direct product, non-abelian, soluble, monomial, A-group
Aliases: C2×C52⋊S3, C52⋊2D6, (C5×C10)⋊S3, C52⋊C3⋊2C22, (C2×C52⋊C3)⋊1C2, SmallGroup(300,26)
Series: Derived ►Chief ►Lower central ►Upper central
| C52⋊C3 — C2×C52⋊S3 |
Generators and relations for C2×C52⋊S3
G = < a,b,c,d,e | a2=b5=c5=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc3, be=eb, dcd-1=b-1c3, ece=b-1c-1, ede=d-1 >
Character table of C2×C52⋊S3
| class | 1 | 2A | 2B | 2C | 3 | 5A | 5B | 5C | 5D | 5E | 5F | 6 | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | |
| size | 1 | 1 | 15 | 15 | 50 | 3 | 3 | 3 | 3 | 6 | 6 | 50 | 3 | 3 | 3 | 3 | 6 | 6 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ7 | 3 | -3 | -1 | 1 | 0 | 2ζ54+ζ52 | ζ53+2ζ5 | 2ζ52+ζ5 | ζ54+2ζ53 | 1-√5/2 | 1+√5/2 | 0 | -2ζ54-ζ52 | -2ζ52-ζ5 | -ζ54-2ζ53 | -ζ53-2ζ5 | -1-√5/2 | -1+√5/2 | ζ52 | ζ53 | ζ54 | ζ5 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | complex faithful |
| ρ8 | 3 | 3 | -1 | -1 | 0 | ζ54+2ζ53 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ53+2ζ5 | 1+√5/2 | 1-√5/2 | 0 | ζ54+2ζ53 | 2ζ54+ζ52 | ζ53+2ζ5 | 2ζ52+ζ5 | 1-√5/2 | 1+√5/2 | -ζ54 | -ζ5 | -ζ53 | -ζ52 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | complex lifted from C52⋊S3 |
| ρ9 | 3 | 3 | 1 | 1 | 0 | 2ζ52+ζ5 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ54+ζ52 | 1+√5/2 | 1-√5/2 | 0 | 2ζ52+ζ5 | ζ53+2ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | 1-√5/2 | 1+√5/2 | ζ5 | ζ54 | ζ52 | ζ53 | ζ53 | ζ5 | ζ54 | ζ52 | complex lifted from C52⋊S3 |
| ρ10 | 3 | -3 | 1 | -1 | 0 | 2ζ52+ζ5 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ54+ζ52 | 1+√5/2 | 1-√5/2 | 0 | -2ζ52-ζ5 | -ζ53-2ζ5 | -2ζ54-ζ52 | -ζ54-2ζ53 | -1+√5/2 | -1-√5/2 | -ζ5 | -ζ54 | -ζ52 | -ζ53 | ζ53 | ζ5 | ζ54 | ζ52 | complex faithful |
| ρ11 | 3 | -3 | 1 | -1 | 0 | ζ54+2ζ53 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ53+2ζ5 | 1+√5/2 | 1-√5/2 | 0 | -ζ54-2ζ53 | -2ζ54-ζ52 | -ζ53-2ζ5 | -2ζ52-ζ5 | -1+√5/2 | -1-√5/2 | -ζ54 | -ζ5 | -ζ53 | -ζ52 | ζ52 | ζ54 | ζ5 | ζ53 | complex faithful |
| ρ12 | 3 | 3 | -1 | -1 | 0 | 2ζ54+ζ52 | ζ53+2ζ5 | 2ζ52+ζ5 | ζ54+2ζ53 | 1-√5/2 | 1+√5/2 | 0 | 2ζ54+ζ52 | 2ζ52+ζ5 | ζ54+2ζ53 | ζ53+2ζ5 | 1+√5/2 | 1-√5/2 | -ζ52 | -ζ53 | -ζ54 | -ζ5 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | complex lifted from C52⋊S3 |
| ρ13 | 3 | 3 | -1 | -1 | 0 | 2ζ52+ζ5 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ54+ζ52 | 1+√5/2 | 1-√5/2 | 0 | 2ζ52+ζ5 | ζ53+2ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | 1-√5/2 | 1+√5/2 | -ζ5 | -ζ54 | -ζ52 | -ζ53 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | complex lifted from C52⋊S3 |
