metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C11⋊F5, C55⋊1C4, C5⋊Dic11, D5.D11, (D5×C11).1C2, SmallGroup(220,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C55 — C11⋊F5 |
Generators and relations for C11⋊F5
G = < a,b,c | a11=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b3 >
Character table of C11⋊F5
| class | 1 | 2 | 4A | 4B | 5 | 11A | 11B | 11C | 11D | 11E | 22A | 22B | 22C | 22D | 22E | 55A | 55B | 55C | 55D | 55E | 55F | 55G | 55H | 55I | 55J | |
| size | 1 | 5 | 55 | 55 | 4 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 2 | 2 | 0 | 0 | 2 | ζ118+ζ113 | ζ1110+ζ11 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ1110+ζ11 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ118+ζ113 | ζ119+ζ112 | ζ119+ζ112 | ζ1110+ζ11 | ζ1110+ζ11 | ζ118+ζ113 | ζ116+ζ115 | ζ117+ζ114 | ζ117+ζ114 | ζ116+ζ115 | orthogonal lifted from D11 |
| ρ6 | 2 | 2 | 0 | 0 | 2 | ζ1110+ζ11 | ζ117+ζ114 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ117+ζ114 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ1110+ζ11 | ζ118+ζ113 | ζ118+ζ113 | ζ117+ζ114 | ζ117+ζ114 | ζ1110+ζ11 | ζ119+ζ112 | ζ116+ζ115 | ζ116+ζ115 | ζ119+ζ112 | orthogonal lifted from D11 |
| ρ7 | 2 | 2 | 0 | 0 | 2 | ζ117+ζ114 | ζ116+ζ115 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ116+ζ115 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ117+ζ114 | ζ1110+ζ11 | ζ1110+ζ11 | ζ116+ζ115 | ζ116+ζ115 | ζ117+ζ114 | ζ118+ζ113 | ζ119+ζ112 | ζ119+ζ112 | ζ118+ζ113 | orthogonal lifted from D11 |
| ρ8 | 2 | 2 | 0 | 0 | 2 | ζ119+ζ112 | ζ118+ζ113 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ118+ζ113 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ119+ζ112 | ζ116+ζ115 | ζ116+ζ115 | ζ118+ζ113 | ζ118+ζ113 | ζ119+ζ112 | ζ117+ζ114 | ζ1110+ζ11 | ζ1110+ζ11 | ζ117+ζ114 | orthogonal lifted from D11 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | ζ116+ζ115 | ζ119+ζ112 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ119+ζ112 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ116+ζ115 | ζ117+ζ114 | ζ117+ζ114 | ζ119+ζ112 | ζ119+ζ112 | ζ116+ζ115 | ζ1110+ζ11 | ζ118+ζ113 | ζ118+ζ113 | ζ1110+ζ11 | orthogonal lifted from D11 |
