metabelian, soluble, monomial, A-group
Aliases: (C3×C6).F5, (C3×C15)⋊4C8, C5⋊(C32⋊2C8), C32⋊2(C5⋊C8), C2.(C32⋊F5), (C3×C30).1C4, C10.(C32⋊C4), C3⋊Dic15.1C2, SmallGroup(360,57)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — (C3×C6).F5 |
Generators and relations for (C3×C6).F5
G = < a,b,c,d | a3=b6=c5=1, d4=b3, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, dbd-1=a-1b, dcd-1=c3 >
Character table of (C3×C6).F5
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5 | 6A | 6B | 8A | 8B | 8C | 8D | 10 | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | |
| size | 1 | 1 | 4 | 4 | 45 | 45 | 4 | 4 | 4 | 45 | 45 | 45 | 45 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 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 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 1 | -1 | 1 | 1 | i | -i | 1 | -1 | -1 | ζ8 | ζ83 | ζ85 | ζ87 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ6 | 1 | -1 | 1 | 1 | -i | i | 1 | -1 | -1 | ζ87 | ζ85 | ζ83 | ζ8 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ7 | 1 | -1 | 1 | 1 | -i | i | 1 | -1 | -1 | ζ83 | ζ8 | ζ87 | ζ85 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ8 | 1 | -1 | 1 | 1 | i | -i | 1 | -1 | -1 | ζ85 | ζ87 | ζ8 | ζ83 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ9 | 4 | 4 | 4 | 4 | 0 | 0 | -1 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ10 | 4 | 4 | -2 | 1 | 0 | 0 | 4 | 1 | -2 | 0 | 0 | 0 | 0 | 4 | -2 | -2 | -2 | 1 | 1 | 1 | 1 | -2 | 1 | 1 | -2 | -2 | -2 | -2 | 1 | 1 | orthogonal lifted from C32⋊C4 |
| ρ11 | 4 | 4 | 1 | -2 | 0 | 0 | 4 | -2 | 1 | 0 | 0 | 0 | 0 | 4 | 1 | 1 | 1 | -2 | -2 | -2 | -2 | 1 | -2 | -2 | 1 | 1 | 1 | 1 | -2 | -2 | orthogonal lifted from C32⋊C4 |
| ρ12 | 4 | 4 | -2 | 1 | 0 | 0 | -1 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | orthogonal lifted from C32⋊F5 |
| ρ13 | 4 | 4 | 1 | -2 | 0 | 0 | -1 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | orthogonal lifted from C32⋊F5 |
| ρ14 | 4 | 4 | -2 | 1 | 0 | 0 | -1 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | orthogonal lifted from C32⋊F5 |
| ρ15 | 4 | 4 | 1 | -2 | 0 | 0 | -1 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | orthogonal lifted from C32⋊F5 |
| ρ16 | 4 | 4 | 1 | -2 | 0 | 0 | -1 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | orthogonal lifted from C32⋊F5 |
| ρ17 | 4 | 4 | -2 | 1 | 0 | 0 | -1 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | orthogonal lifted from C32⋊F5 |
| ρ18 | 4 | 4 | -2 | 1 | 0 | 0 | -1 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | orthogonal lifted from C32⋊F5 |
| ρ19 | 4 | 4 | 1 | -2 | 0 | 0 | -1 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | orthogonal lifted from C32⋊F5 |
| ρ20 | 4 | -4 | -2 | 1 | 0 | 0 | 4 | -1 | 2 | 0 | 0 | 0 | 0 | -4 | -2 | -2 | -2 | 1 | 1 | 1 | 1 | -2 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | symplectic lifted from C32⋊2C8, Schur index 2 |
