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G = C331C18order 486 = 2·35

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

metabelian, supersoluble, monomial

Aliases: C331C18, C34.1C6, C32⋊C93S3, C33⋊C21C9, C33⋊C91C2, 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

C1C33 — C331C18
C1C3C32C33C34C33⋊C9 — C331C18
C33 — C331C18
C1C3

Generators and relations for C331C18
 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 [×2], C3 [×9], S3 [×5], C6, C9 [×2], C32 [×2], C32 [×26], C18, C3×S3 [×5], C3⋊S3 [×5], C3×C9 [×2], C33 [×2], C33 [×9], S3×C9, C3×C3⋊S3 [×5], C33⋊C2, C32⋊C9, C32⋊C9, C34, C32⋊C18, C3×C33⋊C2, C33⋊C9, C331C18
Quotients: C1, C2, C3, S3, C6, C9, C18, C3×S3, S3×C9, C32⋊C6, C32⋊C18, C33⋊C6, C331C18

Permutation representations of C331C18
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 19 10)(4 13 22)(7 25 16)
(1 10 19)(3 21 12)(4 22 13)(6 15 24)(7 16 25)(9 27 18)
(1 19 10)(2 11 20)(3 21 12)(4 13 22)(5 23 14)(6 15 24)(7 25 16)(8 17 26)(9 27 18)
(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)| (1,19,10)(4,13,22)(7,25,16), (1,10,19)(3,21,12)(4,22,13)(6,15,24)(7,16,25)(9,27,18), (1,19,10)(2,11,20)(3,21,12)(4,13,22)(5,23,14)(6,15,24)(7,25,16)(8,17,26)(9,27,18), (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( (1,19,10)(4,13,22)(7,25,16), (1,10,19)(3,21,12)(4,22,13)(6,15,24)(7,16,25)(9,27,18), (1,19,10)(2,11,20)(3,21,12)(4,13,22)(5,23,14)(6,15,24)(7,25,16)(8,17,26)(9,27,18), (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([(1,19,10),(4,13,22),(7,25,16)], [(1,10,19),(3,21,12),(4,22,13),(6,15,24),(7,16,25),(9,27,18)], [(1,19,10),(2,11,20),(3,21,12),(4,13,22),(5,23,14),(6,15,24),(7,25,16),(8,17,26),(9,27,18)], [(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⋊C6C331C18
kernelC331C18C33⋊C9C3×C33⋊C2C34C33⋊C2C33C32⋊C9C33C32C32C3C3C1
# reps1122661261236

Matrix representation of C331C18 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] >;

C331C18 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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