direct product, non-abelian, soluble, monomial
Aliases: C5×S4, A4⋊C10, C22⋊(C5×S3), (C5×A4)⋊3C2, (C2×C10)⋊1S3, SmallGroup(120,37)
Series: Derived ►Chief ►Lower central ►Upper central
| 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, ae=ea, 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 | 5C | 5D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | |
| size | 1 | 3 | 6 | 8 | 6 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 6 | 6 | 6 | 6 | |
| 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 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | 1 | -1 | ζ53 | ζ52 | ζ54 | ζ5 | ζ54 | ζ5 | ζ52 | ζ53 | -ζ53 | -ζ52 | -ζ54 | -ζ5 | ζ52 | ζ5 | ζ54 | ζ53 | -ζ52 | -ζ5 | -ζ54 | -ζ53 | linear of order 10 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | ζ54 | ζ5 | ζ52 | ζ53 | ζ52 | ζ53 | ζ5 | ζ54 | ζ54 | ζ5 | ζ52 | ζ53 | ζ5 | ζ53 | ζ52 | ζ54 | ζ5 | ζ53 | ζ52 | ζ54 | linear of order 5 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | ζ53 | ζ52 | ζ54 | ζ5 | ζ54 | ζ5 | ζ52 | ζ53 | ζ53 | ζ52 | ζ54 | ζ5 | ζ52 | ζ5 | ζ54 | ζ53 | ζ52 | ζ5 | ζ54 | ζ53 | linear of order 5 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | ζ54 | ζ5 | ζ52 | ζ53 | ζ52 | ζ53 | ζ5 | ζ54 | -ζ54 | -ζ5 | -ζ52 | -ζ53 | ζ5 | ζ53 | ζ52 | ζ54 | -ζ5 | -ζ53 | -ζ52 | -ζ54 | linear of order 10 |
| ρ7 | 1 | 1 | -1 | 1 | -1 | ζ52 | ζ53 | ζ5 | ζ54 | ζ5 | ζ54 | ζ53 | ζ52 | -ζ52 | -ζ53 | -ζ5 | -ζ54 | ζ53 | ζ54 | ζ5 | ζ52 | -ζ53 | -ζ54 | -ζ5 | -ζ52 | linear of order 10 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | ζ5 | ζ54 | ζ53 | ζ52 | ζ53 | ζ52 | ζ54 | ζ5 | -ζ5 | -ζ54 | -ζ53 | -ζ52 | ζ54 | ζ52 | ζ53 | ζ5 | -ζ54 | -ζ52 | -ζ53 | -ζ5 | linear of order 10 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | ζ5 | ζ54 | ζ53 | ζ52 | ζ53 | ζ52 | ζ54 | ζ5 | ζ5 | ζ54 | ζ53 | ζ52 | ζ54 | ζ52 | ζ53 | ζ5 | ζ54 | ζ52 | ζ53 | ζ5 | linear of order 5 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | ζ52 | ζ53 | ζ5 | ζ54 | ζ5 | ζ54 | ζ53 | ζ52 | ζ52 | ζ53 | ζ5 | ζ54 | ζ53 | ζ54 | ζ5 | ζ52 | ζ53 | ζ54 | ζ5 | ζ52 | linear of order 5 |
| ρ11 | 2 | 2 | 0 | -1 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 0 | -1 | 0 | 2ζ54 | 2ζ5 | 2ζ52 | 2ζ53 | 2ζ52 | 2ζ53 | 2ζ5 | 2ζ54 | 0 | 0 | 0 | 0 | -ζ5 | -ζ53 | -ζ52 | -ζ54 | 0 | 0 | 0 | 0 | complex lifted from C5×S3 |