| ρ21 | 4 | -4 | -2 | 1 | 0 | 0 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | -ζ32ζ54+ζ32ζ5+ζ54+2ζ5+1 | -ζ3ζ53+ζ3ζ52+ζ53+2ζ52+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52 | 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52 | 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5 | ζ3ζ53-ζ3ζ52+2ζ53+ζ52+1 | -ζ3ζ54+ζ3ζ5+ζ54+2ζ5+1 | symplectic faithful, Schur index 2 |
| ρ22 | 4 | -4 | 1 | -2 | 0 | 0 | 4 | 2 | -1 | 0 | 0 | 0 | 0 | -4 | 1 | 1 | 1 | -2 | -2 | -2 | -2 | 1 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | 2 | symplectic lifted from C32⋊2C8, Schur index 2 |
| ρ23 | 4 | -4 | 1 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52 | -ζ3ζ54+ζ3ζ5+ζ54+2ζ5+1 | -ζ32ζ54+ζ32ζ5+ζ54+2ζ5+1 | -ζ3ζ53+ζ3ζ52+ζ53+2ζ52+1 | ζ3ζ53-ζ3ζ52+2ζ53+ζ52+1 | 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5 | symplectic faithful, Schur index 2 |
| ρ24 | 4 | -4 | 1 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5 | 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52 | -ζ32ζ54+ζ32ζ5+ζ54+2ζ5+1 | -ζ3ζ54+ζ3ζ5+ζ54+2ζ5+1 | ζ3ζ53-ζ3ζ52+2ζ53+ζ52+1 | -ζ3ζ53+ζ3ζ52+ζ53+2ζ52+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52 | 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53 | symplectic faithful, Schur index 2 |
| ρ25 | 4 | -4 | 1 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52 | 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53 | -ζ3ζ53+ζ3ζ52+ζ53+2ζ52+1 | ζ3ζ53-ζ3ζ52+2ζ53+ζ52+1 | -ζ32ζ54+ζ32ζ5+ζ54+2ζ5+1 | -ζ3ζ54+ζ3ζ5+ζ54+2ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52 | symplectic faithful, Schur index 2 |
| ρ26 | 4 | -4 | -2 | 1 | 0 | 0 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | -ζ3ζ54+ζ3ζ5+ζ54+2ζ5+1 | ζ3ζ53-ζ3ζ52+2ζ53+ζ52+1 | 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5 | 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53 | -ζ3ζ53+ζ3ζ52+ζ53+2ζ52+1 | -ζ32ζ54+ζ32ζ5+ζ54+2ζ5+1 | symplectic faithful, Schur index 2 |
| ρ27 | 4 | -4 | -2 | 1 | 0 | 0 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | -ζ3ζ53+ζ3ζ52+ζ53+2ζ52+1 | -ζ3ζ54+ζ3ζ5+ζ54+2ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5 | 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52 | 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52 | -ζ32ζ54+ζ32ζ5+ζ54+2ζ5+1 | ζ3ζ53-ζ3ζ52+2ζ53+ζ52+1 | symplectic faithful, Schur index 2 |
| ρ28 | 4 | -4 | -2 | 1 | 0 | 0 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | ζ3ζ53-ζ3ζ52+2ζ53+ζ52+1 | -ζ32ζ54+ζ32ζ5+ζ54+2ζ5+1 | 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5 | 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52 | -ζ3ζ54+ζ3ζ5+ζ54+2ζ5+1 | -ζ3ζ53+ζ3ζ52+ζ53+2ζ52+1 | symplectic faithful, Schur index 2 |
| ρ29 | 4 | -4 | 4 | 4 | 0 | 0 | -1 | -4 | -4 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ30 | 4 | -4 | 1 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5 | ζ3ζ53-ζ3ζ52+2ζ53+ζ52+1 | -ζ3ζ53+ζ3ζ52+ζ53+2ζ52+1 | -ζ3ζ54+ζ3ζ5+ζ54+2ζ5+1 | -ζ32ζ54+ζ32ζ5+ζ54+2ζ5+1 | 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53 | 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52 | symplectic faithful, Schur index 2 |
