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G = C33⋊C32order 243 = 35

2nd semidirect product of C33 and C32 acting faithfully

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

Aliases: C332C32, He3.3C32, C32.6C33, C32.13He3, 3- 1+23C32, C3≀C31C3, (C3×He3)⋊5C3, C3.12(C3×He3), (C3×3- 1+2)⋊7C3, SmallGroup(243,56)

Series: Derived Chief Lower central Upper central Jennings

C1C32 — C33⋊C32
C1C3C32C33C3×He3 — C33⋊C32
C1C3C32 — C33⋊C32
C1C3C33 — C33⋊C32
C1C3C32 — C33⋊C32

Generators and relations for C33⋊C32
 G = < a,b,c,d,e | a3=b3=c3=d3=e3=1, ab=ba, eae-1=ac=ca, dad-1=ab-1c, bc=cb, dbd-1=bc-1, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 234 in 74 conjugacy classes, 33 normal (8 characteristic)
C1, C3, C3 [×10], C9 [×6], C32 [×2], C32 [×2], C32 [×13], C3×C9 [×2], He3 [×3], He3 [×5], 3- 1+2 [×6], 3- 1+2 [×4], C33, C33 [×3], C33, C3≀C3 [×9], C3×He3 [×2], C3×3- 1+2 [×2], C33⋊C32
Quotients: C1, C3 [×13], C32 [×13], He3 [×3], C33, C3×He3, C33⋊C32

Permutation representations of C33⋊C32
On 27 points - transitive group 27T100
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 9 27)(5 7 25)(6 8 26)(16 20 24)(17 21 22)(18 19 23)
(1 13 11)(2 14 12)(3 15 10)(4 27 9)(5 25 7)(6 26 8)(16 20 24)(17 21 22)(18 19 23)
(1 9 20)(2 25 21)(3 6 19)(4 24 13)(5 17 12)(7 22 14)(8 18 10)(11 27 16)(15 26 23)
(1 2 10)(3 13 14)(4 7 6)(5 26 27)(8 9 25)(11 12 15)(16 17 23)(18 20 21)(19 24 22)

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

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

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

G:=TransitiveGroup(27,100);

On 27 points - transitive group 27T102
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 26 8)(5 27 9)(6 25 7)(16 22 20)(17 23 21)(18 24 19)
(1 14 10)(2 15 11)(3 13 12)(4 8 26)(5 9 27)(6 7 25)(16 22 20)(17 23 21)(18 24 19)
(1 9 16)(2 6 17)(3 26 18)(4 24 13)(5 20 10)(7 23 15)(8 19 12)(11 25 21)(14 27 22)
(2 11 15)(3 13 12)(4 8 26)(6 25 7)(17 21 23)(18 24 19)

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

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

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

G:=TransitiveGroup(27,102);

On 27 points - transitive group 27T114
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 6 14)(2 4 15)(3 5 13)(7 24 16)(8 22 17)(9 23 18)(10 27 19)(11 25 20)(12 26 21)
(1 26 8)(2 27 9)(3 25 7)(4 19 23)(5 20 24)(6 21 22)(10 18 15)(11 16 13)(12 17 14)
(2 4 10)(3 13 24)(5 25 11)(6 21 22)(7 16 20)(9 23 15)(12 14 17)(18 27 19)
(1 26 8)(3 7 25)(5 24 20)(6 21 22)(11 13 16)(12 17 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), (1,6,14)(2,4,15)(3,5,13)(7,24,16)(8,22,17)(9,23,18)(10,27,19)(11,25,20)(12,26,21), (1,26,8)(2,27,9)(3,25,7)(4,19,23)(5,20,24)(6,21,22)(10,18,15)(11,16,13)(12,17,14), (2,4,10)(3,13,24)(5,25,11)(6,21,22)(7,16,20)(9,23,15)(12,14,17)(18,27,19), (1,26,8)(3,7,25)(5,24,20)(6,21,22)(11,13,16)(12,17,14)>;

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

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

G:=TransitiveGroup(27,114);

C33⋊C32 is a maximal subgroup of   C3≀C3⋊C6  (C3×He3)⋊C6  C33⋊(C3×S3)

35 conjugacy classes

class 1 3A3B3C···3J3K···3V9A···9L
order1333···33···39···9
size1113···39···99···9

35 irreducible representations

dim111139
type+
imageC1C3C3C3He3C33⋊C32
kernelC33⋊C32C3≀C3C3×He3C3×3- 1+2C32C1
# reps1184462

Matrix representation of C33⋊C32 in GL9(𝔽19)

000100000
000010000
000001000
000000100
000000010
000000001
100000000
010000000
001000000
,
010000000
001000000
100000000
000010000
000001000
000100000
000000010
000000001
000000100
,
700000000
070000000
007000000
000700000
000070000
000007000
000000700
000000070
000000007
,
100000000
0110000000
007000000
000007000
000100000
0000110000
0000000110
000000007
000000100
,
1100000000
0110000000
0011000000
000100000
000010000
000001000
000000700
000000070
000000007

G:=sub<GL(9,GF(19))| [0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0],[11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7] >;

C33⋊C32 in GAP, Magma, Sage, TeX

C_3^3\rtimes C_3^2
% in TeX

G:=Group("C3^3:C3^2");
// GroupNames label

G:=SmallGroup(243,56);
// by ID

G=gap.SmallGroup(243,56);
# by ID

G:=PCGroup([5,-3,3,3,-3,-3,301,457,2163]);
// Polycyclic

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

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