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## G = S3×C11⋊C5order 330 = 2·3·5·11

### Direct product of S3 and C11⋊C5

Aliases: S3×C11⋊C5, C333C10, (S3×C11)⋊C5, C112(C5×S3), C3⋊(C2×C11⋊C5), (C3×C11⋊C5)⋊3C2, SmallGroup(330,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — S3×C11⋊C5
 Chief series C1 — C11 — C33 — C3×C11⋊C5 — S3×C11⋊C5
 Lower central C33 — S3×C11⋊C5
 Upper central C1

Generators and relations for S3×C11⋊C5
G = < a,b,c,d | a3=b2=c11=d5=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Character table of S3×C11⋊C5

 class 1 2 3 5A 5B 5C 5D 10A 10B 10C 10D 11A 11B 15A 15B 15C 15D 22A 22B 33A 33B size 1 3 2 11 11 11 11 33 33 33 33 5 5 22 22 22 22 15 15 10 10 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 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 linear of order 2 ρ3 1 -1 1 ζ5 ζ52 ζ53 ζ54 -ζ53 -ζ52 -ζ54 -ζ5 1 1 ζ53 ζ52 ζ54 ζ5 -1 -1 1 1 linear of order 10 ρ4 1 1 1 ζ53 ζ5 ζ54 ζ52 ζ54 ζ5 ζ52 ζ53 1 1 ζ54 ζ5 ζ52 ζ53 1 1 1 1 linear of order 5 ρ5 1 1 1 ζ5 ζ52 ζ53 ζ54 ζ53 ζ52 ζ54 ζ5 1 1 ζ53 ζ52 ζ54 ζ5 1 1 1 1 linear of order 5 ρ6 1 -1 1 ζ53 ζ5 ζ54 ζ52 -ζ54 -ζ5 -ζ52 -ζ53 1 1 ζ54 ζ5 ζ52 ζ53 -1 -1 1 1 linear of order 10 ρ7 1 1 1 ζ54 ζ53 ζ52 ζ5 ζ52 ζ53 ζ5 ζ54 1 1 ζ52 ζ53 ζ5 ζ54 1 1 1 1 linear of order 5 ρ8 1 1 1 ζ52 ζ54 ζ5 ζ53 ζ5 ζ54 ζ53 ζ52 1 1 ζ5 ζ54 ζ53 ζ52 1 1 1 1 linear of order 5 ρ9 1 -1 1 ζ52 ζ54 ζ5 ζ53 -ζ5 -ζ54 -ζ53 -ζ52 1 1 ζ5 ζ54 ζ53 ζ52 -1 -1 1 1 linear of order 10 ρ10 1 -1 1 ζ54 ζ53 ζ52 ζ5 -ζ52 -ζ53 -ζ5 -ζ54 1 1 ζ52 ζ53 ζ5 ζ54 -1 -1 1 1 linear of order 10 ρ11 2 0 -1 2 2 2 2 0 0 0 0 2 2 -1 -1 -1 -1 0 0 -1 -1 orthogonal lifted from S3 ρ12 2 0 -1 2ζ54 2ζ53 2ζ52 2ζ5 0 0 0 0 2 2 -ζ52 -ζ53 -ζ5 -ζ54 0 0 -1 -1 complex lifted from C5×S3 ρ13 2 0 -1 2ζ53 2ζ5 2ζ54 2ζ52 0 0 0 0 2 2 -ζ54 -ζ5 -ζ52 -ζ53 0 0 -1 -1 complex lifted from C5×S3 ρ14 2 0 -1 2ζ5 2ζ52 2ζ53 2ζ54 0 0 0 0 2 2 -ζ53 -ζ52 -ζ54 -ζ5 0 0 -1 -1 complex lifted from C5×S3 ρ15 2 0 -1 2ζ52 2ζ54 2ζ5 2ζ53 0 0 0 0 2 2 -ζ5 -ζ54 -ζ53 -ζ52 0 0 -1 -1 complex lifted from C5×S3 ρ16 5 -5 5 0 0 0 0 0 0 0 0 -1-√-11/2 -1+√-11/2 0 0 0 0 1-√-11/2 1+√-11/2 -1+√-11/2 -1-√-11/2 complex lifted from C2×C11⋊C5 ρ17 5 -5 5 0 0 0 0 0 0 0 0 -1+√-11/2 -1-√-11/2 0 0 0 0 1+√-11/2 1-√-11/2 -1-√-11/2 -1+√-11/2 complex lifted from C2×C11⋊C5 ρ18 5 5 5 0 0 0 0 0 0 0 0 -1+√-11/2 -1-√-11/2 0 0 0 0 -1-√-11/2 -1+√-11/2 -1-√-11/2 -1+√-11/2 complex lifted from C11⋊C5 ρ19 5 5 5 0 0 0 0 0 0 0 0 -1-√-11/2 -1+√-11/2 0 0 0 0 -1+√-11/2 -1-√-11/2 -1+√-11/2 -1-√-11/2 complex lifted from C11⋊C5 ρ20 10 0 -5 0 0 0 0 0 0 0 0 -1+√-11 -1-√-11 0 0 0 0 0 0 1+√-11/2 1-√-11/2 complex faithful ρ21 10 0 -5 0 0 0 0 0 0 0 0 -1-√-11 -1+√-11 0 0 0 0 0 0 1-√-11/2 1+√-11/2 complex faithful

