non-abelian, soluble, monomial
Aliases: C5⋊S4, A4⋊D5, C22⋊D15, (C5×A4)⋊1C2, (C2×C10)⋊2S3, SmallGroup(120,38)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×A4 — C5⋊S4 |
Generators and relations for C5⋊S4
G = < a,b,c,d,e | a5=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >
Character table of C5⋊S4
| class | 1 | 2A | 2B | 3 | 4 | 5A | 5B | 10A | 10B | 15A | 15B | 15C | 15D | |
| size | 1 | 3 | 30 | 8 | 30 | 2 | 2 | 6 | 6 | 8 | 8 | 8 | 8 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 2 | 2 | 0 | -1 | 0 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ4 | 2 | 2 | 0 | 2 | 0 | -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 |
| ρ5 | 2 | 2 | 0 | 2 | 0 | -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 |
| ρ6 | 2 | 2 | 0 | -1 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ53-ζ3ζ52-ζ52 | ζ32ζ54-ζ32ζ5-ζ5 | ζ3ζ54-ζ3ζ5-ζ5 | orthogonal lifted from D15 |
| ρ7 | 2 | 2 | 0 | -1 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ54-ζ3ζ5-ζ5 | ζ32ζ54-ζ32ζ5-ζ5 | orthogonal lifted from D15 |
| ρ8 | 2 | 2 | 0 | -1 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | ζ3ζ54-ζ3ζ5-ζ5 | ζ32ζ54-ζ32ζ5-ζ5 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ53-ζ3ζ52-ζ52 | orthogonal lifted from D15 |
| ρ9 | 2 | 2 | 0 | -1 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | ζ32ζ54-ζ32ζ5-ζ5 | ζ3ζ54-ζ3ζ5-ζ5 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ53+ζ3ζ52-ζ53 | orthogonal lifted from D15 |
| ρ10 | 3 | -1 | -1 | 0 | 1 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ11 | 3 | -1 | 1 | 0 | -1 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ12 | 6 | -2 | 0 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ13 | 6 | -2 | 0 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 20 points - transitive group 20T33
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)
(6 11 16)(7 12 17)(8 13 18)(9 14 19)(10 15 20)
(2 5)(3 4)(6 11)(7 15)(8 14)(9 13)(10 12)(17 20)(18 19)
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20), (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15), (6,11,16)(7,12,17)(8,13,18)(9,14,19)(10,15,20), (2,5)(3,4)(6,11)(7,15)(8,14)(9,13)(10,12)(17,20)(18,19)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20), (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15), (6,11,16)(7,12,17)(8,13,18)(9,14,19)(10,15,20), (2,5)(3,4)(6,11)(7,15)(8,14)(9,13)(10,12)(17,20)(18,19) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,11),(7,12),(8,13),(9,14),(10,15)], [(6,11,16),(7,12,17),(8,13,18),(9,14,19),(10,15,20)], [(2,5),(3,4),(6,11),(7,15),(8,14),(9,13),(10,12),(17,20),(18,19)]])
G:=TransitiveGroup(20,33);
►On 30 points - transitive group 30T19
Generators in S30
(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 11)(2 12)(3 13)(4 14)(5 15)(16 25)(17 21)(18 22)(19 23)(20 24)
(6 30)(7 26)(8 27)(9 28)(10 29)(16 25)(17 21)(18 22)(19 23)(20 24)
(1 17 27)(2 18 28)(3 19 29)(4 20 30)(5 16 26)(6 14 24)(7 15 25)(8 11 21)(9 12 22)(10 13 23)
(2 5)(3 4)(6 23)(7 22)(8 21)(9 25)(10 24)(12 15)(13 14)(16 28)(17 27)(18 26)(19 30)(20 29)
G:=sub<Sym(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), (1,11)(2,12)(3,13)(4,14)(5,15)(16,25)(17,21)(18,22)(19,23)(20,24), (6,30)(7,26)(8,27)(9,28)(10,29)(16,25)(17,21)(18,22)(19,23)(20,24), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,14,24)(7,15,25)(8,11,21)(9,12,22)(10,13,23), (2,5)(3,4)(6,23)(7,22)(8,21)(9,25)(10,24)(12,15)(13,14)(16,28)(17,27)(18,26)(19,30)(20,29)>;
