(1,7,13),(2,14,8),(4,16,10),(5,11,17)], [(2,8,14),(3,9,15),(5,17,11),(6,18,12)], [(1,13,7),(2,8,14),(3,15,9),(4,10,16),(5,17,11),(6,12,18)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)]])
 
G:=TransitiveGroup(18,159);
 

►On 27 points - transitive group 27T200
Generators in S27
(1 27 18)(4 21 12)(7 15 24)

(1 18 27)(3 11 20)(4 12 21)(6 23 14)(7 24 15)(9 17 26)

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

(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)
 
G:=sub<Sym(27)| (1,27,18)(4,21,12)(7,15,24), (1,18,27)(3,11,20)(4,12,21)(6,23,14)(7,24,15)(9,17,26), (1,27,18)(2,19,10)(3,11,20)(4,21,12)(5,13,22)(6,23,14)(7,15,24)(8,25,16)(9,17,26), (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)>;
 
G:=Group( (1,27,18)(4,21,12)(7,15,24), (1,18,27)(3,11,20)(4,12,21)(6,23,14)(7,24,15)(9,17,26), (1,27,18)(2,19,10)(3,11,20)(4,21,12)(5,13,22)(6,23,14)(7,15,24)(8,25,16)(9,17,26), (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) );
 
G=PermutationGroup([[(1,27,18),(4,21,12),(7,15,24)], [(1,18,27),(3,11,20),(4,12,21),(6,23,14),(7,24,15),(9,17,26)], [(1,27,18),(2,19,10),(3,11,20),(4,21,12),(5,13,22),(6,23,14),(7,15,24),(8,25,16),(9,17,26)], [(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)]])
 
G:=TransitiveGroup(27,200);
 

39 conjugacy classes

class 1  2 3A3B3C3D3E3F···3Q6A6B9A···9F9G···9L18A···18F
order12333333···3669···99···918···18
size127112226···627279···918···1827···27

39 irreducible representations

dim1111112226666
type+++++
imageC1C2C3C6C9C18S3C3×S3S3×C9C32⋊C6C32⋊C18C33⋊C6C33⋊1C18
kernelC33⋊1C18C33⋊C9C3×C33⋊C2C34C33⋊C2C33C32⋊C9C33C32C32C3C3C1
# reps1122661261236

Matrix representation of C33⋊1C18 ►in GL6(𝔽19)

1100000
0110000
001000
000700
000070
000001
,
100000
070000
0011000
000100
0000110
000007
,
700000
070000
007000
0001100
0000110
0000011
,
0000011
000700
000070
0011000
700000
070000

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

C33⋊1C18 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_1C_{18}
 
% in TeX
 
G:=Group("C3^3:1C18");
 
// GroupNames label
 
G:=SmallGroup(486,18);
 
// by ID
 
G=gap.SmallGroup(486,18);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,867,873,3244,3250,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^3=d^18=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^-1,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

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