direct product, metabelian, soluble, monomial, A-group
Aliases: C2×C32⋊F5, (C3×C6)⋊F5, (C3×C30)⋊1C4, C3⋊D15⋊4C4, C10⋊(C32⋊C4), C32⋊4(C2×F5), C3⋊D15.2C22, C5⋊2(C2×C32⋊C4), (C3×C15)⋊7(C2×C4), (C2×C3⋊D15).1C2, SmallGroup(360,150)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — C2×C32⋊F5 |
Generators and relations for C2×C32⋊F5
G = < a,b,c,d,e | a2=b3=c3=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, ece-1=bc=cb, bd=db, ebe-1=b-1c, cd=dc, ede-1=d3 >
45C2
45C2
2C3
2C3
45C4
45C22
45C4
2C6
2C6
30S3
30S3
30S3
30S3
9D5
9D5
2C15
2C15
45C2×C4
30D6
30D6
9D10
9F5
9F5
2C30
2C30
6D15
6D15
6D15
6D15
6D30
6D30
Character table of C2×C32⋊F5
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 5 | 6A | 6B | 10 | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | |
| size | 1 | 1 | 45 | 45 | 4 | 4 | 45 | 45 | 45 | 45 | 4 | 4 | 4 | 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 | -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 |
| ρ4 | 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 |
| ρ5 | 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 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 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 | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 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 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 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 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 4 | 4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 4 | -2 | 1 | 4 | 1 | -2 | -2 | -2 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | -2 | -2 | -2 | 1 | orthogonal lifted from C32⋊C4 |
| ρ10 | 4 | 4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | 4 | 4 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ11 | 4 | 4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 4 | 1 | -2 | 4 | -2 | 1 | 1 | 1 | 1 | -2 | -2 | -2 | -2 | -2 | -2 | 1 | 1 | 1 | 1 | -2 | orthogonal lifted from C32⋊C4 |
| ρ12 | 4 | -4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | -4 | -4 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ13 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 4 | 2 | -1 | -4 | 1 | -2 | -2 | -2 | -2 | 1 | 1 | 1 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | orthogonal lifted from C2×C32⋊C4 |
| ρ14 | 4 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 4 | -1 | 2 | -4 | -2 | 1 | 1 | 1 | 1 | -2 | -2 | -2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from C2×C32⋊C4 |
| ρ15 | 4 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | 1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5 | 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 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5 | orthogonal faithful |
| ρ16 | 4 | 4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | -2 | -1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+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 | orthogonal lifted from C32⋊F5 |
| ρ17 | 4 | 4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | -2 | -1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+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 | orthogonal lifted from C32⋊F5 |
| ρ18 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ53-ζ3ζ52+2ζ53+ζ52+1 | -ζ3ζ53+ζ3ζ52+ζ53+2ζ52+1 | -ζ3ζ54+ζ3ζ5+ζ54+2ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5 | -ζ32ζ54+ζ32ζ5+ζ54+2ζ5+1 | orthogonal faithful |
| ρ19 | 4 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | 1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5 | -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 | orthogonal faithful |
| ρ20 | 4 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | 1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5 | -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 | orthogonal faithful |
| ρ21 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | -ζ3ζ53+ζ3ζ52+ζ53+2ζ52+1 | ζ3ζ53-ζ3ζ52+2ζ53+ζ52+1 | -ζ32ζ54+ζ32ζ5+ζ54+2ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52 | -ζ3ζ54+ζ3ζ5+ζ54+2ζ5+1 | orthogonal faithful |
| ρ22 | 4 | 4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | -2 | -1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+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 | orthogonal lifted from C32⋊F5 |
| ρ23 | 4 | 4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 1 | -1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | 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 | orthogonal lifted from C32⋊F5 |
| ρ24 | 4 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | 1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5 | 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 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5 | orthogonal faithful |
| ρ25 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 1 | ζ3ζ54-ζ3ζ5-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+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | -ζ32ζ54+ζ32ζ5+ζ54+2ζ5+1 | -ζ3ζ54+ζ3ζ5+ζ54+2ζ5+1 | ζ3ζ53-ζ3ζ52+2ζ53+ζ52+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5 | -ζ3ζ53+ζ3ζ52+ζ53+2ζ52+1 | orthogonal faithful |
