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G = C3×C11⋊C5  order 165 = 3·5·11

Direct product of C3 and C11⋊C5

direct product, metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary

Aliases: C3×C11⋊C5, C33⋊C5, C11⋊C15, SmallGroup(165,1)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C3×C11⋊C5
 G = < a,b,c | a3=b11=c5=1, ab=ba, ac=ca, cbc-1=b3 >

C1
C3
11C5
C11
11C15
C33
C11⋊C5
C3×C11⋊C5

Character table of C3×C11⋊C5

 class 13A3B5A5B5C5D11A11B15A15B15C15D15E15F15G15H33A33B33C33D
 size 111111111115511111111111111115555
ρ1111111111111111111111    trivial
ρ21ζ3ζ32111111ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ31ζ32ζ3111111ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ4111ζ52ζ54ζ5ζ5311ζ53ζ5ζ53ζ52ζ52ζ54ζ5ζ541111    linear of order 5
ρ5111ζ53ζ5ζ54ζ5211ζ52ζ54ζ52ζ53ζ53ζ5ζ54ζ51111    linear of order 5
ρ6111ζ54ζ53ζ52ζ511ζ5ζ52ζ5ζ54ζ54ζ53ζ52ζ531111    linear of order 5
ρ7111ζ5ζ52ζ53ζ5411ζ54ζ53ζ54ζ5ζ5ζ52ζ53ζ521111    linear of order 5
ρ81ζ32ζ3ζ54ζ53ζ52ζ511ζ32ζ5ζ3ζ52ζ3ζ5ζ3ζ54ζ32ζ54ζ32ζ53ζ32ζ52ζ3ζ53ζ3ζ3ζ32ζ32    linear of order 15
ρ91ζ32ζ3ζ5ζ52ζ53ζ5411ζ32ζ54ζ3ζ53ζ3ζ54ζ3ζ5ζ32ζ5ζ32ζ52ζ32ζ53ζ3ζ52ζ3ζ3ζ32ζ32    linear of order 15
ρ101ζ32ζ3ζ52ζ54ζ5ζ5311ζ32ζ53ζ3ζ5ζ3ζ53ζ3ζ52ζ32ζ52ζ32ζ54ζ32ζ5ζ3ζ54ζ3ζ3ζ32ζ32    linear of order 15
ρ111ζ3ζ32ζ54ζ53ζ52ζ511ζ3ζ5ζ32ζ52ζ32ζ5ζ32ζ54ζ3ζ54ζ3ζ53ζ3ζ52ζ32ζ53ζ32ζ32ζ3ζ3    linear of order 15
ρ121ζ3ζ32ζ52ζ54ζ5ζ5311ζ3ζ53ζ32ζ5ζ32ζ53ζ32ζ52ζ3ζ52ζ3ζ54ζ3ζ5ζ32ζ54ζ32ζ32ζ3ζ3    linear of order 15
ρ131ζ3ζ32ζ53ζ5ζ54ζ5211ζ3ζ52ζ32ζ54ζ32ζ52ζ32ζ53ζ3ζ53ζ3ζ5ζ3ζ54ζ32ζ5ζ32ζ32ζ3ζ3    linear of order 15
ρ141ζ3ζ32ζ5ζ52ζ53ζ5411ζ3ζ54ζ32ζ53ζ32ζ54ζ32ζ5ζ3ζ5ζ3ζ52ζ3ζ53ζ32ζ52ζ32ζ32ζ3ζ3    linear of order 15
ρ151ζ32ζ3ζ53ζ5ζ54ζ5211ζ32ζ52ζ3ζ54ζ3ζ52ζ3ζ53ζ32ζ53ζ32ζ5ζ32ζ54ζ3ζ5ζ3ζ3ζ32ζ32    linear of order 15
ρ165550000-1+√-11/2-1-√-11/200000000-1-√-11/2-1+√-11/2-1+√-11/2-1-√-11/2    complex lifted from C11⋊C5
ρ175550000-1-√-11/2-1+√-11/200000000-1+√-11/2-1-√-11/2-1-√-11/2-1+√-11/2    complex lifted from C11⋊C5
ρ185-5-5√-3/2-5+5√-3/20000-1-√-11/2-1+√-11/200000000ζ3ζ119+ζ3ζ115+ζ3ζ114+ζ3ζ113+ζ3ζ11ζ3ζ1110+ζ3ζ118+ζ3ζ117+ζ3ζ116+ζ3ζ112ζ32ζ1110+ζ32ζ118+ζ32ζ117+ζ32ζ116+ζ32ζ112ζ32ζ119+ζ32ζ115+ζ32ζ114+ζ32ζ113+ζ32ζ11    complex faithful
ρ195-5+5√-3/2-5-5√-3/20000-1+√-11/2-1-√-11/200000000ζ32ζ1110+ζ32ζ118+ζ32ζ117+ζ32ζ116+ζ32ζ112ζ32ζ119+ζ32ζ115+ζ32ζ114+ζ32ζ113+ζ32ζ11ζ3ζ119+ζ3ζ115+ζ3ζ114+ζ3ζ113+ζ3ζ11ζ3ζ1110+ζ3ζ118+ζ3ζ117+ζ3ζ116+ζ3ζ112    complex faithful
ρ205-5+5√-3/2-5-5√-3/20000-1-√-11/2-1+√-11/200000000ζ32ζ119+ζ32ζ115+ζ32ζ114+ζ32ζ113+ζ32ζ11ζ32ζ1110+ζ32ζ118+ζ32ζ117+ζ32ζ116+ζ32ζ112ζ3ζ1110+ζ3ζ118+ζ3ζ117+ζ3ζ116+ζ3ζ112ζ3ζ119+ζ3ζ115+ζ3ζ114+ζ3ζ113+ζ3ζ11    complex faithful
ρ215-5-5√-3/2-5+5√-3/20000-1+√-11/2-1-√-11/200000000ζ3ζ1110+ζ3ζ118+ζ3ζ117+ζ3ζ116+ζ3ζ112ζ3ζ119+ζ3ζ115+ζ3ζ114+ζ3ζ113+ζ3ζ11ζ32ζ119+ζ32ζ115+ζ32ζ114+ζ32ζ113+ζ32ζ11ζ32ζ1110+ζ32ζ118+ζ32ζ117+ζ32ζ116+ζ32ζ112    complex faithful