| ρ10 | 2 | -2 | 0 | 0 | 2 | ζ1110+ζ11 | ζ117+ζ114 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | -ζ117-ζ114 | -ζ118-ζ113 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ1110-ζ11 | ζ1110+ζ11 | ζ118+ζ113 | ζ118+ζ113 | ζ117+ζ114 | ζ117+ζ114 | ζ1110+ζ11 | ζ119+ζ112 | ζ116+ζ115 | ζ116+ζ115 | ζ119+ζ112 | symplectic lifted from Dic11, Schur index 2 |
| ρ11 | 2 | -2 | 0 | 0 | 2 | ζ118+ζ113 | ζ1110+ζ11 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | -ζ1110-ζ11 | -ζ119-ζ112 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ118-ζ113 | ζ118+ζ113 | ζ119+ζ112 | ζ119+ζ112 | ζ1110+ζ11 | ζ1110+ζ11 | ζ118+ζ113 | ζ116+ζ115 | ζ117+ζ114 | ζ117+ζ114 | ζ116+ζ115 | symplectic lifted from Dic11, Schur index 2 |
| ρ12 | 2 | -2 | 0 | 0 | 2 | ζ117+ζ114 | ζ116+ζ115 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | -ζ116-ζ115 | -ζ1110-ζ11 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ117-ζ114 | ζ117+ζ114 | ζ1110+ζ11 | ζ1110+ζ11 | ζ116+ζ115 | ζ116+ζ115 | ζ117+ζ114 | ζ118+ζ113 | ζ119+ζ112 | ζ119+ζ112 | ζ118+ζ113 | symplectic lifted from Dic11, Schur index 2 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | ζ119+ζ112 | ζ118+ζ113 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | -ζ118-ζ113 | -ζ116-ζ115 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ119-ζ112 | ζ119+ζ112 | ζ116+ζ115 | ζ116+ζ115 | ζ118+ζ113 | ζ118+ζ113 | ζ119+ζ112 | ζ117+ζ114 | ζ1110+ζ11 | ζ1110+ζ11 | ζ117+ζ114 | symplectic lifted from Dic11, Schur index 2 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | ζ116+ζ115 | ζ119+ζ112 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | -ζ119-ζ112 | -ζ117-ζ114 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ116-ζ115 | ζ116+ζ115 | ζ117+ζ114 | ζ117+ζ114 | ζ119+ζ112 | ζ119+ζ112 | ζ116+ζ115 | ζ1110+ζ11 | ζ118+ζ113 | ζ118+ζ113 | ζ1110+ζ11 | symplectic lifted from Dic11, Schur index 2 |