| ρ13 | 2 | 2 | 0 | -1 | 0 | 2ζ53 | 2ζ52 | 2ζ54 | 2ζ5 | 2ζ54 | 2ζ5 | 2ζ52 | 2ζ53 | 0 | 0 | 0 | 0 | -ζ52 | -ζ5 | -ζ54 | -ζ53 | 0 | 0 | 0 | 0 | complex lifted from C5×S3 |
| ρ14 | 2 | 2 | 0 | -1 | 0 | 2ζ52 | 2ζ53 | 2ζ5 | 2ζ54 | 2ζ5 | 2ζ54 | 2ζ53 | 2ζ52 | 0 | 0 | 0 | 0 | -ζ53 | -ζ54 | -ζ5 | -ζ52 | 0 | 0 | 0 | 0 | complex lifted from C5×S3 |
| ρ15 | 2 | 2 | 0 | -1 | 0 | 2ζ5 | 2ζ54 | 2ζ53 | 2ζ52 | 2ζ53 | 2ζ52 | 2ζ54 | 2ζ5 | 0 | 0 | 0 | 0 | -ζ54 | -ζ52 | -ζ53 | -ζ5 | 0 | 0 | 0 | 0 | complex lifted from C5×S3 |
| ρ16 | 3 | -1 | -1 | 0 | 1 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ17 | 3 | -1 | 1 | 0 | -1 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ18 | 3 | -1 | -1 | 0 | 1 | 3ζ54 | 3ζ5 | 3ζ52 | 3ζ53 | -ζ52 | -ζ53 | -ζ5 | -ζ54 | -ζ54 | -ζ5 | -ζ52 | -ζ53 | 0 | 0 | 0 | 0 | ζ5 | ζ53 | ζ52 | ζ54 | complex faithful |
| ρ19 | 3 | -1 | -1 | 0 | 1 | 3ζ53 | 3ζ52 | 3ζ54 | 3ζ5 | -ζ54 | -ζ5 | -ζ52 | -ζ53 | -ζ53 | -ζ52 | -ζ54 | -ζ5 | 0 | 0 | 0 | 0 | ζ52 | ζ5 | ζ54 | ζ53 | complex faithful |
| ρ20 | 3 | -1 | -1 | 0 | 1 | 3ζ5 | 3ζ54 | 3ζ53 | 3ζ52 | -ζ53 | -ζ52 | -ζ54 | -ζ5 | -ζ5 | -ζ54 | -ζ53 | -ζ52 | 0 | 0 | 0 | 0 | ζ54 | ζ52 | ζ53 | ζ5 | complex faithful |
| ρ21 | 3 | -1 | 1 | 0 | -1 | 3ζ5 | 3ζ54 | 3ζ53 | 3ζ52 | -ζ53 | -ζ52 | -ζ54 | -ζ5 | ζ5 | ζ54 | ζ53 | ζ52 | 0 | 0 | 0 | 0 | -ζ54 | -ζ52 | -ζ53 | -ζ5 | complex faithful |
| ρ22 | 3 | -1 | 1 | 0 | -1 | 3ζ53 | 3ζ52 | 3ζ54 | 3ζ5 | -ζ54 | -ζ5 | -ζ52 | -ζ53 | ζ53 | ζ52 | ζ54 | ζ5 | 0 | 0 | 0 | 0 | -ζ52 | -ζ5 | -ζ54 | -ζ53 | complex faithful |
| ρ23 | 3 | -1 | -1 | 0 | 1 | 3ζ52 | 3ζ53 | 3ζ5 | 3ζ54 | -ζ5 | -ζ54 | -ζ53 | -ζ52 | -ζ52 | -ζ53 | -ζ5 | -ζ54 | 0 | 0 | 0 | 0 | ζ53 | ζ54 | ζ5 | ζ52 | complex faithful |
| ρ24 | 3 | -1 | 1 | 0 | -1 | 3ζ54 | 3ζ5 | 3ζ52 | 3ζ53 | -ζ52 | -ζ53 | -ζ5 | -ζ54 | ζ54 | ζ5 | ζ52 | ζ53 | 0 | 0 | 0 | 0 | -ζ5 | -ζ53 | -ζ52 | -ζ54 | complex faithful |
| ρ25 | 3 | -1 | 1 | 0 | -1 | 3ζ52 | 3ζ53 | 3ζ5 | 3ζ54 | -ζ5 | -ζ54 | -ζ53 | -ζ52 | ζ52 | ζ53 | ζ5 | ζ54 | 0 | 0 | 0 | 0 | -ζ53 | -ζ54 | -ζ5 | -ζ52 | complex faithful |
►On 20 points - transitive group 20T34
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 7)(2 8)(3 9)(4 10)(5 6)(11 17)(12 18)(13 19)(14 20)(15 16)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 14)(7 15)(8 11)(9 12)(10 13)
(6 20 14)(7 16 15)(8 17 11)(9 18 12)(10 19 13)