| ρ14 | 3 | -3 | -1 | 1 | 0 | 2ζ52+ζ5 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ54+ζ52 | 1+√5/2 | 1-√5/2 | 0 | -2ζ52-ζ5 | -ζ53-2ζ5 | -2ζ54-ζ52 | -ζ54-2ζ53 | -1+√5/2 | -1-√5/2 | ζ5 | ζ54 | ζ52 | ζ53 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | complex faithful |
| ρ15 | 3 | -3 | 1 | -1 | 0 | ζ53+2ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | 2ζ52+ζ5 | 1-√5/2 | 1+√5/2 | 0 | -ζ53-2ζ5 | -ζ54-2ζ53 | -2ζ52-ζ5 | -2ζ54-ζ52 | -1-√5/2 | -1+√5/2 | -ζ53 | -ζ52 | -ζ5 | -ζ54 | ζ54 | ζ53 | ζ52 | ζ5 | complex faithful |
| ρ16 | 3 | 3 | 1 | 1 | 0 | ζ53+2ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | 2ζ52+ζ5 | 1-√5/2 | 1+√5/2 | 0 | ζ53+2ζ5 | ζ54+2ζ53 | 2ζ52+ζ5 | 2ζ54+ζ52 | 1+√5/2 | 1-√5/2 | ζ53 | ζ52 | ζ5 | ζ54 | ζ54 | ζ53 | ζ52 | ζ5 | complex lifted from C52⋊S3 |
| ρ17 | 3 | -3 | -1 | 1 | 0 | ζ53+2ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | 2ζ52+ζ5 | 1-√5/2 | 1+√5/2 | 0 | -ζ53-2ζ5 | -ζ54-2ζ53 | -2ζ52-ζ5 | -2ζ54-ζ52 | -1-√5/2 | -1+√5/2 | ζ53 | ζ52 | ζ5 | ζ54 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | complex faithful |
| ρ18 | 3 | 3 | -1 | -1 | 0 | ζ53+2ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | 2ζ52+ζ5 | 1-√5/2 | 1+√5/2 | 0 | ζ53+2ζ5 | ζ54+2ζ53 | 2ζ52+ζ5 | 2ζ54+ζ52 | 1+√5/2 | 1-√5/2 | -ζ53 | -ζ52 | -ζ5 | -ζ54 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | complex lifted from C52⋊S3 |
| ρ19 | 3 | -3 | 1 | -1 | 0 | 2ζ54+ζ52 | ζ53+2ζ5 | 2ζ52+ζ5 | ζ54+2ζ53 | 1-√5/2 | 1+√5/2 | 0 | -2ζ54-ζ52 | -2ζ52-ζ5 | -ζ54-2ζ53 | -ζ53-2ζ5 | -1-√5/2 | -1+√5/2 | -ζ52 | -ζ53 | -ζ54 | -ζ5 | ζ5 | ζ52 | ζ53 | ζ54 | complex faithful |
| ρ20 | 3 | 3 | 1 | 1 | 0 | ζ54+2ζ53 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ53+2ζ5 | 1+√5/2 | 1-√5/2 | 0 | ζ54+2ζ53 | 2ζ54+ζ52 | ζ53+2ζ5 | 2ζ52+ζ5 | 1-√5/2 | 1+√5/2 | ζ54 | ζ5 | ζ53 | ζ52 | ζ52 | ζ54 | ζ5 | ζ53 | complex lifted from C52⋊S3 |
| ρ21 | 3 | -3 | -1 | 1 | 0 | ζ54+2ζ53 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ53+2ζ5 | 1+√5/2 | 1-√5/2 | 0 | -ζ54-2ζ53 | -2ζ54-ζ52 | -ζ53-2ζ5 | -2ζ52-ζ5 | -1+√5/2 | -1-√5/2 | ζ54 | ζ5 | ζ53 | ζ52 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | complex faithful |
| ρ22 | 3 | 3 | 1 | 1 | 0 | 2ζ54+ζ52 | ζ53+2ζ5 | 2ζ52+ζ5 | ζ54+2ζ53 | 1-√5/2 | 1+√5/2 | 0 | 2ζ54+ζ52 | 2ζ52+ζ5 | ζ54+2ζ53 | ζ53+2ζ5 | 1+√5/2 | 1-√5/2 | ζ52 | ζ53 | ζ54 | ζ5 | ζ5 | ζ52 | ζ53 | ζ54 | complex lifted from C52⋊S3 |
| ρ23 | 6 | 6 | 0 | 0 | 0 | 1+√5 | 1+√5 | 1-√5 | 1-√5 | -3+√5/2 | -3-√5/2 | 0 | 1+√5 | 1-√5 | 1-√5 | 1+√5 | -3-√5/2 | -3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊S3 |
