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G = C3⋊F11  order 330 = 2·3·5·11

The semidirect product of C3 and F11 acting via F11/C11⋊C5=C2

metacyclic, supersoluble, monomial, Z-group

Aliases: C3⋊F11, D33⋊C5, C33⋊1C10, C11⋊C5⋊S3, C11⋊(C5×S3), (C3×C11⋊C5)⋊1C2, SmallGroup(330,3)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C3⋊F11
Chief series
C1 — C11 — C33 — C3×C11⋊C5 — C3⋊F11
Lower central
C33 — C3⋊F11
Upper central
C1

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

C1
33C2
C3
11C5
C11
11S3
33C10
11C15
3D11
C33
C11⋊C5
11C5×S3
D33
3F11
C3×C11⋊C5
C3⋊F11

Character table of C3⋊F11

 class 1235A5B5C5D10A10B10C10D1115A15B15C15D33A33B
 size 1332111111113333333310222222221010
ρ1111111111111111111    trivial
ρ21-111111-1-1-1-11111111    linear of order 2
ρ31-11ζ52ζ54ζ5ζ53-ζ5-ζ54-ζ53-ζ521ζ5ζ53ζ54ζ5211    linear of order 10
ρ41-11ζ5ζ52ζ53ζ54-ζ53-ζ52-ζ54-ζ51ζ53ζ54ζ52ζ511    linear of order 10
ρ5111ζ53ζ5ζ54ζ52ζ54ζ5ζ52ζ531ζ54ζ52ζ5ζ5311    linear of order 5
ρ6111ζ5ζ52ζ53ζ54ζ53ζ52ζ54ζ51ζ53ζ54ζ52ζ511    linear of order 5
ρ7111ζ52ζ54ζ5ζ53ζ5ζ54ζ53ζ521ζ5ζ53ζ54ζ5211    linear of order 5
ρ8111ζ54ζ53ζ52ζ5ζ52ζ53ζ5ζ541ζ52ζ5ζ53ζ5411    linear of order 5
ρ91-11ζ53ζ5ζ54ζ52-ζ54-ζ5-ζ52-ζ531ζ54ζ52ζ5ζ5311    linear of order 10
ρ101-11ζ54ζ53ζ52ζ5-ζ52-ζ53-ζ5-ζ541ζ52ζ5ζ53ζ5411    linear of order 10
ρ1120-1222200002-1-1-1-1-1-1    orthogonal lifted from S3
ρ1220-12ζ542ζ532ζ522ζ500002-ζ52-ζ5-ζ53-ζ54-1-1    complex lifted from C5×S3
ρ1320-12ζ532ζ52ζ542ζ5200002-ζ54-ζ52-ζ5-ζ53-1-1    complex lifted from C5×S3
ρ1420-12ζ52ζ522ζ532ζ5400002-ζ53-ζ54-ζ52-ζ5-1-1    complex lifted from C5×S3
ρ1520-12ζ522ζ542ζ52ζ5300002-ζ5-ζ53-ζ54-ζ52-1-1    complex lifted from C5×S3
ρ161001000000000-10000-1-1    orthogonal lifted from F11
ρ17100-500000000-100001-√33/21+√33/2    orthogonal faithful
ρ18100-500000000-100001+√33/21-√33/2    orthogonal faithful

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)(12 23)(13 25 16 31 17 33 21 30 15 29)(14 27 20 28 22 32 19 26 18 24)
 
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)(12,23)(13,25,16,31,17,33,21,30,15,29)(14,27,20,28,22,32,19,26,18,24)>;
 
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)(12,23)(13,25,16,31,17,33,21,30,15,29)(14,27,20,28,22,32,19,26,18,24) );
 
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),(12,23),(13,25,16,31,17,33,21,30,15,29),(14,27,20,28,22,32,19,26,18,24)]])
 

Matrix representation of C3⋊F11 ►in GL10(𝔽2)

0001100011
0111110011
1010000011
0110010001
1100010001
0111100000
0011000110
0001111110
1110100010
1101000001
,
0000001110
1011101111
1101100000
0010100101
1111000001
1010101110
0001101100
0000001111
0111110010
0110001100
,
1000110000
0000110101
0100100100
0000110010
0000111100
0001001000
0000010100
0000001000
0010010100
0000111000

G:=sub<GL(10,GF(2))| [0,0,1,0,1,0,0,0,1,1,0,1,0,1,1,1,0,0,1,1,0,1,1,1,0,1,1,0,1,0,1,1,0,0,0,1,1,1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,1,1,1,1,1,0,0,0,0,1],[0,1,1,0,1,1,0,0,0,0,0,0,1,0,1,0,0,0,1,1,0,1,0,1,1,1,0,0,1,1,0,1,1,0,1,0,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,1,1,1,0,1,1,1,0,1,0,1,1,1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,1,1,1,1,0,0,0,0,1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0] >;
 

C3⋊F11 in GAP, Magma, Sage, TeX 

C_3\rtimes F_{11}
 
% in TeX
 
G:=Group("C3:F11");
 
// GroupNames label
 
G:=SmallGroup(330,3);
 
// by ID
 
G=gap.SmallGroup(330,3);
 
# by ID
 
G:=PCGroup([4,-2,-5,-3,-11,242,4803,967]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^11=c^10=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^6>;
 
// generators/relations
 

Export

Subgroup lattice of C3⋊F11 in TeX
Character table of C3⋊F11 in TeX

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