direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3⋊S3×F5, (C3×F5)⋊S3, C3⋊1(S3×F5), C15⋊1(C4×S3), C3⋊D15⋊2C4, C32⋊6(C2×F5), (C3×D5).2D6, (C32×F5)⋊2C2, C32⋊3F5⋊1C2, (C32×D5).2C22, C5⋊(C4×C3⋊S3), (C5×C3⋊S3)⋊2C4, (C3×C15)⋊4(C2×C4), D5.1(C2×C3⋊S3), (D5×C3⋊S3).3C2, SmallGroup(360,127)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — C3⋊S3×F5 |
Generators and relations for C3⋊S3×F5
G = < a,b,c,d,e | a3=b3=c2=d5=e4=1, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >
Subgroups: 656 in 96 conjugacy classes, 30 normal (14 characteristic)
C1, C2, C3, C4, C22, C5, S3, C6, C2×C4, C32, D5, D5, C10, Dic3, C12, D6, C15, C3⋊S3, C3⋊S3, C3×C6, F5, F5, D10, C4×S3, C5×S3, C3×D5, D15, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×F5, C3×C15, C3×F5, C3⋊F5, S3×D5, C4×C3⋊S3, C32×D5, C5×C3⋊S3, C3⋊D15, S3×F5, C32×F5, C32⋊3F5, D5×C3⋊S3, C3⋊S3×F5
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, C3⋊S3, F5, C4×S3, C2×C3⋊S3, C2×F5, C4×C3⋊S3, S3×F5, C3⋊S3×F5
Character table of C3⋊S3×F5
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 5 | 6A | 6B | 6C | 6D | 10 | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 15C | 15D | |
| size | 1 | 5 | 9 | 45 | 2 | 2 | 2 | 2 | 5 | 5 | 45 | 45 | 4 | 10 | 10 | 10 | 10 | 36 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -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 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -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 |
| ρ5 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | i | -i | -i | i | 1 | -1 | -1 | -1 | -1 | -1 | i | -i | -i | -i | -i | i | i | i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | i | -i | 1 | -1 | -1 | -1 | -1 | 1 | i | -i | -i | -i | -i | i | i | i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | -i | i | 1 | -1 | -1 | -1 | -1 | 1 | -i | i | i | i | i | -i | -i | -i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -i | i | i | -i | 1 | -1 | -1 | -1 | -1 | -1 | -i | i | i | i | i | -i | -i | -i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | 2 | -1 | 2 | -1 | -1 | 0 | 1 | 1 | -2 | 1 | 1 | 1 | -2 | 1 | 2 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 1 | -2 | 1 | 1 | 1 | -2 | 1 | 1 | -1 | -1 | 2 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | 2 | -1 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 0 | -2 | 1 | 1 | 1 | -2 | 1 | 1 | 1 | -1 | 2 | -1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | 2 | -1 | 2 | -1 | -1 | 0 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | 2 | -1 | 0 | 1 | 1 | 1 | -2 | 1 | 1 | 1 | -2 | -1 | -1 | -1 | 2 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ17 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 2i | -2i | 0 | 0 | 2 | 1 | 1 | 1 | -2 | 0 | 2i | i | i | i | -2i | -i | -i | -i | -1 | 2 | -1 | -1 | complex lifted from C4×S3 |
| ρ18 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | -2i | 2i | 0 | 0 | 2 | -2 | 1 | 1 | 1 | 0 | i | 2i | -i | -i | -i | -2i | i | i | -1 | -1 | 2 | -1 | complex lifted from C4×S3 |
| ρ19 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 2i | -2i | 0 | 0 | 2 | -2 | 1 | 1 | 1 | 0 | -i | -2i | i | i | i | 2i | -i | -i | -1 | -1 | 2 | -1 | complex lifted from C4×S3 |
| ρ20 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 2i | -2i | 0 | 0 | 2 | 1 | 1 | -2 | 1 | 0 | -i | i | i | -2i | i | -i | -i | 2i | -1 | -1 | -1 | 2 | complex lifted from C4×S3 |
| ρ21 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | -2i | 2i | 0 | 0 | 2 | 1 | 1 | -2 | 1 | 0 | i | -i | -i | 2i | -i | i | i | -2i | -1 | -1 | -1 | 2 | complex lifted from C4×S3 |
| ρ22 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | -2i | 2i | 0 | 0 | 2 | 1 | -2 | 1 | 1 | 0 | i | -i | 2i | -i | -i | i | -2i | i | 2 | -1 | -1 | -1 | complex lifted from C4×S3 |
| ρ23 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 2i | -2i | 0 | 0 | 2 | 1 | -2 | 1 | 1 | 0 | -i | i | -2i | i | i | -i | 2i | -i | 2 | -1 | -1 | -1 | complex lifted from C4×S3 |