Smallest permutation representation of C3×C11⋊C5
►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 5 6 10 4)(3 9 11 8 7)(13 16 17 21 15)(14 20 22 19 18)(24 27 28 32 26)(25 31 33 30 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), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)>;
 
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), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29) );
 
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)], [(2,5,6,10,4),(3,9,11,8,7),(13,16,17,21,15),(14,20,22,19,18),(24,27,28,32,26),(25,31,33,30,29)]])
 

C3×C11⋊C5 is a maximal subgroup of   C3⋊F11

Matrix representation of C3×C11⋊C5 ►in GL5(𝔽331)

310000
031000
003100
000310
000031
,
10413301051
10000
01000
00100
00010
,
10000
00010
2263291225103
2251031329104
01000

G:=sub<GL(5,GF(331))| [31,0,0,0,0,0,31,0,0,0,0,0,31,0,0,0,0,0,31,0,0,0,0,0,31],[104,1,0,0,0,1,0,1,0,0,330,0,0,1,0,105,0,0,0,1,1,0,0,0,0],[1,0,226,225,0,0,0,329,103,1,0,0,1,1,0,0,1,225,329,0,0,0,103,104,0] >;
 

C3×C11⋊C5 in GAP, Magma, Sage, TeX 

C_3\times C_{11}\rtimes C_5
 
% in TeX
 
G:=Group("C3xC11:C5");
 
// GroupNames label
 
G:=SmallGroup(165,1);
 
// by ID
 
G=gap.SmallGroup(165,1);
 
# by ID
 
G:=PCGroup([3,-3,-5,-11,185]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^11=c^5=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
 
// generators/relations
 

Export

Subgroup lattice of C3×C11⋊C5 in TeX
Character table of C3×C11⋊C5 in TeX

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