| ρ15 | 4 | 0 | 0 | 0 | -1 | 4 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ16 | 4 | 0 | 0 | 0 | -1 | 2ζ119+2ζ112 | 2ζ118+2ζ113 | 2ζ116+2ζ115 | 2ζ1110+2ζ11 | 2ζ117+2ζ114 | 0 | 0 | 0 | 0 | 0 | -ζ53ζ119+ζ53ζ112-ζ52ζ119+ζ52ζ112-ζ119 | ζ54ζ116-ζ54ζ115+ζ5ζ116-ζ5ζ115-ζ115 | -ζ54ζ116+ζ54ζ115-ζ5ζ116+ζ5ζ115-ζ116 | ζ53ζ118-ζ53ζ113+ζ52ζ118-ζ52ζ113-ζ113 | -ζ53ζ118+ζ53ζ113-ζ52ζ118+ζ52ζ113-ζ118 | -ζ54ζ119+ζ54ζ112-ζ5ζ119+ζ5ζ112-ζ119 | -ζ53ζ117+ζ53ζ114-ζ52ζ117+ζ52ζ114-ζ117 | ζ53ζ1110-ζ53ζ11+ζ52ζ1110-ζ52ζ11-ζ11 | -ζ53ζ1110+ζ53ζ11-ζ52ζ1110+ζ52ζ11-ζ1110 | -ζ54ζ117+ζ54ζ114-ζ5ζ117+ζ5ζ114-ζ117 | complex faithful |
| ρ17 | 4 | 0 | 0 | 0 | -1 | 2ζ1110+2ζ11 | 2ζ117+2ζ114 | 2ζ118+2ζ113 | 2ζ116+2ζ115 | 2ζ119+2ζ112 | 0 | 0 | 0 | 0 | 0 | -ζ53ζ1110+ζ53ζ11-ζ52ζ1110+ζ52ζ11-ζ1110 | ζ53ζ118-ζ53ζ113+ζ52ζ118-ζ52ζ113-ζ113 | -ζ53ζ118+ζ53ζ113-ζ52ζ118+ζ52ζ113-ζ118 | -ζ53ζ117+ζ53ζ114-ζ52ζ117+ζ52ζ114-ζ117 | -ζ54ζ117+ζ54ζ114-ζ5ζ117+ζ5ζ114-ζ117 | ζ53ζ1110-ζ53ζ11+ζ52ζ1110-ζ52ζ11-ζ11 | -ζ53ζ119+ζ53ζ112-ζ52ζ119+ζ52ζ112-ζ119 | ζ54ζ116-ζ54ζ115+ζ5ζ116-ζ5ζ115-ζ115 | -ζ54ζ116+ζ54ζ115-ζ5ζ116+ζ5ζ115-ζ116 | -ζ54ζ119+ζ54ζ112-ζ5ζ119+ζ5ζ112-ζ119 | complex faithful |
| ρ18 | 4 | 0 | 0 | 0 | -1 | 2ζ117+2ζ114 | 2ζ116+2ζ115 | 2ζ1110+2ζ11 | 2ζ119+2ζ112 | 2ζ118+2ζ113 | 0 | 0 | 0 | 0 | 0 | -ζ54ζ117+ζ54ζ114-ζ5ζ117+ζ5ζ114-ζ117 | -ζ53ζ1110+ζ53ζ11-ζ52ζ1110+ζ52ζ11-ζ1110 | ζ53ζ1110-ζ53ζ11+ζ52ζ1110-ζ52ζ11-ζ11 | -ζ54ζ116+ζ54ζ115-ζ5ζ116+ζ5ζ115-ζ116 | ζ54ζ116-ζ54ζ115+ζ5ζ116-ζ5ζ115-ζ115 | -ζ53ζ117+ζ53ζ114-ζ52ζ117+ζ52ζ114-ζ117 | -ζ53ζ118+ζ53ζ113-ζ52ζ118+ζ52ζ113-ζ118 | -ζ53ζ119+ζ53ζ112-ζ52ζ119+ζ52ζ112-ζ119 | -ζ54ζ119+ζ54ζ112-ζ5ζ119+ζ5ζ112-ζ119 | ζ53ζ118-ζ53ζ113+ζ52ζ118-ζ52ζ113-ζ113 | complex faithful |
| ρ19 | 4 | 0 | 0 | 0 | -1 | 2ζ1110+2ζ11 | 2ζ117+2ζ114 | 2ζ118+2ζ113 | 2ζ116+2ζ115 | 2ζ119+2ζ112 | 0 | 0 | 0 | 0 | 0 | ζ53ζ1110-ζ53ζ11+ζ52ζ1110-ζ52ζ11-ζ11 | -ζ53ζ118+ζ53ζ113-ζ52ζ118+ζ52ζ113-ζ118 | ζ53ζ118-ζ53ζ113+ζ52ζ118-ζ52ζ113-ζ113 | -ζ54ζ117+ζ54ζ114-ζ5ζ117+ζ5ζ114-ζ117 | -ζ53ζ117+ζ53ζ114-ζ52ζ117+ζ52ζ114-ζ117 | -ζ53ζ1110+ζ53ζ11-ζ52ζ1110+ζ52ζ11-ζ1110 | -ζ54ζ119+ζ54ζ112-ζ5ζ119+ζ5ζ112-ζ119 | -ζ54ζ116+ζ54ζ115-ζ5ζ116+ζ5ζ115-ζ116 | ζ54ζ116-ζ54ζ115+ζ5ζ116-ζ5ζ115-ζ115 | -ζ53ζ119+ζ53ζ112-ζ52ζ119+ζ52ζ112-ζ119 | complex faithful |