Smallest permutation representation of S3×C11⋊C5
On 33 points
Generators in S33
(1 12 23)(2 13 24)(3 14 25)(4 15 26)(5 16 27)(6 17 28)(7 18 29)(8 19 30)(9 20 31)(10 21 32)(11 22 33)
(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)
(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)| (1,12,23)(2,13,24)(3,14,25)(4,15,26)(5,16,27)(6,17,28)(7,18,29)(8,19,30)(9,20,31)(10,21,32)(11,22,33), (12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33), (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( (1,12,23)(2,13,24)(3,14,25)(4,15,26)(5,16,27)(6,17,28)(7,18,29)(8,19,30)(9,20,31)(10,21,32)(11,22,33), (12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33), (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([[(1,12,23),(2,13,24),(3,14,25),(4,15,26),(5,16,27),(6,17,28),(7,18,29),(8,19,30),(9,20,31),(10,21,32),(11,22,33)], [(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33)], [(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)]])

Matrix representation of S3×C11⋊C5 in GL7(𝔽331)

 0 330 0 0 0 0 0 1 330 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 330 330 330 330 104 0 0 1 0 0 0 228 0 0 0 1 0 0 329 0 0 0 0 1 0 106 0 0 0 0 0 1 227
,
 150 0 0 0 0 0 0 0 150 0 0 0 0 0 0 0 0 330 1 0 106 0 0 0 105 0 0 227 0 0 0 227 0 0 103 0 0 1 330 0 0 2 0 0 0 1 0 1 226

G:=sub<GL(7,GF(331))| [0,1,0,0,0,0,0,330,330,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,330,1,0,0,0,0,0,330,0,1,0,0,0,0,330,0,0,1,0,0,0,330,0,0,0,1,0,0,104,228,329,106,227],[150,0,0,0,0,0,0,0,150,0,0,0,0,0,0,0,0,0,0,1,0,0,0,330,105,227,330,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,106,227,103,2,226] >;

S3×C11⋊C5 in GAP, Magma, Sage, TeX

S_3\times C_{11}\rtimes C_5
% in TeX

G:=Group("S3xC11:C5");
// GroupNames label

G:=SmallGroup(330,2);
// by ID

G=gap.SmallGroup(330,2);
# by ID

G:=PCGroup([4,-2,-5,-3,-11,242,967]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^11=d^5=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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