G:=Group( (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), (1,11)(2,12)(3,13)(4,14)(5,15)(16,25)(17,21)(18,22)(19,23)(20,24), (6,30)(7,26)(8,27)(9,28)(10,29)(16,25)(17,21)(18,22)(19,23)(20,24), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,14,24)(7,15,25)(8,11,21)(9,12,22)(10,13,23), (2,5)(3,4)(6,23)(7,22)(8,21)(9,25)(10,24)(12,15)(13,14)(16,28)(17,27)(18,26)(19,30)(20,29) );
G=PermutationGroup([[(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)], [(1,11),(2,12),(3,13),(4,14),(5,15),(16,25),(17,21),(18,22),(19,23),(20,24)], [(6,30),(7,26),(8,27),(9,28),(10,29),(16,25),(17,21),(18,22),(19,23),(20,24)], [(1,17,27),(2,18,28),(3,19,29),(4,20,30),(5,16,26),(6,14,24),(7,15,25),(8,11,21),(9,12,22),(10,13,23)], [(2,5),(3,4),(6,23),(7,22),(8,21),(9,25),(10,24),(12,15),(13,14),(16,28),(17,27),(18,26),(19,30),(20,29)]])
G:=TransitiveGroup(30,19);
►On 30 points - transitive group 30T31
Generators in S30
(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 11)(2 12)(3 13)(4 14)(5 15)(16 25)(17 21)(18 22)(19 23)(20 24)
(6 30)(7 26)(8 27)(9 28)(10 29)(16 25)(17 21)(18 22)(19 23)(20 24)
(1 17 27)(2 18 28)(3 19 29)(4 20 30)(5 16 26)(6 14 24)(7 15 25)(8 11 21)(9 12 22)(10 13 23)
(1 11)(2 15)(3 14)(4 13)(5 12)(6 19)(7 18)(8 17)(9 16)(10 20)(21 27)(22 26)(23 30)(24 29)(25 28)
G:=sub<Sym(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), (1,11)(2,12)(3,13)(4,14)(5,15)(16,25)(17,21)(18,22)(19,23)(20,24), (6,30)(7,26)(8,27)(9,28)(10,29)(16,25)(17,21)(18,22)(19,23)(20,24), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,14,24)(7,15,25)(8,11,21)(9,12,22)(10,13,23), (1,11)(2,15)(3,14)(4,13)(5,12)(6,19)(7,18)(8,17)(9,16)(10,20)(21,27)(22,26)(23,30)(24,29)(25,28)>;
G:=Group( (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), (1,11)(2,12)(3,13)(4,14)(5,15)(16,25)(17,21)(18,22)(19,23)(20,24), (6,30)(7,26)(8,27)(9,28)(10,29)(16,25)(17,21)(18,22)(19,23)(20,24), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,14,24)(7,15,25)(8,11,21)(9,12,22)(10,13,23), (1,11)(2,15)(3,14)(4,13)(5,12)(6,19)(7,18)(8,17)(9,16)(10,20)(21,27)(22,26)(23,30)(24,29)(25,28) );
G=PermutationGroup([[(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)], [(1,11),(2,12),(3,13),(4,14),(5,15),(16,25),(17,21),(18,22),(19,23),(20,24)], [(6,30),(7,26),(8,27),(9,28),(10,29),(16,25),(17,21),(18,22),(19,23),(20,24)], [(1,17,27),(2,18,28),(3,19,29),(4,20,30),(5,16,26),(6,14,24),(7,15,25),(8,11,21),(9,12,22),(10,13,23)], [(1,11),(2,15),(3,14),(4,13),(5,12),(6,19),(7,18),(8,17),(9,16),(10,20),(21,27),(22,26),(23,30),(24,29),(25,28)]])
G:=TransitiveGroup(30,31);
C5⋊S4 is a maximal subgroup of
D5×S4 A4⋊D15 C42⋊D15 C24⋊4D15
C5⋊S4 is a maximal quotient of Q8.D15 Q8⋊D15 A4⋊Dic5 C22⋊D45 A4⋊D15 C42⋊D15 C24⋊4D15
Matrix representation of C5⋊S4 ►in GL5(𝔽61)
| 51 | 24 | 0 | 0 | 0 |
| 37 | 27 | 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 | 0 | 60 | 1 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 1 | 60 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 1 | 0 | 60 |
| 0 | 0 | 0 | 0 | 60 |
| 60 | 60 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(61))| [51,37,0,0,0,24,27,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,60,60,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,60,60,60],[60,1,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;
C5⋊S4 in GAP, Magma, Sage, TeX
C_5\rtimes S_4
% in TeX
G:=Group("C5:S4"); // GroupNames label
G:=SmallGroup(120,38);
// by ID
G=gap.SmallGroup(120,38);
# by ID
G:=PCGroup([5,-2,-3,-5,-2,2,41,362,1203,608,754,1134]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations
Export
Subgroup lattice of C5⋊S4 in TeX
Character table of C5⋊S4 in TeX