| ρ20 | 4 | 0 | 0 | 0 | -1 | 2ζ118+2ζ113 | 2ζ1110+2ζ11 | 2ζ119+2ζ112 | 2ζ117+2ζ114 | 2ζ116+2ζ115 | 0 | 0 | 0 | 0 | 0 | ζ53ζ118-ζ53ζ113+ζ52ζ118-ζ52ζ113-ζ113 | -ζ54ζ119+ζ54ζ112-ζ5ζ119+ζ5ζ112-ζ119 | -ζ53ζ119+ζ53ζ112-ζ52ζ119+ζ52ζ112-ζ119 | ζ53ζ1110-ζ53ζ11+ζ52ζ1110-ζ52ζ11-ζ11 | -ζ53ζ1110+ζ53ζ11-ζ52ζ1110+ζ52ζ11-ζ1110 | -ζ53ζ118+ζ53ζ113-ζ52ζ118+ζ52ζ113-ζ118 | ζ54ζ116-ζ54ζ115+ζ5ζ116-ζ5ζ115-ζ115 | -ζ54ζ117+ζ54ζ114-ζ5ζ117+ζ5ζ114-ζ117 | -ζ53ζ117+ζ53ζ114-ζ52ζ117+ζ52ζ114-ζ117 | -ζ54ζ116+ζ54ζ115-ζ5ζ116+ζ5ζ115-ζ116 | complex faithful |
| ρ21 | 4 | 0 | 0 | 0 | -1 | 2ζ117+2ζ114 | 2ζ116+2ζ115 | 2ζ1110+2ζ11 | 2ζ119+2ζ112 | 2ζ118+2ζ113 | 0 | 0 | 0 | 0 | 0 | -ζ53ζ117+ζ53ζ114-ζ52ζ117+ζ52ζ114-ζ117 | ζ53ζ1110-ζ53ζ11+ζ52ζ1110-ζ52ζ11-ζ11 | -ζ53ζ1110+ζ53ζ11-ζ52ζ1110+ζ52ζ11-ζ1110 | ζ54ζ116-ζ54ζ115+ζ5ζ116-ζ5ζ115-ζ115 | -ζ54ζ116+ζ54ζ115-ζ5ζ116+ζ5ζ115-ζ116 | -ζ54ζ117+ζ54ζ114-ζ5ζ117+ζ5ζ114-ζ117 | ζ53ζ118-ζ53ζ113+ζ52ζ118-ζ52ζ113-ζ113 | -ζ54ζ119+ζ54ζ112-ζ5ζ119+ζ5ζ112-ζ119 | -ζ53ζ119+ζ53ζ112-ζ52ζ119+ζ52ζ112-ζ119 | -ζ53ζ118+ζ53ζ113-ζ52ζ118+ζ52ζ113-ζ118 | complex faithful |
| ρ22 | 4 | 0 | 0 | 0 | -1 | 2ζ116+2ζ115 | 2ζ119+2ζ112 | 2ζ117+2ζ114 | 2ζ118+2ζ113 | 2ζ1110+2ζ11 | 0 | 0 | 0 | 0 | 0 | -ζ54ζ116+ζ54ζ115-ζ5ζ116+ζ5ζ115-ζ116 | -ζ53ζ117+ζ53ζ114-ζ52ζ117+ζ52ζ114-ζ117 | -ζ54ζ117+ζ54ζ114-ζ5ζ117+ζ5ζ114-ζ117 | -ζ53ζ119+ζ53ζ112-ζ52ζ119+ζ52ζ112-ζ119 | -ζ54ζ119+ζ54ζ112-ζ5ζ119+ζ5ζ112-ζ119 | ζ54ζ116-ζ54ζ115+ζ5ζ116-ζ5ζ115-ζ115 | -ζ53ζ1110+ζ53ζ11-ζ52ζ1110+ζ52ζ11-ζ1110 | ζ53ζ118-ζ53ζ113+ζ52ζ118-ζ52ζ113-ζ113 | -ζ53ζ118+ζ53ζ113-ζ52ζ118+ζ52ζ113-ζ118 | ζ53ζ1110-ζ53ζ11+ζ52ζ1110-ζ52ζ11-ζ11 | complex faithful |
| ρ23 | 4 | 0 | 0 | 0 | -1 | 2ζ116+2ζ115 | 2ζ119+2ζ112 | 2ζ117+2ζ114 | 2ζ118+2ζ113 | 2ζ1110+2ζ11 | 0 | 0 | 0 | 0 | 0 | ζ54ζ116-ζ54ζ115+ζ5ζ116-ζ5ζ115-ζ115 | -ζ54ζ117+ζ54ζ114-ζ5ζ117+ζ5ζ114-ζ117 | -ζ53ζ117+ζ53ζ114-ζ52ζ117+ζ52ζ114-ζ117 | -ζ54ζ119+ζ54ζ112-ζ5ζ119+ζ5ζ112-ζ119 | -ζ53ζ119+ζ53ζ112-ζ52ζ119+ζ52ζ112-ζ119 | -ζ54ζ116+ζ54ζ115-ζ5ζ116+ζ5ζ115-ζ116 | ζ53ζ1110-ζ53ζ11+ζ52ζ1110-ζ52ζ11-ζ11 | -ζ53ζ118+ζ53ζ113-ζ52ζ118+ζ52ζ113-ζ118 | ζ53ζ118-ζ53ζ113+ζ52ζ118-ζ52ζ113-ζ113 | -ζ53ζ1110+ζ53ζ11-ζ52ζ1110+ζ52ζ11-ζ1110 | complex faithful |