| ρ26 | 4 | 4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | -2 | -1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+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 | orthogonal lifted from C32⋊F5 |
| ρ27 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 1 | ζ32ζ54-ζ32ζ5-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-ζ53-2ζ52-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | -ζ3ζ54+ζ3ζ5+ζ54+2ζ5+1 | -ζ32ζ54+ζ32ζ5+ζ54+2ζ5+1 | -ζ3ζ53+ζ3ζ52+ζ53+2ζ52+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5 | ζ3ζ53-ζ3ζ52+2ζ53+ζ52+1 | orthogonal faithful |
| ρ28 | 4 | 4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 1 | -1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | 2ζ3ζ53+2ζ3ζ5+ζ3+ζ53+ζ5+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ32ζ52+2ζ32ζ5+ζ32+ζ52+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | -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 | orthogonal lifted from C32⋊F5 |
| ρ29 | 4 | 4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 1 | -1 | ζ32ζ54-ζ32ζ5-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-ζ53-2ζ52-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | ζ3ζ54-ζ3ζ5-ζ54-2ζ5-1 | 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 | orthogonal lifted from C32⋊F5 |
| ρ30 | 4 | 4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 1 | -1 | ζ3ζ54-ζ3ζ5-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+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | ζ32ζ54-ζ32ζ5-ζ54-2ζ5-1 | 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 | orthogonal lifted from C32⋊F5 |
►On 60 points
Generators in S60
(1 19)(2 20)(3 16)(4 17)(5 18)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)
(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 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)(56 57 58 59 60)
(1 47)(2 49 5 50)(3 46 4 48)(6 56 12 53)(7 58 11 51)(8 60 15 54)(9 57 14 52)(10 59 13 55)(16 31 17 33)(18 35 20 34)(19 32)(21 41 27 38)(22 43 26 36)(23 45 30 39)(24 42 29 37)(25 44 28 40)
G:=sub<Sym(60)| (1,19)(2,20)(3,16)(4,17)(5,18)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60), (31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55), (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)(56,57,58,59,60), (1,47)(2,49,5,50)(3,46,4,48)(6,56,12,53)(7,58,11,51)(8,60,15,54)(9,57,14,52)(10,59,13,55)(16,31,17,33)(18,35,20,34)(19,32)(21,41,27,38)(22,43,26,36)(23,45,30,39)(24,42,29,37)(25,44,28,40)>;
G:=Group( (1,19)(2,20)(3,16)(4,17)(5,18)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60), (31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55), (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)(56,57,58,59,60), (1,47)(2,49,5,50)(3,46,4,48)(6,56,12,53)(7,58,11,51)(8,60,15,54)(9,57,14,52)(10,59,13,55)(16,31,17,33)(18,35,20,34)(19,32)(21,41,27,38)(22,43,26,36)(23,45,30,39)(24,42,29,37)(25,44,28,40) );
G=PermutationGroup([[(1,19),(2,20),(3,16),(4,17),(5,18),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,56),(42,57),(43,58),(44,59),(45,60)], [(31,41,36),(32,42,37),(33,43,38),(34,44,39),(35,45,40),(46,56,51),(47,57,52),(48,58,53),(49,59,54),(50,60,55)], [(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30),(31,41,36),(32,42,37),(33,43,38),(34,44,39),(35,45,40),(46,56,51),(47,57,52),(48,58,53),(49,59,54),(50,60,55)], [(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),(56,57,58,59,60)], [(1,47),(2,49,5,50),(3,46,4,48),(6,56,12,53),(7,58,11,51),(8,60,15,54),(9,57,14,52),(10,59,13,55),(16,31,17,33),(18,35,20,34),(19,32),(21,41,27,38),(22,43,26,36),(23,45,30,39),(24,42,29,37),(25,44,28,40)]])
Matrix representation of C2×C32⋊F5 ►in GL5(𝔽61)
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 36 | 16 |
| 0 | 0 | 0 | 12 | 24 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 24 | 45 | 0 | 0 |
| 0 | 49 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 16 |
| 0 | 0 | 0 | 12 | 24 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 60 | 1 | 0 | 0 |
| 0 | 16 | 44 | 0 | 0 |
| 0 | 0 | 0 | 17 | 43 |
| 0 | 0 | 0 | 17 | 0 |
| 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 17 | 1 | 0 | 0 |
| 0 | 17 | 44 | 0 | 0 |
G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,36,12,0,0,0,16,24],[1,0,0,0,0,0,24,49,0,0,0,45,36,0,0,0,0,0,36,12,0,0,0,16,24],[1,0,0,0,0,0,60,16,0,0,0,1,44,0,0,0,0,0,17,17,0,0,0,43,0],[11,0,0,0,0,0,0,0,17,17,0,0,0,1,44,0,1,0,0,0,0,0,1,0,0] >;
C2×C32⋊F5 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes F_5
% in TeX
G:=Group("C2xC3^2:F5"); // GroupNames label
G:=SmallGroup(360,150);
// by ID
G=gap.SmallGroup(360,150);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,3,-5,24,1347,111,1924,376,5189,2609]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*c*e^-1=b*c=c*b,b*d=d*b,e*b*e^-1=b^-1*c,c*d=d*c,e*d*e^-1=d^3>;
// generators/relations
Export
Subgroup lattice of C2×C32⋊F5 in TeX
Character table of C2×C32⋊F5 in TeX