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## G = C3×C3≀S3order 486 = 2·35

### Direct product of C3 and C3≀S3

Aliases: C3×C3≀S3, C342S3, C3≀C37C6, He3⋊(C3×C6), (C3×He3)⋊4C6, C337(C3×S3), He3⋊C21C32, C32.1(S3×C32), C32.49(C32⋊C6), (C3×C3≀C3)⋊3C2, (C3×He3⋊C2)⋊1C3, C3.15(C3×C32⋊C6), SmallGroup(486,115)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C3×C3≀S3
 Chief series C1 — C3 — C32 — He3 — C3×He3 — C3×C3≀C3 — C3×C3≀S3
 Lower central He3 — C3×C3≀S3
 Upper central C1 — C32

Generators and relations for C3×C3≀S3
G = < a,b,c,d,e | a3=b3=c3=d3=e6=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=b-1c, cd=dc, ce=ec, ede-1=b-1d-1 >

Subgroups: 738 in 165 conjugacy classes, 24 normal (12 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, C3×S3, C3×C6, C3×C9, He3, He3, 3- 1+2, C33, C33, C33, He3⋊C2, S3×C32, C32×C6, C3≀C3, C3≀C3, C3×He3, C3×3- 1+2, C34, C3≀S3, C3×He3⋊C2, S3×C33, C3×C3≀C3, C3×C3≀S3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C32⋊C6, S3×C32, C3≀S3, C3×C32⋊C6, C3×C3≀S3

Permutation representations of C3×C3≀S3
On 27 points - transitive group 27T177
Generators in S27
(1 7 6)(2 8 4)(3 9 5)(10 12 14)(11 13 15)(16 18 20)(17 19 21)(22 24 26)(23 25 27)
(1 5 8)(2 6 9)(3 4 7)(11 19 23)(13 21 25)(15 17 27)
(1 8 5)(2 9 6)(3 7 4)(10 18 22)(11 19 23)(12 20 24)(13 21 25)(14 16 26)(15 17 27)
(1 13 10)(2 19 16)(3 27 24)(4 17 20)(5 25 22)(6 11 14)(7 15 12)(8 21 18)(9 23 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)| (1,7,6)(2,8,4)(3,9,5)(10,12,14)(11,13,15)(16,18,20)(17,19,21)(22,24,26)(23,25,27), (1,5,8)(2,6,9)(3,4,7)(11,19,23)(13,21,25)(15,17,27), (1,8,5)(2,9,6)(3,7,4)(10,18,22)(11,19,23)(12,20,24)(13,21,25)(14,16,26)(15,17,27), (1,13,10)(2,19,16)(3,27,24)(4,17,20)(5,25,22)(6,11,14)(7,15,12)(8,21,18)(9,23,26), (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( (1,7,6)(2,8,4)(3,9,5)(10,12,14)(11,13,15)(16,18,20)(17,19,21)(22,24,26)(23,25,27), (1,5,8)(2,6,9)(3,4,7)(11,19,23)(13,21,25)(15,17,27), (1,8,5)(2,9,6)(3,7,4)(10,18,22)(11,19,23)(12,20,24)(13,21,25)(14,16,26)(15,17,27), (1,13,10)(2,19,16)(3,27,24)(4,17,20)(5,25,22)(6,11,14)(7,15,12)(8,21,18)(9,23,26), (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([[(1,7,6),(2,8,4),(3,9,5),(10,12,14),(11,13,15),(16,18,20),(17,19,21),(22,24,26),(23,25,27)], [(1,5,8),(2,6,9),(3,4,7),(11,19,23),(13,21,25),(15,17,27)], [(1,8,5),(2,9,6),(3,7,4),(10,18,22),(11,19,23),(12,20,24),(13,21,25),(14,16,26),(15,17,27)], [(1,13,10),(2,19,16),(3,27,24),(4,17,20),(5,25,22),(6,11,14),(7,15,12),(8,21,18),(9,23,26)], [(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(27,177);

66 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Z 3AA 3AB 3AC 3AD 3AE 3AF 6A ··· 6Z 9A ··· 9F order 1 2 3 ··· 3 3 ··· 3 3 3 3 3 3 3 6 ··· 6 9 ··· 9 size 1 9 1 ··· 1 3 ··· 3 6 6 6 18 18 18 9 ··· 9 18 ··· 18

66 irreducible representations

 dim 1 1 1 1 1 1 2 2 3 6 6 type + + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3≀S3 C32⋊C6 C3×C32⋊C6 kernel C3×C3≀S3 C3×C3≀C3 C3≀S3 C3×He3⋊C2 C3≀C3 C3×He3 C34 C33 C3 C32 C3 # reps 1 1 6 2 6 2 1 8 36 1 2

Matrix representation of C3×C3≀S3 in GL5(𝔽19)

 7 0 0 0 0 0 7 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 1 0 0 0 0 0 7
,
 1 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 0 1 0 0 0 18 18 0 0 0 0 0 0 11 0 0 0 0 0 1 0 0 7 0 0
,
 0 7 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(19))| [7,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[0,18,0,0,0,1,18,0,0,0,0,0,0,0,7,0,0,11,0,0,0,0,0,1,0],[0,7,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,1,0] >;

C3×C3≀S3 in GAP, Magma, Sage, TeX

C_3\times C_3\wr S_3
% in TeX

G:=Group("C3xC3wrS3");
// GroupNames label

G:=SmallGroup(486,115);
// by ID

G=gap.SmallGroup(486,115);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,867,873,8104,382]);
// 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,d*b*d^-1=b*c^-1,e*b*e^-1=b^-1*c,c*d=d*c,c*e=e*c,e*d*e^-1=b^-1*d^-1>;
// generators/relations

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