direct product, metacyclic, supersoluble, monomial, Z-group
Aliases: S3×C11⋊C5, C33⋊3C10, (S3×C11)⋊C5, C11⋊2(C5×S3), C3⋊(C2×C11⋊C5), (C3×C11⋊C5)⋊3C2, SmallGroup(330,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — S3×C11⋊C5 |
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 |
►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
Export
Subgroup lattice of S3×C11⋊C5 in TeX
Character table of S3×C11⋊C5 in TeX