| ρ24 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | -2i | 2i | 0 | 0 | 2 | 1 | 1 | 1 | -2 | 0 | -2i | -i | -i | -i | 2i | i | i | i | -1 | 2 | -1 | -1 | complex lifted from C4×S3 |
| ρ25 | 4 | 0 | -4 | 0 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from C2×F5 |
| ρ26 | 4 | 0 | 4 | 0 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ27 | 8 | 0 | 0 | 0 | -4 | -4 | 8 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | 1 | orthogonal lifted from S3×F5 |
| ρ28 | 8 | 0 | 0 | 0 | 8 | -4 | -4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | orthogonal lifted from S3×F5 |
| ρ29 | 8 | 0 | 0 | 0 | -4 | 8 | -4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | orthogonal lifted from S3×F5 |
| ρ30 | 8 | 0 | 0 | 0 | -4 | -4 | -4 | 8 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -2 | 1 | orthogonal lifted from S3×F5 |
►On 45 points
Generators in S45
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)
(1 16 31)(2 17 32)(3 18 33)(4 19 34)(5 20 35)(6 21 36)(7 22 37)(8 23 38)(9 24 39)(10 25 40)(11 26 41)(12 27 42)(13 28 43)(14 29 44)(15 30 45)
(6 11)(7 12)(8 13)(9 14)(10 15)(16 31)(17 32)(18 33)(19 34)(20 35)(21 41)(22 42)(23 43)(24 44)(25 45)(26 36)(27 37)(28 38)(29 39)(30 40)
(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 34 35)(36 37 38 39 40)(41 42 43 44 45)
(2 3 5 4)(7 8 10 9)(12 13 15 14)(17 18 20 19)(22 23 25 24)(27 28 30 29)(32 33 35 34)(37 38 40 39)(42 43 45 44)
G:=sub<Sym(45)| (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40), (1,16,31)(2,17,32)(3,18,33)(4,19,34)(5,20,35)(6,21,36)(7,22,37)(8,23,38)(9,24,39)(10,25,40)(11,26,41)(12,27,42)(13,28,43)(14,29,44)(15,30,45), (6,11)(7,12)(8,13)(9,14)(10,15)(16,31)(17,32)(18,33)(19,34)(20,35)(21,41)(22,42)(23,43)(24,44)(25,45)(26,36)(27,37)(28,38)(29,39)(30,40), (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)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45), (2,3,5,4)(7,8,10,9)(12,13,15,14)(17,18,20,19)(22,23,25,24)(27,28,30,29)(32,33,35,34)(37,38,40,39)(42,43,45,44)>;
G:=Group( (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40), (1,16,31)(2,17,32)(3,18,33)(4,19,34)(5,20,35)(6,21,36)(7,22,37)(8,23,38)(9,24,39)(10,25,40)(11,26,41)(12,27,42)(13,28,43)(14,29,44)(15,30,45), (6,11)(7,12)(8,13)(9,14)(10,15)(16,31)(17,32)(18,33)(19,34)(20,35)(21,41)(22,42)(23,43)(24,44)(25,45)(26,36)(27,37)(28,38)(29,39)(30,40), (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)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45), (2,3,5,4)(7,8,10,9)(12,13,15,14)(17,18,20,19)(22,23,25,24)(27,28,30,29)(32,33,35,34)(37,38,40,39)(42,43,45,44) );
G=PermutationGroup([[(1,11,6),(2,12,7),(3,13,8),(4,14,9),(5,15,10),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25),(31,41,36),(32,42,37),(33,43,38),(34,44,39),(35,45,40)], [(1,16,31),(2,17,32),(3,18,33),(4,19,34),(5,20,35),(6,21,36),(7,22,37),(8,23,38),(9,24,39),(10,25,40),(11,26,41),(12,27,42),(13,28,43),(14,29,44),(15,30,45)], [(6,11),(7,12),(8,13),(9,14),(10,15),(16,31),(17,32),(18,33),(19,34),(20,35),(21,41),(22,42),(23,43),(24,44),(25,45),(26,36),(27,37),(28,38),(29,39),(30,40)], [(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),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45)], [(2,3,5,4),(7,8,10,9),(12,13,15,14),(17,18,20,19),(22,23,25,24),(27,28,30,29),(32,33,35,34),(37,38,40,39),(42,43,45,44)]])
Matrix representation of C3⋊S3×F5 ►in GL8(𝔽61)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 60 | 60 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 60 | 60 | 60 |
G:=sub<GL(8,GF(61))| [0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,60,0,1,0,0,0,0,0,60,0,0,1,0,0,0,0,60,0,0,0],[11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,60,0,0,0,0,0,1,0,60] >;
C3⋊S3×F5 in GAP, Magma, Sage, TeX
C_3\rtimes S_3\times F_5
% in TeX
G:=Group("C3:S3xF5"); // GroupNames label
G:=SmallGroup(360,127);
// by ID
G=gap.SmallGroup(360,127);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,201,730,5189,2609]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^5=e^4=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations
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