| ρ24 | 4 | 0 | 0 | 0 | -1 | 2ζ119+2ζ112 | 2ζ118+2ζ113 | 2ζ116+2ζ115 | 2ζ1110+2ζ11 | 2ζ117+2ζ114 | 0 | 0 | 0 | 0 | 0 | -ζ54ζ119+ζ54ζ112-ζ5ζ119+ζ5ζ112-ζ119 | -ζ54ζ116+ζ54ζ115-ζ5ζ116+ζ5ζ115-ζ116 | ζ54ζ116-ζ54ζ115+ζ5ζ116-ζ5ζ115-ζ115 | -ζ53ζ118+ζ53ζ113-ζ52ζ118+ζ52ζ113-ζ118 | ζ53ζ118-ζ53ζ113+ζ52ζ118-ζ52ζ113-ζ113 | -ζ53ζ119+ζ53ζ112-ζ52ζ119+ζ52ζ112-ζ119 | -ζ54ζ117+ζ54ζ114-ζ5ζ117+ζ5ζ114-ζ117 | -ζ53ζ1110+ζ53ζ11-ζ52ζ1110+ζ52ζ11-ζ1110 | ζ53ζ1110-ζ53ζ11+ζ52ζ1110-ζ52ζ11-ζ11 | -ζ53ζ117+ζ53ζ114-ζ52ζ117+ζ52ζ114-ζ117 | complex faithful |
| ρ25 | 4 | 0 | 0 | 0 | -1 | 2ζ118+2ζ113 | 2ζ1110+2ζ11 | 2ζ119+2ζ112 | 2ζ117+2ζ114 | 2ζ116+2ζ115 | 0 | 0 | 0 | 0 | 0 | -ζ53ζ118+ζ53ζ113-ζ52ζ118+ζ52ζ113-ζ118 | -ζ53ζ119+ζ53ζ112-ζ52ζ119+ζ52ζ112-ζ119 | -ζ54ζ119+ζ54ζ112-ζ5ζ119+ζ5ζ112-ζ119 | -ζ53ζ1110+ζ53ζ11-ζ52ζ1110+ζ52ζ11-ζ1110 | ζ53ζ1110-ζ53ζ11+ζ52ζ1110-ζ52ζ11-ζ11 | ζ53ζ118-ζ53ζ113+ζ52ζ118-ζ52ζ113-ζ113 | -ζ54ζ116+ζ54ζ115-ζ5ζ116+ζ5ζ115-ζ116 | -ζ53ζ117+ζ53ζ114-ζ52ζ117+ζ52ζ114-ζ117 | -ζ54ζ117+ζ54ζ114-ζ5ζ117+ζ5ζ114-ζ117 | ζ54ζ116-ζ54ζ115+ζ5ζ116-ζ5ζ115-ζ115 | complex faithful |
►On 55 points
Generators in S55
(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 31 32 33)(34 35 36 37 38 39 40 41 42 43 44)(45 46 47 48 49 50 51 52 53 54 55)
(1 21 32 43 54)(2 22 33 44 55)(3 12 23 34 45)(4 13 24 35 46)(5 14 25 36 47)(6 15 26 37 48)(7 16 27 38 49)(8 17 28 39 50)(9 18 29 40 51)(10 19 30 41 52)(11 20 31 42 53)
(2 11)(3 10)(4 9)(5 8)(6 7)(12 30 45 41)(13 29 46 40)(14 28 47 39)(15 27 48 38)(16 26 49 37)(17 25 50 36)(18 24 51 35)(19 23 52 34)(20 33 53 44)(21 32 54 43)(22 31 55 42)
G:=sub<Sym(55)| (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33)(34,35,36,37,38,39,40,41,42,43,44)(45,46,47,48,49,50,51,52,53,54,55), (1,21,32,43,54)(2,22,33,44,55)(3,12,23,34,45)(4,13,24,35,46)(5,14,25,36,47)(6,15,26,37,48)(7,16,27,38,49)(8,17,28,39,50)(9,18,29,40,51)(10,19,30,41,52)(11,20,31,42,53), (2,11)(3,10)(4,9)(5,8)(6,7)(12,30,45,41)