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## G = C3×C11⋊C5order 165 = 3·5·11

### Direct product of C3 and C11⋊C5

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 >

Character table of C3×C11⋊C5

 class 1 3A 3B 5A 5B 5C 5D 11A 11B 15A 15B 15C 15D 15E 15F 15G 15H 33A 33B 33C 33D size 1 1 1 11 11 11 11 5 5 11 11 11 11 11 11 11 11 5 5 5 5 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ3 ζ32 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ3 1 ζ32 ζ3 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ4 1 1 1 ζ52 ζ54 ζ5 ζ53 1 1 ζ53 ζ5 ζ53 ζ52 ζ52 ζ54 ζ5 ζ54 1 1 1 1 linear of order 5 ρ5 1 1 1 ζ53 ζ5 ζ54 ζ52 1 1 ζ52 ζ54 ζ52 ζ53 ζ53 ζ5 ζ54 ζ5 1 1 1 1 linear of order 5 ρ6 1 1 1 ζ54 ζ53 ζ52 ζ5 1 1 ζ5 ζ52 ζ5 ζ54 ζ54 ζ53 ζ52 ζ53 1 1 1 1 linear of order 5 ρ7 1 1 1 ζ5 ζ52 ζ53 ζ54 1 1 ζ54 ζ53 ζ54 ζ5 ζ5 ζ52 ζ53 ζ52 1 1 1 1 linear of order 5 ρ8 1 ζ32 ζ3 ζ54 ζ53 ζ52 ζ5 1 1 ζ32ζ5 ζ3ζ52 ζ3ζ5 ζ3ζ54 ζ32ζ54 ζ32ζ53 ζ32ζ52 ζ3ζ53 ζ3 ζ3 ζ32 ζ32 linear of order 15 ρ9 1 ζ32 ζ3 ζ5 ζ52 ζ53 ζ54 1 1 ζ32ζ54 ζ3ζ53 ζ3ζ54 ζ3ζ5 ζ32ζ5 ζ32ζ52 ζ32ζ53 ζ3ζ52 ζ3 ζ3 ζ32 ζ32 linear of order 15 ρ10 1 ζ32 ζ3 ζ52 ζ54 ζ5 ζ53 1 1 ζ32ζ53 ζ3ζ5 ζ3ζ53 ζ3ζ52 ζ32ζ52 ζ32ζ54 ζ32ζ5 ζ3ζ54 ζ3 ζ3 ζ32 ζ32 linear of order 15 ρ11 1 ζ3 ζ32 ζ54 ζ53 ζ52 ζ5 1 1 ζ3ζ5 ζ32ζ52 ζ32ζ5 ζ32ζ54 ζ3ζ54 ζ3ζ53 ζ3ζ52 ζ32ζ53 ζ32 ζ32 ζ3 ζ3 linear of order 15 ρ12 1 ζ3 ζ32 ζ52 ζ54 ζ5 ζ53 1 1 ζ3ζ53 ζ32ζ5 ζ32ζ53 ζ32ζ52 ζ3ζ52 ζ3ζ54 ζ3ζ5 ζ32ζ54 ζ32 ζ32 ζ3 ζ3 linear of order 15 ρ13 1 ζ3 ζ32 ζ53 ζ5 ζ54 ζ52 1 1 ζ3ζ52 ζ32ζ54 ζ32ζ52 ζ32ζ53 ζ3ζ53 ζ3ζ5 ζ3ζ54 ζ32ζ5 ζ32 ζ32 ζ3 ζ3 linear of order 15 ρ14 1 ζ3 ζ32 ζ5 ζ52 ζ53 ζ54 1 1 ζ3ζ54 ζ32ζ53 ζ32ζ54 ζ32ζ5 ζ3ζ5 ζ3ζ52 ζ3ζ53 ζ32ζ52 ζ32 ζ32 ζ3 ζ3 linear of order 15 ρ15 1 ζ32 ζ3 ζ53 ζ5 ζ54 ζ52 1 1 ζ32ζ52 ζ3ζ54 ζ3ζ52 ζ3ζ53 ζ32ζ53 ζ32ζ5 ζ32ζ54 ζ3ζ5 ζ3 ζ3 ζ32 ζ32 linear of order 15 ρ16 5 5 5 0 0 0 0 -1+√-11/2 -1-√-11/2 0 0 0 0 0 0 0 0 -1-√-11/2 -1+√-11/2 -1+√-11/2 -1-√-11/2 complex lifted from C11⋊C5 ρ17 5 5 5 0 0 0 0 -1-√-11/2 -1+√-11/2 0 0 0 0 0 0 0 0 -1+√-11/2 -1-√-11/2 -1-√-11/2 -1+√-11/2 complex lifted from C11⋊C5 ρ18 5 -5-5√-3/2 -5+5√-3/2 0 0 0 0 -1-√-11/2 -1+√-11/2 0 0 0 0 0 0 0 0 ζ3ζ119+ζ3ζ115+ζ3ζ114+ζ3ζ113+ζ3ζ11 ζ3ζ1110+ζ3ζ118+ζ3ζ117+ζ3ζ116+ζ3ζ112 ζ32ζ1110+ζ32ζ118+ζ32ζ117+ζ32ζ116+ζ32ζ112 ζ32ζ119+ζ32ζ115+ζ32ζ114+ζ32ζ113+ζ32ζ11 complex faithful ρ19 5 -5+5√-3/2 -5-5√-3/2 0 0 0 0 -1+√-11/2 -1-√-11/2 0 0 0 0 0 0 0 0 ζ32ζ1110+ζ32ζ118+ζ32ζ117+ζ32ζ116+ζ32ζ112 ζ32ζ119+ζ32ζ115+ζ32ζ114+ζ32ζ113+ζ32ζ11 ζ3ζ119+ζ3ζ115+ζ3ζ114+ζ3ζ113+ζ3ζ11 ζ3ζ1110+ζ3ζ118+ζ3ζ117+ζ3ζ116+ζ3ζ112 complex faithful ρ20 5 -5+5√-3/2 -5-5√-3/2 0 0 0 0 -1-√-11/2 -1+√-11/2 0 0 0 0 0 0 0 0 ζ32ζ119+ζ32ζ115+ζ32ζ114+ζ32ζ113+ζ32ζ11 ζ32ζ1110+ζ32ζ118+ζ32ζ117+ζ32ζ116+ζ32ζ112 ζ3ζ1110+ζ3ζ118+ζ3ζ117+ζ3ζ116+ζ3ζ112 ζ3ζ119+ζ3ζ115+ζ3ζ114+ζ3ζ113+ζ3ζ11 complex faithful ρ21 5 -5-5√-3/2 -5+5√-3/2 0 0 0 0 -1+√-11/2 -1-√-11/2 0 0 0 0 0 0 0 0 ζ3ζ1110+ζ3ζ118+ζ3ζ117+ζ3ζ116+ζ3ζ112 ζ3ζ119+ζ3ζ115+ζ3ζ114+ζ3ζ113+ζ3ζ11 ζ32ζ119+ζ32ζ115+ζ32ζ114+ζ32ζ113+ζ32ζ11 ζ32ζ1110+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)

 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 1 330 105 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 1 0 226 329 1 225 103 225 103 1 329 104 0 1 0 0 0

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

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