Copied to
clipboard

G = C33⋊C9order 243 = 35

1st semidirect product of C33 and C9 acting via C9/C3=C3

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

Aliases: C331C9, C34.1C3, C32.18He3, C33.18C32, C32.43- 1+2, C3.1C3≀C3, C32⋊C94C3, C32.7(C3×C9), C3.3(C32⋊C9), SmallGroup(243,13)

Series: Derived Chief Lower central Upper central Jennings

C1C32 — C33⋊C9
C1C3C32C33C34 — C33⋊C9
C1C3C32 — C33⋊C9
C1C32C33 — C33⋊C9
C1C32C33 — C33⋊C9

Generators and relations for C33⋊C9
 G = < a,b,c,d | a3=b3=c3=d9=1, ab=ba, ac=ca, dad-1=ab-1c, bc=cb, dbd-1=bc-1, cd=dc >

Subgroups: 252 in 90 conjugacy classes, 18 normal (7 characteristic)
C1, C3, C3 [×3], C3 [×12], C9 [×3], C32 [×2], C32 [×2], C32 [×42], C3×C9 [×3], C33, C33 [×3], C33 [×12], C32⋊C9 [×3], C34, C33⋊C9
Quotients: C1, C3 [×4], C9 [×3], C32, C3×C9, He3, 3- 1+2 [×2], C32⋊C9, C3≀C3 [×3], C33⋊C9

Permutation representations of C33⋊C9
On 27 points - transitive group 27T89
Generators in S27
(1 21 17)(2 25 15)(3 10 23)(4 24 11)(5 19 18)(6 13 26)(7 27 14)(8 22 12)(9 16 20)
(1 21 17)(2 8 5)(3 13 20)(4 24 11)(6 16 23)(7 27 14)(9 10 26)(12 18 15)(19 25 22)
(1 24 14)(2 25 15)(3 26 16)(4 27 17)(5 19 18)(6 20 10)(7 21 11)(8 22 12)(9 23 13)
(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)| (1,21,17)(2,25,15)(3,10,23)(4,24,11)(5,19,18)(6,13,26)(7,27,14)(8,22,12)(9,16,20), (1,21,17)(2,8,5)(3,13,20)(4,24,11)(6,16,23)(7,27,14)(9,10,26)(12,18,15)(19,25,22), (1,24,14)(2,25,15)(3,26,16)(4,27,17)(5,19,18)(6,20,10)(7,21,11)(8,22,12)(9,23,13), (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( (1,21,17)(2,25,15)(3,10,23)(4,24,11)(5,19,18)(6,13,26)(7,27,14)(8,22,12)(9,16,20), (1,21,17)(2,8,5)(3,13,20)(4,24,11)(6,16,23)(7,27,14)(9,10,26)(12,18,15)(19,25,22), (1,24,14)(2,25,15)(3,26,16)(4,27,17)(5,19,18)(6,20,10)(7,21,11)(8,22,12)(9,23,13), (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([(1,21,17),(2,25,15),(3,10,23),(4,24,11),(5,19,18),(6,13,26),(7,27,14),(8,22,12),(9,16,20)], [(1,21,17),(2,8,5),(3,13,20),(4,24,11),(6,16,23),(7,27,14),(9,10,26),(12,18,15),(19,25,22)], [(1,24,14),(2,25,15),(3,26,16),(4,27,17),(5,19,18),(6,20,10),(7,21,11),(8,22,12),(9,23,13)], [(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(27,89);

On 27 points - transitive group 27T98
Generators in S27
(1 7 4)(2 22 18)(3 9 6)(5 25 12)(8 19 15)(10 16 13)(11 17 14)(20 26 23)(21 27 24)
(2 15 25)(3 26 16)(5 18 19)(6 20 10)(8 12 22)(9 23 13)
(1 24 14)(2 25 15)(3 26 16)(4 27 17)(5 19 18)(6 20 10)(7 21 11)(8 22 12)(9 23 13)
(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)| (1,7,4)(2,22,18)(3,9,6)(5,25,12)(8,19,15)(10,16,13)(11,17,14)(20,26,23)(21,27,24), (2,15,25)(3,26,16)(5,18,19)(6,20,10)(8,12,22)(9,23,13), (1,24,14)(2,25,15)(3,26,16)(4,27,17)(5,19,18)(6,20,10)(7,21,11)(8,22,12)(9,23,13), (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( (1,7,4)(2,22,18)(3,9,6)(5,25,12)(8,19,15)(10,16,13)(11,17,14)(20,26,23)(21,27,24), (2,15,25)(3,26,16)(5,18,19)(6,20,10)(8,12,22)(9,23,13), (1,24,14)(2,25,15)(3,26,16)(4,27,17)(5,19,18)(6,20,10)(7,21,11)(8,22,12)(9,23,13), (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([(1,7,4),(2,22,18),(3,9,6),(5,25,12),(8,19,15),(10,16,13),(11,17,14),(20,26,23),(21,27,24)], [(2,15,25),(3,26,16),(5,18,19),(6,20,10),(8,12,22),(9,23,13)], [(1,24,14),(2,25,15),(3,26,16),(4,27,17),(5,19,18),(6,20,10),(7,21,11),(8,22,12),(9,23,13)], [(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(27,98);

C33⋊C9 is a maximal subgroup of   C331C18  C331D9  C332D9

51 conjugacy classes

class 1 3A···3H3I···3AF9A···9R
order13···33···39···9
size11···13···39···9

51 irreducible representations

dim1111333
type+
imageC1C3C3C9He33- 1+2C3≀C3
kernelC33⋊C9C32⋊C9C34C33C32C32C3
# reps162182418

Matrix representation of C33⋊C9 in GL4(𝔽19) generated by

7000
01101
001111
0007
,
1000
0108
001111
0007
,
1000
01100
00110
00011
,
5000
01211
018011
0907
G:=sub<GL(4,GF(19))| [7,0,0,0,0,11,0,0,0,0,11,0,0,1,11,7],[1,0,0,0,0,1,0,0,0,0,11,0,0,8,11,7],[1,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[5,0,0,0,0,12,18,9,0,1,0,0,0,1,11,7] >;

C33⋊C9 in GAP, Magma, Sage, TeX

C_3^3\rtimes C_9
% in TeX

G:=Group("C3^3:C9");
// GroupNames label

G:=SmallGroup(243,13);
// by ID

G=gap.SmallGroup(243,13);
# by ID

G:=PCGroup([5,-3,3,-3,3,-3,135,121,1352]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^9=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1*c,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations

׿
×
𝔽