direct product, non-abelian, soluble, monomial
Aliases: C2×C5⋊S4, C10⋊S4, C23⋊D15, C22⋊D30, A4⋊2D10, C5⋊2(C2×S4), (C2×A4)⋊D5, (C2×C10)⋊3D6, (C10×A4)⋊1C2, (C5×A4)⋊2C22, (C22×C10)⋊2S3, SmallGroup(240,197)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×A4 — C2×C5⋊S4 |
Generators and relations for C2×C5⋊S4
G = < a,b,c,d,e,f | a2=b5=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >
Subgroups: 484 in 66 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, D4, C23, C23, D5, C10, C10, A4, D6, C15, C2×D4, Dic5, D10, C2×C10, C2×C10, S4, C2×A4, D15, C30, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C2×S4, C5×A4, D30, C2×C5⋊D4, C5⋊S4, C10×A4, C2×C5⋊S4
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, D15, C2×S4, D30, C5⋊S4, C2×C5⋊S4
Character table of C2×C5⋊S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 5A | 5B | 6 | 10A | 10B | 10C | 10D | 10E | 10F | 15A | 15B | 15C | 15D | 30A | 30B | 30C | 30D | |
| size | 1 | 1 | 3 | 3 | 30 | 30 | 8 | 30 | 30 | 2 | 2 | 8 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 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 | 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 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 2 | 2 | 1 | -2 | -2 | 2 | -2 | -2 | 2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ8 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ10 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1 | 1+√5/2 | 1-√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ32ζ54+ζ32ζ5+ζ5 | ζ3ζ53-ζ3ζ52+ζ53 | -ζ3ζ53+ζ3ζ52+ζ52 | -ζ3ζ54+ζ3ζ5+ζ5 | orthogonal lifted from D30 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | orthogonal lifted from D15 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ32ζ54+ζ32ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | orthogonal lifted from D15 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1 | 1-√5/2 | 1+√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ32ζ54+ζ32ζ5-ζ54 | ζ3ζ53-ζ3ζ52+ζ53 | -ζ3ζ54+ζ3ζ5+ζ5 | -ζ32ζ54+ζ32ζ5+ζ5 | -ζ3ζ53+ζ3ζ52+ζ52 | orthogonal lifted from D30 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1 | 1+√5/2 | 1-√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ32ζ54+ζ32ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ54+ζ3ζ5+ζ5 | -ζ3ζ53+ζ3ζ52+ζ52 | ζ3ζ53-ζ3ζ52+ζ53 | -ζ32ζ54+ζ32ζ5+ζ5 | orthogonal lifted from D30 |
| ρ16 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | orthogonal lifted from D15 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | orthogonal lifted from D15 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1 | 1-√5/2 | 1+√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ3ζ53+ζ3ζ52+ζ52 | -ζ32ζ54+ζ32ζ5+ζ5 | -ζ3ζ54+ζ3ζ5+ζ5 | ζ3ζ53-ζ3ζ52+ζ53 | orthogonal lifted from D30 |
| ρ19 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | 3 | 3 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ20 | 3 | 3 | -1 | -1 | 1 | 1 | 0 | -1 | -1 | 3 | 3 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ21 | 3 | -3 | -1 | 1 | 1 | -1 | 0 | 1 | -1 | 3 | 3 | 0 | -3 | -3 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ22 | 3 | -3 | -1 | 1 | -1 | 1 | 0 | -1 | 1 | 3 | 3 | 0 | -3 | -3 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ23 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | -3+3√5/2 | -3-3√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C5⋊S4 |
| ρ24 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | -3-3√5/2 | -3+3√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C5⋊S4 |