(6 14)(7 15)(8 11)(9 12)(10 13)
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,7)(2,8)(3,9)(4,10)(5,6)(11,17)(12,18)(13,19)(14,20)(15,16), (1,16)(2,17)(3,18)(4,19)(5,20)(6,14)(7,15)(8,11)(9,12)(10,13), (6,20,14)(7,16,15)(8,17,11)(9,18,12)(10,19,13), (6,14)(7,15)(8,11)(9,12)(10,13)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,7)(2,8)(3,9)(4,10)(5,6)(11,17)(12,18)(13,19)(14,20)(15,16), (1,16)(2,17)(3,18)(4,19)(5,20)(6,14)(7,15)(8,11)(9,12)(10,13), (6,20,14)(7,16,15)(8,17,11)(9,18,12)(10,19,13), (6,14)(7,15)(8,11)(9,12)(10,13) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,7),(2,8),(3,9),(4,10),(5,6),(11,17),(12,18),(13,19),(14,20),(15,16)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,14),(7,15),(8,11),(9,12),(10,13)], [(6,20,14),(7,16,15),(8,17,11),(9,18,12),(10,19,13)], [(6,14),(7,15),(8,11),(9,12),(10,13)]])
G:=TransitiveGroup(20,34);
►On 30 points - transitive group 30T33
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)(6 28)(7 29)(8 30)(9 26)(10 27)
(6 28)(7 29)(8 30)(9 26)(10 27)(16 22)(17 23)(18 24)(19 25)(20 21)
(1 9 18)(2 10 19)(3 6 20)(4 7 16)(5 8 17)(11 26 24)(12 27 25)(13 28 21)(14 29 22)(15 30 23)
(6 20)(7 16)(8 17)(9 18)(10 19)(21 28)(22 29)(23 30)(24 26)(25 27)
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)(6,28)(7,29)(8,30)(9,26)(10,27), (6,28)(7,29)(8,30)(9,26)(10,27)(16,22)(17,23)(18,24)(19,25)(20,21), (1,9,18)(2,10,19)(3,6,20)(4,7,16)(5,8,17)(11,26,24)(12,27,25)(13,28,21)(14,29,22)(15,30,23), (6,20)(7,16)(8,17)(9,18)(10,19)(21,28)(22,29)(23,30)(24,26)(25,27)>;
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)(6,28)(7,29)(8,30)(9,26)(10,27), (6,28)(7,29)(8,30)(9,26)(10,27)(16,22)(17,23)(18,24)(19,25)(20,21), (1,9,18)(2,10,19)(3,6,20)(4,7,16)(5,8,17)(11,26,24)(12,27,25)(13,28,21)(14,29,22)(15,30,23), (6,20)(7,16)(8,17)(9,18)(10,19)(21,28)(22,29)(23,30)(24,26)(25,27) );
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),(6,28),(7,29),(8,30),(9,26),(10,27)], [(6,28),(7,29),(8,30),(9,26),(10,27),(16,22),(17,23),(18,24),(19,25),(20,21)], [(1,9,18),(2,10,19),(3,6,20),(4,7,16),(5,8,17),(11,26,24),(12,27,25),(13,28,21),(14,29,22),(15,30,23)], [(6,20),(7,16),(8,17),(9,18),(10,19),(21,28),(22,29),(23,30),(24,26),(25,27)]])
G:=TransitiveGroup(30,33);
►On 30 points - transitive group 30T34