| ρ24 | 6 | -6 | 0 | 0 | 0 | 1+√5 | 1+√5 | 1-√5 | 1-√5 | -3+√5/2 | -3-√5/2 | 0 | -1-√5 | -1+√5 | -1+√5 | -1-√5 | 3+√5/2 | 3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ25 | 6 | -6 | 0 | 0 | 0 | 1-√5 | 1-√5 | 1+√5 | 1+√5 | -3-√5/2 | -3+√5/2 | 0 | -1+√5 | -1-√5 | -1-√5 | -1+√5 | 3-√5/2 | 3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ26 | 6 | 6 | 0 | 0 | 0 | 1-√5 | 1-√5 | 1+√5 | 1+√5 | -3-√5/2 | -3+√5/2 | 0 | 1-√5 | 1+√5 | 1+√5 | 1-√5 | -3+√5/2 | -3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊S3 |
►On 30 points - transitive group 30T77
Generators in S30
(1 7)(2 6)(3 9)(4 8)(5 10)(11 30)(12 26)(13 27)(14 28)(15 29)(16 24)(17 25)(18 21)(19 22)(20 23)
(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 5 4 3 2)(6 7 10 8 9)(11 14 12 15 13)(16 17 18 19 20)(21 22 23 24 25)(26 29 27 30 28)
(1 20 27)(2 18 29)(3 16 26)(4 19 28)(5 17 30)(6 21 15)(7 23 13)(8 22 14)(9 24 12)(10 25 11)
(1 7)(2 10)(3 8)(4 9)(5 6)(11 18)(12 19)(13 20)(14 16)(15 17)(21 30)(22 26)(23 27)(24 28)(25 29)
G:=sub<Sym(30)| (1,7)(2,6)(3,9)(4,8)(5,10)(11,30)(12,26)(13,27)(14,28)(15,29)(16,24)(17,25)(18,21)(19,22)(20,23), (11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,5,4,3,2)(6,7,10,8,9)(11,14,12,15,13)(16,17,18,19,20)(21,22,23,24,25)(26,29,27,30,28), (1,20,27)(2,18,29)(3,16,26)(4,19,28)(5,17,30)(6,21,15)(7,23,13)(8,22,14)(9,24,12)(10,25,11), (1,7)(2,10)(3,8)(4,9)(5,6)(11,18)(12,19)(13,20)(14,16)(15,17)(21,30)(22,26)(23,27)(24,28)(25,29)>;
G:=Group( (1,7)(2,6)(3,9)(4,8)(5,10)(11,30)(12,26)(13,27)(14,28)(15,29)(16,24)(17,25)(18,21)(19,22)(20,23), (11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,5,4,3,2)(6,7,10,8,9)(11,14,12,15,13)(16,17,18,19,20)(21,22,23,24,25)(26,29,27,30,28), (1,20,27)(2,18,29)(3,16,26)(4,19,28)(5,17,30)(6,21,15)(7,23,13)(8,22,14)(9,24,12)(10,25,11), (1,7)(2,10)(3,8)(4,9)(5,6)(11,18)(12,19)(13,20)(14,16)(15,17)(21,30)(22,26)(23,27)(24,28)(25,29) );
G=PermutationGroup([[(1,7),(2,6),(3,9),(4,8),(5,10),(11,30),(12,26),(13,27),(14,28),(15,29),(16,24),(17,25),(18,21),(19,22),(20,23)], [(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30)], [(1,5,4,3,2),(6,7,10,8,9),(11,14,12,15,13),(16,17,18,19,20),(21,22,23,24,25),(26,29,27,30,28)], [(1,20,27),(2,18,29),(3,16,26),(4,19,28),(5,17,30),(6,21,15),(7,23,13),(8,22,14),(9,24,12),(10,25,11)], [(1,7),(2,10),(3,8),(4,9),(5,6),(11,18),(12,19),(13,20),(14,16),(15,17),(21,30),(22,26),(23,27),(24,28),(25,29)]])
G:=TransitiveGroup(30,77);
►On 30 points - transitive group 30T81
Generators in S30
(1 17)(2 18)(3 19)(4 20)(5 16)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)