►On 120 points
Generators in S120
(2 51 12)(4 14 53)(6 55 16)(8 10 49)(17 29 106)(19 108 31)(21 25 110)(23 112 27)(34 114 80)(36 74 116)(38 118 76)(40 78 120)(42 104 59)(44 61 98)(46 100 63)(48 57 102)(65 95 87)(67 81 89)(69 91 83)(71 85 93)
(1 15 50 5 11 54)(2 16 51 6 12 55)(3 56 13 7 52 9)(4 49 14 8 53 10)(17 110 29 21 106 25)(18 26 107 22 30 111)(19 27 108 23 31 112)(20 105 32 24 109 28)(33 75 113 37 79 117)(34 76 114 38 80 118)(35 119 73 39 115 77)(36 120 74 40 116 78)(41 62 103 45 58 99)(42 63 104 46 59 100)(43 101 60 47 97 64)(44 102 61 48 98 57)(65 83 95 69 87 91)(66 92 88 70 96 84)(67 93 81 71 89 85)(68 86 90 72 82 94)
(1 99 88 113 28)(2 114 100 29 81)(3 30 115 82 101)(4 83 31 102 116)(5 103 84 117 32)(6 118 104 25 85)(7 26 119 86 97)(8 87 27 98 120)(9 22 39 72 43)(10 65 23 44 40)(11 45 66 33 24)(12 34 46 17 67)(13 18 35 68 47)(14 69 19 48 36)(15 41 70 37 20)(16 38 42 21 71)(49 95 112 61 78)(50 62 96 79 105)(51 80 63 106 89)(52 107 73 90 64)(53 91 108 57 74)(54 58 92 75 109)(55 76 59 110 93)(56 111 77 94 60)
(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 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)
G:=sub<Sym(120)| (2,51,12)(4,14,53)(6,55,16)(8,10,49)(17,29,106)(19,108,31)(21,25,110)(23,112,27)(34,114,80)(36,74,116)(38,118,76)(40,78,120)(42,104,59)(44,61,98)(46,100,63)(48,57,102)(65,95,87)(67,81,89)(69,91,83)(71,85,93), (1,15,50,5,11,54)(2,16,51,6,12,55)(3,56,13,7,52,9)(4,49,14,8,53,10)(17,110,29,21,106,25)(18,26,107,22,30,111)(19,27,108,23,31,112)(20,105,32,24,109,28)(33,75,113,37,79,117)(34,76,114,38,80,118)(35,119,73,39,115,77)(36,120,74,40,116,78)(41,62,103,45,58,99)(42,63,104,46,59,100)(43,101,60,47,97,64)(44,102,61,48,98,57)(65,83,95,69,87,91)(66,92,88,70,96,84)(67,93,81,71,89,85)(68,86,90,72,82,94), (1,99,88,113,28)(2,114,100,29,81)(3,30,115,82,101)(4,83,31,102,116)(5,103,84,117,32)(6,118,104,25,85)(7,26,119,86,97)(8,87,27,98,120)(9,22,39,72,43)(10,65,23,44,40)(11,45,66,33,24)(12,34,46,17,67)(13,18,35,68,47)(14,69,19,48,36)(15,41,70,37,20)(16,38,42,21,71)(49,95,112,61,78)(50,62,96,79,105)(51,80,63,106,89)(52,107,73,90,64)(53,91,108,57,74)(54,58,92,75,109)(55,76,59,110,93)(56,111,77,94,60), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)>;
G:=Group( (2,51,12)(4,14,53)(6,55,16)(8,10,49)(17,29,106)(19,108,31)(21,25,110)(23,112,27)(34,114,80)(36,74,116)(38,118,76)(40,78,120)(42,104,59)(44,61,98)(46,100,63)(48,57,102)(65,95,87)(67,81,89)(69,91,83)(71,85,93), (1,15,50,5,11,54)(2,16,51,6,12,55)(3,56,13,7,52,9)(4,49,14,8,53,10)(17,110,29,21,106,25)(18,26,107,22,30,111)(19,27,108,23,31,112)(20,105,32,24,109,28)(33,75,113,37,79,117)(34,76,114,38,80,118)(35,119,73,39,115,77)(36,120,74,40,116,78)(41,62,103,45,58,99