(13,29,46,40)(14,28,47,39)(15,27,48,38)(16,26,49,37)(17,25,50,36)(18,24,51,35)(19,23,52,34)(20,33,53,44)(21,32,54,43)(22,31,55,42)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33)(34,35,36,37,38,39,40,41,42,43,44)(45,46,47,48,49,50,51,52,53,54,55), (1,21,32,43,54)(2,22,33,44,55)(3,12,23,34,45)(4,13,24,35,46)(5,14,25,36,47)(6,15,26,37,48)(7,16,27,38,49)(8,17,28,39,50)(9,18,29,40,51)(10,19,30,41,52)(11,20,31,42,53), (2,11)(3,10)(4,9)(5,8)(6,7)(12,30,45,41)(13,29,46,40)(14,28,47,39)(15,27,48,38)(16,26,49,37)(17,25,50,36)(18,24,51,35)(19,23,52,34)(20,33,53,44)(21,32,54,43)(22,31,55,42) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22),(23,24,25,26,27,28,29,30,31,32,33),(34,35,36,37,38,39,40,41,42,43,44),(45,46,47,48,49,50,51,52,53,54,55)], [(1,21,32,43,54),(2,22,33,44,55),(3,12,23,34,45),(4,13,24,35,46),(5,14,25,36,47),(6,15,26,37,48),(7,16,27,38,49),(8,17,28,39,50),(9,18,29,40,51),(10,19,30,41,52),(11,20,31,42,53)], [(2,11),(3,10),(4,9),(5,8),(6,7),(12,30,45,41),(13,29,46,40),(14,28,47,39),(15,27,48,38),(16,26,49,37),(17,25,50,36),(18,24,51,35),(19,23,52,34),(20,33,53,44),(21,32,54,43),(22,31,55,42)]])
C11⋊F5 is a maximal subgroup of
F5×D11
C11⋊F5 is a maximal quotient of C55⋊C8
Matrix representation of C11⋊F5 ►in GL4(𝔽661) generated by
| 0 | 1 | 0 | 0 |
| 660 | 486 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 660 | 486 |
| 524 | 633 | 660 | 0 |
| 28 | 136 | 0 | 660 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 486 | 660 | 0 | 0 |
| 137 | 28 | 137 | 28 |
| 510 | 524 | 510 | 524 |
G:=sub<GL(4,GF(661))| [0,660,0,0,1,486,0,0,0,0,0,660,0,0,1,486],[524,28,1,0,633,136,0,1,660,0,0,0,0,660,0,0],[1,486,137,510,0,660,28,524,0,0,137,510,0,0,28,524] >;
C11⋊F5 in GAP, Magma, Sage, TeX
C_{11}\rtimes F_5 % in TeX
G:=Group("C11:F5"); // GroupNames label
G:=SmallGroup(220,10);
// by ID
G=gap.SmallGroup(220,10);
# by ID
G:=PCGroup([4,-2,-2,-5,-11,8,146,102,3203]);
// Polycyclic
G:=Group<a,b,c|a^11=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C11⋊F5 in TeX
Character table of C11⋊F5 in TeX