| ρ25 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 3+3√5/2 | 3-3√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ26 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 3-3√5/2 | 3+3√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 30 points - transitive group 30T61
Generators in S30
(1 13)(2 14)(3 15)(4 11)(5 12)(6 29)(7 30)(8 26)(9 27)(10 28)(16 21)(17 22)(18 23)(19 24)(20 25)
(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 13)(2 14)(3 15)(4 11)(5 12)(6 29)(7 30)(8 26)(9 27)(10 28)
(6 29)(7 30)(8 26)(9 27)(10 28)(16 21)(17 22)(18 23)(19 24)(20 25)
(1 28 18)(2 29 19)(3 30 20)(4 26 16)(5 27 17)(6 24 14)(7 25 15)(8 21 11)(9 22 12)(10 23 13)
(2 5)(3 4)(6 22)(7 21)(8 25)(9 24)(10 23)(11 15)(12 14)(16 30)(17 29)(18 28)(19 27)(20 26)
G:=sub<Sym(30)| (1,13)(2,14)(3,15)(4,11)(5,12)(6,29)(7,30)(8,26)(9,27)(10,28)(16,21)(17,22)(18,23)(19,24)(20,25), (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), (1,13)(2,14)(3,15)(4,11)(5,12)(6,29)(7,30)(8,26)(9,27)(10,28), (6,29)(7,30)(8,26)(9,27)(10,28)(16,21)(17,22)(18,23)(19,24)(20,25), (1,28,18)(2,29,19)(3,30,20)(4,26,16)(5,27,17)(6,24,14)(7,25,15)(8,21,11)(9,22,12)(10,23,13), (2,5)(3,4)(6,22)(7,21)(8,25)(9,24)(10,23)(11,15)(12,14)(16,30)(17,29)(18,28)(19,27)(20,26)>;
G:=Group( (1,13)(2,14)(3,15)(4,11)(5,12)(6,29)(7,30)(8,26)(9,27)(10,28)(16,21)(17,22)(18,23)(19,24)(20,25), (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), (1,13)(2,14)(3,15)(4,11)(5,12)(6,29)(7,30)(8,26)(9,27)(10,28), (6,29)(7,30)(8,26)(9,27)(10,28)(16,21)(17,22)(18,23)(19,24)(20,25), (1,28,18)(2,29,19)(3,30,20)(4,26,16)(5,27,17)(6,24,14)(7,25,15)(8,21,11)(9,22,12)(10,23,13), (2,5)(3,4)(6,22)(7,21)(8,25)(9,24)(10,23)(11,15)(12,14)(16,30)(17,29)(18,28)(19,27)(20,26) );
G=PermutationGroup([[(1,13),(2,14),(3,15),(4,11),(5,12),(6,29),(7,30),(8,26),(9,27),(10,28),(16,21),(17,22),(18,23),(19,24),(20,25)], [(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)], [(1,13),(2,14),(3,15),(4,11),(5,12),(6,29),(7,30),(8,26),(9,27),(10,28)], [(6,29),(7,30),(8,26),(9,27),(10,28),(16,21),(17,22),(18,23),(19,24),(20,25)], [(1,28,18),(2,29,19),(3,30,20),(4,26,16),(5,27,17),(6,24,14),(7,25,15),(8,21,11),(9,22,12),(10,23,13)], [(2,5),(3,4),(6,22),(7,21),(8,25),(9,24),(10,23),(11,15),(12,14),(16,30),(17,29),(18,28),(19,27),(20,26)]])
G:=TransitiveGroup(30,61);
C2×C5⋊S4 is a maximal subgroup of
Dic5⋊2S4 Dic5⋊S4 A4⋊D20 C20⋊S4 C24⋊2D15 C2×D5×S4
C2×C5⋊S4 is a maximal quotient of C20.1S4 C20⋊S4 Q8.D30 C20.2S4 C20.6S4 C20.3S4 C24⋊2D15
Matrix representation of C2×C5⋊S4 ►in GL5(𝔽61)
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 17 | 60 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 59 | 60 | 1 |
| 36 | 28 | 0 | 0 | 0 |
| 33 | 24 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 | 59 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 32 | 32 | 59 |
| 37 | 28 | 0 | 0 | 0 |
| 47 | 24 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 32 | 60 |
G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[17,1,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,1,1,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,59,0,0,0,60,60,0,0,0,0,1],[36,33,0,0,0,28,24,0,0,0,0,0,2,60,32,0,0,1,0,32,0,0,59,0,59],[37,47,0,0,0,28,24,0,0,0,0,0,0,1,32,0,0,1,0,32,0,0,0,0,60] >;
C2×C5⋊S4 in GAP, Magma, Sage, TeX
C_2\times C_5\rtimes S_4
% in TeX
G:=Group("C2xC5:S4"); // GroupNames label
G:=SmallGroup(240,197);
// by ID
G=gap.SmallGroup(240,197);
# by ID
G:=PCGroup([6,-2,-2,-3,-5,-2,2,146,1155,3604,916,2165,1637]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^5=c^2=d^2=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations
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