(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)
(1 3 5 2 4)(6 10 9 8 7)(16 18 20 17 19)(21 25 24 23 22)
(1 11 6)(2 14 9)(3 12 7)(4 15 10)(5 13 8)(16 28 23)(17 26 21)(18 29 24)(19 27 22)(20 30 25)
(1 17)(2 18)(3 19)(4 20)(5 16)(6 26)(7 27)(8 28)(9 29)(10 30)(11 21)(12 22)(13 23)(14 24)(15 25)
G:=sub<Sym(30)| (1,17)(2,18)(3,19)(4,20)(5,16)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30), (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), (1,3,5,2,4)(6,10,9,8,7)(16,18,20,17,19)(21,25,24,23,22), (1,11,6)(2,14,9)(3,12,7)(4,15,10)(5,13,8)(16,28,23)(17,26,21)(18,29,24)(19,27,22)(20,30,25), (1,17)(2,18)(3,19)(4,20)(5,16)(6,26)(7,27)(8,28)(9,29)(10,30)(11,21)(12,22)(13,23)(14,24)(15,25)>;
G:=Group( (1,17)(2,18)(3,19)(4,20)(5,16)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30), (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), (1,3,5,2,4)(6,10,9,8,7)(16,18,20,17,19)(21,25,24,23,22), (1,11,6)(2,14,9)(3,12,7)(4,15,10)(5,13,8)(16,28,23)(17,26,21)(18,29,24)(19,27,22)(20,30,25), (1,17)(2,18)(3,19)(4,20)(5,16)(6,26)(7,27)(8,28)(9,29)(10,30)(11,21)(12,22)(13,23)(14,24)(15,25) );
G=PermutationGroup([[(1,17),(2,18),(3,19),(4,20),(5,16),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30)], [(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)], [(1,3,5,2,4),(6,10,9,8,7),(16,18,20,17,19),(21,25,24,23,22)], [(1,11,6),(2,14,9),(3,12,7),(4,15,10),(5,13,8),(16,28,23),(17,26,21),(18,29,24),(19,27,22),(20,30,25)], [(1,17),(2,18),(3,19),(4,20),(5,16),(6,26),(7,27),(8,28),(9,29),(10,30),(11,21),(12,22),(13,23),(14,24),(15,25)]])
G:=TransitiveGroup(30,81);
Matrix representation of C2×C52⋊S3 ►in GL3(𝔽11) generated by
| 10 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 1 | 3 | 7 |
| 9 | 7 | 10 |
| 8 | 8 | 2 |
| 1 | 5 | 7 |
| 2 | 4 | 9 |
| 1 | 0 | 10 |
| 6 | 2 | 8 |
| 5 | 10 | 7 |
| 5 | 1 | 6 |
| 1 | 4 | 1 |
| 0 | 1 | 0 |
| 0 | 3 | 10 |
G:=sub<GL(3,GF(11))| [10,0,0,0,10,0,0,0,10],[1,9,8,3,7,8,7,10,2],[1,2,1,5,4,0,7,9,10],[6,5,5,2,10,1,8,7,6],[1,0,0,4,1,3,1,0,10] >;
C2×C52⋊S3 in GAP, Magma, Sage, TeX
C_2\times C_5^2\rtimes S_3
% in TeX
G:=Group("C2xC5^2:S3"); // GroupNames label
G:=SmallGroup(300,26);
// by ID
G=gap.SmallGroup(300,26);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,5,122,973,7204,1439]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^5=c^5=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^3,b*e=e*b,d*c*d^-1=b^-1*c^3,e*c*e=b^-1*c^-1,e*d*e=d^-1>;
// generators/relations
Export
Subgroup lattice of C2×C52⋊S3 in TeX
Character table of C2×C52⋊S3 in TeX