Copied to
clipboard

G = C3⋊F11order 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, C331C10, C11⋊C5⋊S3, C11⋊(C5×S3), (C3×C11⋊C5)⋊1C2, SmallGroup(330,3)

Series: Derived Chief Lower central Upper central

C1C33 — C3⋊F11
C1C11C33C3×C11⋊C5 — C3⋊F11
C33 — C3⋊F11
C1

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

33C2
11C5
11S3
33C10
11C15
3D11
11C5×S3
3F11

Character table of C3⋊F11

 class 1235A5B5C5D10A10B10C10D1115A15B15C15D33A33B
 size 1332111111113333333310222222221010
ρ1111111111111111111    trivial
ρ21-111111-1-1-1-11111111    linear of order 2
ρ31-11ζ52ζ54ζ5ζ5355453521ζ5ζ53ζ54ζ5211    linear of order 10
ρ41-11ζ5ζ52ζ53ζ5453525451ζ53ζ54ζ52ζ511    linear of order 10
ρ5111ζ53ζ5ζ54ζ52ζ54ζ5ζ52ζ531ζ54ζ52ζ5ζ5311    linear of order 5
ρ6111ζ5ζ52ζ53ζ54ζ53ζ52ζ54ζ51ζ53ζ54ζ52ζ511    linear of order 5
ρ7111ζ52ζ54ζ5ζ53ζ5ζ54ζ53ζ521ζ5ζ53ζ54ζ5211    linear of order 5
ρ8111ζ54ζ53ζ52ζ5ζ52ζ53ζ5ζ541ζ52ζ5ζ53ζ5411    linear of order 5
ρ91-11ζ53ζ5ζ54ζ5254552531ζ54ζ52ζ5ζ5311    linear of order 10
ρ101-11ζ54ζ53ζ52ζ552535541ζ52ζ5ζ53ζ5411    linear of order 10
ρ1120-1222200002-1-1-1-1-1-1    orthogonal lifted from S3
ρ1220-15453525000025255354-1-1    complex lifted from C5×S3
ρ1320-15355452000025452553-1-1    complex lifted from C5×S3
ρ1420-15525354000025354525-1-1    complex lifted from C5×S3
ρ1520-15254553000025535452-1-1    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

׿
×
𝔽