)(42,63,104,46,59,100)(43,101,60,47,97,64)(44,102,61,48,98,57)(65,83,95,69,87,91)(66,92,88,70,96,84)(67,93,81,71,89,85)(68,86,90,72,82,94), (1,99,88,113,28)(2,114,100,29,81)(3,30,115,82,101)(4,83,31,102,116)(5,103,84,117,32)(6,118,104,25,85)(7,26,119,86,97)(8,87,27,98,120)(9,22,39,72,43)(10,65,23,44,40)(11,45,66,33,24)(12,34,46,17,67)(13,18,35,68,47)(14,69,19,48,36)(15,41,70,37,20)(16,38,42,21,71)(49,95,112,61,78)(50,62,96,79,105)(51,80,63,106,89)(52,107,73,90,64)(53,91,108,57,74)(54,58,92,75,109)(55,76,59,110,93)(56,111,77,94,60), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120) );
G=PermutationGroup([[(2,51,12),(4,14,53),(6,55,16),(8,10,49),(17,29,106),(19,108,31),(21,25,110),(23,112,27),(34,114,80),(36,74,116),(38,118,76),(40,78,120),(42,104,59),(44,61,98),(46,100,63),(48,57,102),(65,95,87),(67,81,89),(69,91,83),(71,85,93)], [(1,15,50,5,11,54),(2,16,51,6,12,55),(3,56,13,7,52,9),(4,49,14,8,53,10),(17,110,29,21,106,25),(18,26,107,22,30,111),(19,27,108,23,31,112),(20,105,32,24,109,28),(33,75,113,37,79,117),(34,76,114,38,80,118),(35,119,73,39,115,77),(36,120,74,40,116,78),(41,62,103,45,58,99),(42,63,104,46,59,100),(43,101,60,47,97,64),(44,102,61,48,98,57),(65,83,95,69,87,91),(66,92,88,70,96,84),(67,93,81,71,89,85),(68,86,90,72,82,94)], [(1,99,88,113,28),(2,114,100,29,81),(3,30,115,82,101),(4,83,31,102,116),(5,103,84,117,32),(6,118,104,25,85),(7,26,119,86,97),(8,87,27,98,120),(9,22,39,72,43),(10,65,23,44,40),(11,45,66,33,24),(12,34,46,17,67),(13,18,35,68,47),(14,69,19,48,36),(15,41,70,37,20),(16,38,42,21,71),(49,95,112,61,78),(50,62,96,79,105),(51,80,63,106,89),(52,107,73,90,64),(53,91,108,57,74),(54,58,92,75,109),(55,76,59,110,93),(56,111,77,94,60)], [(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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120)]])
Matrix representation of (C3×C6).F5 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 147 | 84 |
| 0 | 0 | 157 | 93 |
| 94 | 157 | 0 | 0 |
| 84 | 148 | 0 | 0 |
| 0 | 0 | 148 | 84 |
| 0 | 0 | 157 | 94 |
| 51 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 190 | 190 |
| 0 | 0 | 51 | 240 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 183 | 113 | 0 | 0 |
| 47 | 58 | 0 | 0 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,147,157,0,0,84,93],[94,84,0,0,157,148,0,0,0,0,148,157,0,0,84,94],[51,1,0,0,240,0,0,0,0,0,190,51,0,0,190,240],[0,0,183,47,0,0,113,58,1,0,0,0,0,1,0,0] >;
(C3×C6).F5 in GAP, Magma, Sage, TeX
(C_3\times C_6).F_5
% in TeX
G:=Group("(C3xC6).F5"); // GroupNames label
G:=SmallGroup(360,57);
// by ID
G=gap.SmallGroup(360,57);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,3,-5,12,31,1347,201,1924,730,5189,5195]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^5=1,d^4=b^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^3>;
// generators/relations
Export
Subgroup lattice of (C3×C6).F5 in TeX
Character table of (C3×C6).F5 in TeX