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G = C3×F11  order 330 = 2·3·5·11

Direct product of C3 and F11

direct product, metacyclic, supersoluble, monomial, Z-group

Aliases: C3×F11, C11⋊C30, D11⋊C15, C33⋊2C10, C11⋊C5⋊C6, (C3×D11)⋊C5, (C3×C11⋊C5)⋊2C2, SmallGroup(330,1)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C11 — C3×F11
Chief series
C1 — C11 — C11⋊C5 — C3×C11⋊C5 — C3×F11
Lower central
C11 — C3×F11
Upper central
C1 — C3

Generators and relations for C3×F11
 G = < a,b,c | a3=b11=c10=1, ab=ba, ac=ca, cbc-1=b6 >

C1
11C2
C3
11C5
C11
11C6
11C10
11C15
D11
C33
C11⋊C5
11C30
C3×D11
F11
C3×C11⋊C5
C3×F11

Smallest permutation representation of C3×F11
►On 33 points
Generators in S33
(1 23 12)(2 24 13)(3 25 14)(4 26 15)(5 27 16)(6 28 17)(7 29 18)(8 30 19)(9 31 20)(10 32 21)(11 33 22)

(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 28 29 30 31 32 33)

(2 3 5 9 6 11 10 8 4 7)(13 14 16 20 17 22 21 19 15 18)(24 25 27 31 28 33 32 30 26 29)
 
G:=sub<Sym(33)| (1,23,12)(2,24,13)(3,25,14)(4,26,15)(5,27,16)(6,28,17)(7,29,18)(8,30,19)(9,31,20)(10,32,21)(11,33,22), (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33), (2,3,5,9,6,11,10,8,4,7)(13,14,16,20,17,22,21,19,15,18)(24,25,27,31,28,33,32,30,26,29)>;
 
G:=Group( (1,23,12)(2,24,13)(3,25,14)(4,26,15)(5,27,16)(6,28,17)(7,29,18)(8,30,19)(9,31,20)(10,32,21)(11,33,22), (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33), (2,3,5,9,6,11,10,8,4,7)(13,14,16,20,17,22,21,19,15,18)(24,25,27,31,28,33,32,30,26,29) );
 
G=PermutationGroup([[(1,23,12),(2,24,13),(3,25,14),(4,26,15),(5,27,16),(6,28,17),(7,29,18),(8,30,19),(9,31,20),(10,32,21),(11,33,22)], [(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22),(23,24,25,26,27,28,29,30,31,32,33)], [(2,3,5,9,6,11,10,8,4,7),(13,14,16,20,17,22,21,19,15,18),(24,25,27,31,28,33,32,30,26,29)]])
 

33 conjugacy classes

class 1  2 3A3B5A5B5C5D6A6B10A10B10C10D 11 15A···15H30A···30H33A33B
order1233555566101010101115···1530···303333
size11111111111111111111111111011···1111···111010

33 irreducible representations

dim111111111010
type+++
imageC1C2C3C5C6C10C15C30F11C3×F11
kernelC3×F11C3×C11⋊C5F11C3×D11C11⋊C5C33D11C11C3C1
# reps1124248812

Matrix representation of C3×F11 ►in GL11(𝔽331)

310000000000
01000000000
00100000000
00010000000
00001000000
00000100000
00000010000
00000001000
00000000100
00000000010
00000000001
,
10000000000
0000000000330
0100000000330
0010000000330
0001000000330
0000100000330
0000010000330
0000001000330
0000000100330
0000000010330
0000000001330
,
80000000000
00000010000
01000000000
00000001000
00100000000
00000000100
00010000000
00000000010
00001000000
00000000001
00000100000

G:=sub<GL(11,GF(331))| [31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,330,330,330,330,330,330,330,330,330,330],[8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0] >;
 

C3×F11 in GAP, Magma, Sage, TeX 

C_3\times F_{11}
 
% in TeX
 
G:=Group("C3xF11");
 
// GroupNames label
 
G:=SmallGroup(330,1);
 
// by ID
 
G=gap.SmallGroup(330,1);
 
# by ID
 
G:=PCGroup([4,-2,-3,-5,-11,4803,731]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^11=c^10=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
 
// generators/relations
 

Export

Subgroup lattice of C3×F11 in TeX

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