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## G = C3×F11order 330 = 2·3·5·11

### Direct product of C3 and F11

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

 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 >

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 3A 3B 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 11 15A ··· 15H 30A ··· 30H 33A 33B order 1 2 3 3 5 5 5 5 6 6 10 10 10 10 11 15 ··· 15 30 ··· 30 33 33 size 1 11 1 1 11 11 11 11 11 11 11 11 11 11 10 11 ··· 11 11 ··· 11 10 10

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 10 10 type + + + image C1 C2 C3 C5 C6 C10 C15 C30 F11 C3×F11 kernel C3×F11 C3×C11⋊C5 F11 C3×D11 C11⋊C5 C33 D11 C11 C3 C1 # reps 1 1 2 4 2 4 8 8 1 2

Matrix representation of C3×F11 in GL11(𝔽331)

 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 0 0 0 0 0 0 0 0 330 0 1 0 0 0 0 0 0 0 0 330 0 0 1 0 0 0 0 0 0 0 330 0 0 0 1 0 0 0 0 0 0 330 0 0 0 0 1 0 0 0 0 0 330 0 0 0 0 0 1 0 0 0 0 330 0 0 0 0 0 0 1 0 0 0 330 0 0 0 0 0 0 0 1 0 0 330 0 0 0 0 0 0 0 0 1 0 330 0 0 0 0 0 0 0 0 0 1 330
,
 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0

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

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