Aliases: 2- 1+4⋊C5, C2.(C24⋊C5), SmallGroup(160,199)
Series: Derived ►Chief ►Lower central ►Upper central
| 2- 1+4 — 2- 1+4⋊C5 |
Generators and relations for 2- 1+4⋊C5
G = < a,b,c,d,e | a4=b2=e5=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, eae-1=a2bcd, bc=cb, bd=db, ebe-1=ab, dcd-1=a2c, ece-1=abc, ede-1=cd >
Character table of 2- 1+4⋊C5
| class | 1 | 2A | 2B | 4A | 4B | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | |
| size | 1 | 1 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | ζ53 | ζ5 | ζ54 | ζ52 | ζ52 | ζ54 | ζ5 | ζ53 | linear of order 5 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | ζ54 | ζ53 | ζ52 | ζ5 | ζ5 | ζ52 | ζ53 | ζ54 | linear of order 5 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | ζ5 | ζ52 | ζ53 | ζ54 | ζ54 | ζ53 | ζ52 | ζ5 | linear of order 5 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | ζ52 | ζ54 | ζ5 | ζ53 | ζ53 | ζ5 | ζ54 | ζ52 | linear of order 5 |
| ρ6 | 4 | -4 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | symplectic faithful, Schur index 2 |
| ρ7 | 4 | -4 | 0 | 0 | 0 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | ζ5 | ζ52 | ζ53 | ζ54 | complex faithful |
| ρ8 | 4 | -4 | 0 | 0 | 0 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | ζ53 | ζ5 | ζ54 | ζ52 | complex faithful |
| ρ9 | 4 | -4 | 0 | 0 | 0 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | ζ52 | ζ54 | ζ5 | ζ53 | complex faithful |
| ρ10 | 4 | -4 | 0 | 0 | 0 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | ζ54 | ζ53 | ζ52 | ζ5 | complex faithful |
| ρ11 | 5 | 5 | 1 | 1 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊C5 |
| ρ12 | 5 | 5 | 1 | -3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊C5 |
| ρ13 | 5 | 5 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊C5 |
►On 32 points
Generators in S32
(1 11 2 30)(3 10 17 29)(4 31 13 12)(5 7 14 16)(6 27 15 21)(8 23 32 22)(9 26 28 20)(18 19 24 25)
(1 25)(2 19)(3 22)(4 28)(5 27)(6 7)(8 29)(9 13)(10 32)(11 24)(12 20)(14 21)(15 16)(17 23)(18 30)(26 31)
(1 6 2 15)(3 12 17 31)(4 29 13 10)(5 18 14 24)(7 19 16 25)(8 9 32 28)(11 27 30 21)(20 23 26 22)
(1 9 2 28)(3 5 17 14)(4 25 13 19)(6 8 15 32)(7 29 16 10)(11 26 30 20)(12 24 31 18)(21 22 27 23)
(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)
G:=sub<Sym(32)| (1,11,2,30)(3,10,17,29)(4,31,13,12)(5,7,14,16)(6,27,15,21)(8,23,32,22)(9,26,28,20)(18,19,24,25), (1,25)(2,19)(3,22)(4,28)(5,27)(6,7)(8,29)(9,13)(10,32)(11,24)(12,20)(14,21)(15,16)(17,23)(18,30)(26,31), (1,6,2,15)(3,12,17,31)(4,29,13,10)(5,18,14,24)(7,19,16,25)(8,9,32,28)(11,27,30,21)(20,23,26,22), (1,9,2,28)(3,5,17,14)(4,25,13,19)(6,8,15,32)(7,29,16,10)(11,26,30,20)(12,24,31,18)(21,22,27,23), (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)>;
G:=Group( (1,11,2,30)(3,10,17,29)(4,31,13,12)(5,7,14,16)(6,27,15,21)(8,23,32,22)(9,26,28,20)(18,19,24,25), (1,25)(2,19)(3,22)(4,28)(5,27)(6,7)(8,29)(9,13)(10,32)(11,24)(12,20)(14,21)(15,16)(17,23)(18,30)(26,31), (1,6,2,15)(3,12,17,31)(4,29,13,10)(5,18,14,24)(7,19,16,25)(8,9,32,28)(11,27,30,21)(20,23,26,22), (1,9,2,28)(3,5,17,14)(4,25,13,19)(6,8,15,32)(7,29,16,10)(11,26,30,20)(12,24,31,18)(21,22,27,23), (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) );
G=PermutationGroup([[(1,11,2,30),(3,10,17,29),(4,31,13,12),(5,7,14,16),(6,27,15,21),(8,23,32,22),(9,26,28,20),(18,19,24,25)], [(1,25),(2,19),(3,22),(4,28),(5,27),(6,7),(8,29),(9,13),(10,32),(11,24),(12,20),(14,21),(15,16),(17,23),(18,30),(26,31)], [(1,6,2,15),(3,12,17,31),(4,29,13,10),(5,18,14,24),(7,19,16,25),(8,9,32,28),(11,27,30,21),(20,23,26,22)], [(1,9,2,28),(3,5,17,14),(4,25,13,19),(6,8,15,32),(7,29,16,10),(11,26,30,20),(12,24,31,18),(21,22,27,23)], [(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)]])
2- 1+4⋊C5 is a maximal subgroup of
2- 1+4.D5 2- 1+4⋊D5 2- 1+4.C10
2- 1+4⋊C5 is a maximal quotient of C22.58C24⋊C5
Matrix representation of 2- 1+4⋊C5 ►in GL4(𝔽3) generated by
| 2 | 0 | 1 | 1 |
| 2 | 0 | 1 | 2 |
| 0 | 1 | 1 | 1 |
| 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 0 | 1 | 2 |
| 1 | 1 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 2 | 1 | 0 | 2 |
| 2 | 1 | 1 | 0 |
| 0 | 2 | 2 | 2 |
| 1 | 0 | 1 | 1 |
| 2 | 1 | 0 | 1 |
| 1 | 1 | 2 | 0 |
| 0 | 0 | 2 | 1 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 2 | 1 |
| 1 | 2 | 0 | 1 |
| 0 | 1 | 2 | 0 |
| 0 | 0 | 2 | 1 |
G:=sub<GL(4,GF(3))| [2,2,0,1,0,0,1,2,1,1,1,0,1,2,1,0],[0,2,1,1,0,0,1,0,0,1,0,0,1,2,1,0],[2,2,0,1,1,1,2,0,0,1,2,1,2,0,2,1],[2,1,0,0,1,1,0,0,0,2,2,1,1,0,1,1],[0,1,0,0,1,2,1,0,2,0,2,2,1,1,0,1] >;
2- 1+4⋊C5 in GAP, Magma, Sage, TeX
2_-^{1+4}\rtimes C_5 % in TeX
G:=Group("ES-(2,2):C5"); // GroupNames label
G:=SmallGroup(160,199);
// by ID
G=gap.SmallGroup(160,199);
# by ID
G:=PCGroup([6,-5,-2,2,2,2,-2,481,812,332,158,1323,489,255,117,2254,730]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=e^5=1,c^2=d^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^2*b*c*d,b*c=c*b,b*d=d*b,e*b*e^-1=a*b,d*c*d^-1=a^2*c,e*c*e^-1=a*b*c,e*d*e^-1=c*d>;
// generators/relations
Export
Subgroup lattice of 2- 1+4⋊C5 in TeX
Character table of 2- 1+4⋊C5 in TeX