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 17)(2 18)(3 19)(4 20)(5 16)(6 28)(7 29)(8 30)(9 26)(10 27)
(6 28)(7 29)(8 30)(9 26)(10 27)(11 23)(12 24)(13 25)(14 21)(15 22)
(1 9 12)(2 10 13)(3 6 14)(4 7 15)(5 8 11)(16 30 23)(17 26 24)(18 27 25)(19 28 21)(20 29 22)
(1 17)(2 18)(3 19)(4 20)(5 16)(6 21)(7 22)(8 23)(9 24)(10 25)(11 30)(12 26)(13 27)(14 28)(15 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), (1,17)(2,18)(3,19)(4,20)(5,16)(6,28)(7,29)(8,30)(9,26)(10,27), (6,28)(7,29)(8,30)(9,26)(10,27)(11,23)(12,24)(13,25)(14,21)(15,22), (1,9,12)(2,10,13)(3,6,14)(4,7,15)(5,8,11)(16,30,23)(17,26,24)(18,27,25)(19,28,21)(20,29,22), (1,17)(2,18)(3,19)(4,20)(5,16)(6,21)(7,22)(8,23)(9,24)(10,25)(11,30)(12,26)(13,27)(14,28)(15,29)>;
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), (1,17)(2,18)(3,19)(4,20)(5,16)(6,28)(7,29)(8,30)(9,26)(10,27), (6,28)(7,29)(8,30)(9,26)(10,27)(11,23)(12,24)(13,25)(14,21)(15,22), (1,9,12)(2,10,13)(3,6,14)(4,7,15)(5,8,11)(16,30,23)(17,26,24)(18,27,25)(19,28,21)(20,29,22), (1,17)(2,18)(3,19)(4,20)(5,16)(6,21)(7,22)(8,23)(9,24)(10,25)(11,30)(12,26)(13,27)(14,28)(15,29) );
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)], [(1,17),(2,18),(3,19),(4,20),(5,16),(6,28),(7,29),(8,30),(9,26),(10,27)], [(6,28),(7,29),(8,30),(9,26),(10,27),(11,23),(12,24),(13,25),(14,21),(15,22)], [(1,9,12),(2,10,13),(3,6,14),(4,7,15),(5,8,11),(16,30,23),(17,26,24),(18,27,25),(19,28,21),(20,29,22)], [(1,17),(2,18),(3,19),(4,20),(5,16),(6,21),(7,22),(8,23),(9,24),(10,25),(11,30),(12,26),(13,27),(14,28),(15,29)]])
G:=TransitiveGroup(30,34);
Matrix representation of C5×S4 ►in GL3(𝔽11) generated by
| 9 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 10 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 0 | 0 | 2 |
| 3 | 0 | 0 |
| 0 | 2 | 0 |
| 10 | 0 | 0 |
| 0 | 0 | 5 |
| 0 | 9 | 0 |
G:=sub<GL(3,GF(11))| [9,0,0,0,9,0,0,0,9],[10,0,0,0,10,0,0,0,1],[1,0,0,0,10,0,0,0,10],[0,3,0,0,0,2,2,0,0],[10,0,0,0,0,9,0,5,0] >;
C5×S4 in GAP, Magma, Sage, TeX
C_5\times S_4
% in TeX
G:=Group("C5xS4"); // GroupNames label
G:=SmallGroup(120,37);
// by ID
G=gap.SmallGroup(120,37);
# by ID
G:=PCGroup([5,-2,-5,-3,-2,2,302,1203,133,754,239]);
// 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,a*e=e*a,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