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G = C3×F11order 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, C332C10, C11⋊C5⋊C6, (C3×D11)⋊C5, (C3×C11⋊C5)⋊2C2, SmallGroup(330,1)

Series: Derived Chief Lower central Upper central

C1C11 — C3×F11
C1C11C11⋊C5C3×C11⋊C5 — C3×F11
C11 — C3×F11
C1C3

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

11C2
11C5
11C6
11C10
11C15
11C30

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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