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## G = C3×C33⋊C6order 486 = 2·35

### Direct product of C3 and C33⋊C6

Aliases: C3×C33⋊C6, C342C6, C3≀C38C6, (C3×He3)⋊4S3, He31(C3×S3), C337(C3×C6), C33.57(C3×S3), C33⋊C24C32, C32.13(S3×C32), C32.42(C32⋊C6), (C3×C3≀C3)⋊4C2, C3.4(C3×C32⋊C6), (C3×C33⋊C2)⋊1C3, SmallGroup(486,116)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C3×C33⋊C6
 Chief series C1 — C3 — C32 — C33 — C34 — C3×C3≀C3 — C3×C33⋊C6
 Lower central C33 — C3×C33⋊C6
 Upper central C1 — C3

Generators and relations for C3×C33⋊C6
G = < a,b,c,d,e | a3=b3=c3=d3=e6=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1c-1, cd=dc, ece-1=c-1d-1, ede-1=d-1 >

Subgroups: 870 in 123 conjugacy classes, 22 normal (13 characteristic)
C1, C2, C3 [×2], C3 [×12], S3 [×5], C6 [×4], C9 [×3], C32 [×2], C32 [×31], C3×S3 [×8], C3⋊S3 [×5], C3×C6, C3×C9, He3 [×3], He3, 3- 1+2 [×5], C33 [×2], C33 [×10], C32⋊C6 [×3], S3×C32, C3×C3⋊S3 [×5], C33⋊C2, C3≀C3 [×3], C3≀C3 [×3], C3×He3, C3×3- 1+2, C34, C33⋊C6 [×3], C3×C32⋊C6, C3×C33⋊C2, C3×C3≀C3, C3×C33⋊C6
Quotients: C1, C2, C3 [×4], S3, C6 [×4], C32, C3×S3 [×4], C3×C6, C32⋊C6, S3×C32, C33⋊C6, C3×C32⋊C6, C3×C33⋊C6

Permutation representations of C3×C33⋊C6
On 18 points - transitive group 18T162
Generators in S18
(1 8 15)(2 9 16)(3 10 17)(4 11 18)(5 12 13)(6 7 14)
(1 8 15)(4 18 11)
(1 15 8)(3 10 17)(4 11 18)(6 14 7)
(1 8 15)(2 16 9)(3 10 17)(4 18 11)(5 12 13)(6 14 7)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

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

G:=Group( (1,8,15)(2,9,16)(3,10,17)(4,11,18)(5,12,13)(6,7,14), (1,8,15)(4,18,11), (1,15,8)(3,10,17)(4,11,18)(6,14,7), (1,8,15)(2,16,9)(3,10,17)(4,18,11)(5,12,13)(6,14,7), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18) );

G=PermutationGroup([(1,8,15),(2,9,16),(3,10,17),(4,11,18),(5,12,13),(6,7,14)], [(1,8,15),(4,18,11)], [(1,15,8),(3,10,17),(4,11,18),(6,14,7)], [(1,8,15),(2,16,9),(3,10,17),(4,18,11),(5,12,13),(6,14,7)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)])

G:=TransitiveGroup(18,162);

On 27 points - transitive group 27T199
Generators in S27
(1 8 4)(2 9 5)(3 7 6)(10 26 19)(11 27 20)(12 22 21)(13 23 16)(14 24 17)(15 25 18)
(2 24 27)(3 22 25)(5 14 11)(6 12 15)(7 21 18)(9 17 20)
(1 26 23)(3 25 22)(4 10 13)(6 15 12)(7 18 21)(8 19 16)
(1 23 26)(2 27 24)(3 25 22)(4 13 10)(5 11 14)(6 15 12)(7 18 21)(8 16 19)(9 20 17)
(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)| (1,8,4)(2,9,5)(3,7,6)(10,26,19)(11,27,20)(12,22,21)(13,23,16)(14,24,17)(15,25,18), (2,24,27)(3,22,25)(5,14,11)(6,12,15)(7,21,18)(9,17,20), (1,26,23)(3,25,22)(4,10,13)(6,15,12)(7,18,21)(8,19,16), (1,23,26)(2,27,24)(3,25,22)(4,13,10)(5,11,14)(6,15,12)(7,18,21)(8,16,19)(9,20,17), (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( (1,8,4)(2,9,5)(3,7,6)(10,26,19)(11,27,20)(12,22,21)(13,23,16)(14,24,17)(15,25,18), (2,24,27)(3,22,25)(5,14,11)(6,12,15)(7,21,18)(9,17,20), (1,26,23)(3,25,22)(4,10,13)(6,15,12)(7,18,21)(8,19,16), (1,23,26)(2,27,24)(3,25,22)(4,13,10)(5,11,14)(6,15,12)(7,18,21)(8,16,19)(9,20,17), (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([(1,8,4),(2,9,5),(3,7,6),(10,26,19),(11,27,20),(12,22,21),(13,23,16),(14,24,17),(15,25,18)], [(2,24,27),(3,22,25),(5,14,11),(6,12,15),(7,21,18),(9,17,20)], [(1,26,23),(3,25,22),(4,10,13),(6,15,12),(7,18,21),(8,19,16)], [(1,23,26),(2,27,24),(3,25,22),(4,13,10),(5,11,14),(6,15,12),(7,18,21),(8,16,19),(9,20,17)], [(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(27,199);

39 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3Q 3R ··· 3W 6A ··· 6H 9A ··· 9F order 1 2 3 3 3 3 3 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 size 1 27 1 1 2 2 2 6 ··· 6 9 ··· 9 27 ··· 27 18 ··· 18

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 6 6 6 6 type + + + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3×S3 C32⋊C6 C33⋊C6 C3×C32⋊C6 C3×C33⋊C6 kernel C3×C33⋊C6 C3×C3≀C3 C33⋊C6 C3×C33⋊C2 C3≀C3 C34 C3×He3 He3 C33 C32 C3 C3 C1 # reps 1 1 6 2 6 2 1 6 2 1 3 2 6

Matrix representation of C3×C33⋊C6 in GL6(𝔽19)

 11 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 0 0 11 0 0 0 0 0 0 11
,
 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 7 0 0 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 7 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 12 0 0 11 0 0 0 7 0 0 7 0 0 0 0 0 0 1
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 7 0 0 7 0 0 0 7 0 0 7 0 0 0 7 0 0 7
,
 0 18 0 0 6 0 0 0 18 0 0 6 18 0 0 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[11,0,0,7,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,0,0,12,0,0,0,11,0,0,7,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1],[11,0,0,7,0,0,0,11,0,0,7,0,0,0,11,0,0,7,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,0,18,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,6,0,0,1,6,0,0,1,0,0,0,6,0,0,1,0] >;

C3×C33⋊C6 in GAP, Magma, Sage, TeX

C_3\times C_3^3\rtimes C_6
% in TeX

G:=Group("C3xC3^3:C6");
// GroupNames label

G:=SmallGroup(486,116);
// by ID

G=gap.SmallGroup(486,116);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,867,873,3244,3250,11669]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^6=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1*c^-1,c*d=d*c,e*c*e^-1=c^-1*d^-1,e*d*e^-1=d^-1>;
// generators/relations

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