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G = C3×C3≀C3order 243 = 35

Direct product of C3 and C3≀C3

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C3×C3≀C3, C342C3, He31C32, C334C32, C32.9He3, C32.1C33, 3- 1+21C32, (C3×He3)⋊3C3, C3.7(C3×He3), (C3×3- 1+2)⋊5C3, 3-Sylow(S12), SmallGroup(243,51)

Series: Derived Chief Lower central Upper central Jennings

C1C32 — C3×C3≀C3
C1C3C32C33C34 — C3×C3≀C3
C1C3C32 — C3×C3≀C3
C1C32C33 — C3×C3≀C3
C1C3C32 — C3×C3≀C3

Generators and relations for C3×C3≀C3
 G = < a,b,c,d,e | a3=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, be=eb, cd=dc, ce=ec, ede-1=bc-1d >

Subgroups: 342 in 126 conjugacy classes, 36 normal (8 characteristic)
C1, C3, C3 [×3], C3 [×15], C9 [×6], C32 [×2], C32 [×2], C32 [×48], C3×C9 [×2], He3 [×3], He3 [×2], 3- 1+2 [×6], 3- 1+2 [×4], C33, C33 [×3], C33 [×13], C3≀C3 [×9], C3×He3, C3×3- 1+2 [×2], C34, C3×C3≀C3
Quotients: C1, C3 [×13], C32 [×13], He3 [×3], C33, C3≀C3 [×3], C3×He3, C3×C3≀C3

Permutation representations of C3×C3≀C3
On 27 points - transitive group 27T105
Generators in S27
(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)
(1 2 3)(4 9 26)(5 7 27)(6 8 25)(10 11 12)(13 14 15)(16 24 21)(17 22 19)(18 23 20)
(1 14 10)(2 15 11)(3 13 12)(4 27 8)(5 25 9)(6 26 7)(16 23 19)(17 24 20)(18 22 21)
(1 7 21)(2 8 19)(3 9 20)(4 16 15)(5 17 13)(6 18 14)(10 26 22)(11 27 23)(12 25 24)
(1 3 2)(4 25 7)(5 26 8)(6 27 9)(10 12 11)(13 15 14)

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

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

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

G:=TransitiveGroup(27,105);

On 27 points - transitive group 27T106
Generators in S27
(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)
(4 7 27)(5 8 25)(6 9 26)(16 22 19)(17 23 20)(18 24 21)
(1 10 15)(2 11 13)(3 12 14)(4 27 7)(5 25 8)(6 26 9)(16 22 19)(17 23 20)(18 24 21)
(1 25 23)(2 26 24)(3 27 22)(4 16 14)(5 17 15)(6 18 13)(7 19 12)(8 20 10)(9 21 11)
(4 27 7)(5 25 8)(6 26 9)

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

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

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

G:=TransitiveGroup(27,106);

C3×C3≀C3 is a maximal subgroup of   C345S3  C345C6  C347S3

51 conjugacy classes

class 1 3A···3H3I···3AF3AG···3AL9A···9L
order13···33···33···39···9
size11···13···39···99···9

51 irreducible representations

dim1111133
type+
imageC1C3C3C3C3He3C3≀C3
kernelC3×C3≀C3C3≀C3C3×He3C3×3- 1+2C34C32C3
# reps118242618

Matrix representation of C3×C3≀C3 in GL4(𝔽19) generated by

11000
01100
00110
00011
,
1000
01100
0010
0007
,
1000
0700
0070
0007
,
1000
0010
0001
0100
,
1000
01100
0070
0007
G:=sub<GL(4,GF(19))| [11,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,11,0,0,0,0,1,0,0,0,0,7],[1,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0],[1,0,0,0,0,11,0,0,0,0,7,0,0,0,0,7] >;

C3×C3≀C3 in GAP, Magma, Sage, TeX

C_3\times C_3\wr C_3
% in TeX

G:=Group("C3xC3wrC3");
// GroupNames label

G:=SmallGroup(243,51);
// by ID

G=gap.SmallGroup(243,51);
# by ID

G:=PCGroup([5,-3,3,3,-3,-3,301,2163]);
// Polycyclic

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

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