metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊2F5, (C4×C20)⋊2C4, C5⋊1(C42⋊3C4), (C2×Dic10)⋊3C4, (C22×D5).8D4, C10.2(C23⋊C4), C4.D20.1C2, (C2×D20).2C22, D10.D4.1C2, C22.9(C22⋊F5), C2.5(D10.D4), (C2×C4).50(C2×F5), (C2×C20).96(C2×C4), (C2×C10).9(C22⋊C4), SmallGroup(320,192)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊2F5
G = < a,b,c,d | a4=b4=c5=d4=1, ab=ba, ac=ca, dad-1=a-1b, bc=cb, dbd-1=a2b-1, dcd-1=c3 >
Subgroups: 474 in 70 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, F5, D10, C2×C10, C23⋊C4, C4.4D4, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C2×F5, C22×D5, C42⋊3C4, D10⋊C4, C4×C20, C22⋊F5, C2×Dic10, C2×D20, D10.D4, C4.D20, C42⋊2F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C23⋊C4, C2×F5, C42⋊3C4, C22⋊F5, D10.D4, C42⋊2F5
Character table of C42⋊2F5
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 20K | 20L | |
| size | 1 | 1 | 2 | 20 | 20 | 4 | 4 | 4 | 40 | 40 | 40 | 40 | 40 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | i | -1 | -i | i | -i | 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 | 1 | -1 | -i | 1 | -i | i | i | 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 | 1 | 1 | -i | -1 | i | -i | i | 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 | 1 | -1 | i | 1 | i | -i | -i | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | 4 | 0 | 0 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ12 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ13 | 4 | 4 | 4 | 0 | 0 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ14 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 2ζ4ζ54+2ζ4ζ52+ζ4 | -√5 | √5 | 2ζ4ζ53+2ζ4ζ5+ζ4 | 2ζ43ζ54+2ζ43ζ53+ζ43 | 2ζ43ζ52+2ζ43ζ5+ζ43 | 2ζ4ζ54+2ζ4ζ52+ζ4 | √5 | -√5 | 2ζ4ζ53+2ζ4ζ5+ζ4 | 2ζ43ζ54+2ζ43ζ53+ζ43 | 2ζ43ζ52+2ζ43ζ5+ζ43 | orthogonal lifted from D10.D4 |
| ρ15 | 4 | 4 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -√5 | 1 | 1 | -√5 | √5 | √5 | -√5 | 1 | 1 | -√5 | √5 | √5 | orthogonal lifted from C22⋊F5 |
| ρ16 | 4 | 4 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | √5 | 1 | 1 | √5 | -√5 | -√5 | √5 | 1 | 1 | √5 | -√5 | -√5 | orthogonal lifted from C22⋊F5 |
| ρ17 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 2ζ4ζ53+2ζ4ζ5+ζ4 | -√5 | √5 | 2ζ4ζ54+2ζ4ζ52+ζ4 | 2ζ43ζ52+2ζ43ζ5+ζ43 | 2ζ43ζ54+2ζ43ζ53+ζ43 | 2ζ4ζ53+2ζ4ζ5+ζ4 | √5 | -√5 | 2ζ4ζ54+2ζ4ζ52+ζ4 | 2ζ43ζ52+2ζ43ζ5+ζ43 | 2ζ43ζ54+2ζ43ζ53+ζ43 | orthogonal lifted from D10.D4 |
| ρ18 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 2ζ43ζ52+2ζ43ζ5+ζ43 | √5 | -√5 | 2ζ43ζ54+2ζ43ζ53+ζ43 | 2ζ4ζ54+2ζ4ζ52+ζ4 | 2ζ4ζ53+2ζ4ζ5+ζ4 | 2ζ43ζ52+2ζ43ζ5+ζ43 | -√5 | √5 | 2ζ43ζ54+2ζ43ζ53+ζ43 | 2ζ4ζ54+2ζ4ζ52+ζ4 | 2ζ4ζ53+2ζ4ζ5+ζ4 | orthogonal lifted from D10.D4 |
| ρ19 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 2ζ43ζ54+2ζ43ζ53+ζ43 | √5 | -√5 | 2ζ43ζ52+2ζ43ζ5+ζ43 | 2ζ4ζ53+2ζ4ζ5+ζ4 | 2ζ4ζ54+2ζ4ζ52+ζ4 | 2ζ43ζ54+2ζ43ζ53+ζ43 | -√5 | √5 | 2ζ43ζ52+2ζ43ζ5+ζ43 | 2ζ4ζ53+2ζ4ζ5+ζ4 | 2ζ4ζ54+2ζ4ζ52+ζ4 | orthogonal lifted from D10.D4 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 2i | 0 | 0 | -2i | -2i | -2i | -2i | 0 | 0 | 2i | 2i | 2i | complex lifted from C42⋊3C4 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | -1 | -√5 | √5 | 1 | ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 | 2ζ4ζ53+2ζ4ζ5+ζ4 | 2ζ43ζ54+2ζ43ζ53+ζ43 | ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 | ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 | ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 | ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 | 2ζ43ζ52+2ζ43ζ5+ζ43 | 2ζ4ζ54+2ζ4ζ52+ζ4 | ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 | ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 | ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | -1 | -√5 | √5 | 1 | ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 | 2ζ4ζ54+2ζ4ζ52+ζ4 | 2ζ43ζ52+2ζ43ζ5+ζ43 | ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 | ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 | ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 | ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 | 2ζ43ζ54+2ζ43ζ53+ζ43 | 2ζ4ζ53+2ζ4ζ5+ζ4 | ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 | ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 | ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | -1 | √5 | -√5 | 1 | ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 | 2ζ43ζ52+2ζ43ζ5+ζ43 | 2ζ4ζ53+2ζ4ζ5+ζ4 | ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 | ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 | ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 | ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 | 2ζ4ζ54+2ζ4ζ52+ζ4 | 2ζ43ζ54+2ζ43ζ53+ζ43 | ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 | ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 | ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | -1 | √5 | -√5 | 1 | ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 | 2ζ43ζ54+2ζ43ζ53+ζ43 | 2ζ4ζ54+2ζ4ζ52+ζ4 | ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 | ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 | ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 | ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 | 2ζ4ζ53+2ζ4ζ5+ζ4 | 2ζ43ζ52+2ζ43ζ5+ζ43 | ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 | ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 | ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | -1 | -√5 | √5 | 1 | ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 | 2ζ4ζ54+2ζ4ζ52+ζ4 | 2ζ43ζ52+2ζ43ζ5+ζ43 | ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 | ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 | ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 | ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 | 2ζ43ζ54+2ζ43ζ53+ζ43 | 2ζ4ζ53+2ζ4ζ5+ζ4 | ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 | ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 | ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | -1 | √5 | -√5 | 1 | ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 | 2ζ43ζ54+2ζ43ζ53+ζ43 | 2ζ4ζ54+2ζ4ζ52+ζ4 | ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 | ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 | ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 | ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 | 2ζ4ζ53+2ζ4ζ5+ζ4 | 2ζ43ζ52+2ζ43ζ5+ζ43 | ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 | ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 | ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | -2i | 0 | 0 | 2i | 2i | 2i | 2i | 0 | 0 | -2i | -2i | -2i | complex lifted from C42⋊3C4 |
| ρ28 | 4 | -4 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | -1 | -√5 | √5 | 1 | ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 | 2ζ4ζ53+2ζ4ζ5+ζ4 | 2ζ43ζ54+2ζ43ζ53+ζ43 | ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 | ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 | ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 | ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 | 2ζ43ζ52+2ζ43ζ5+ζ43 | 2ζ4ζ54+2ζ4ζ52+ζ4 | ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 | ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 | ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 | complex faithful |
| ρ29 | 4 | -4 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | -1 | √5 | -√5 | 1 | ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 | 2ζ43ζ52+2ζ43ζ5+ζ43 | 2ζ4ζ53+2ζ4ζ5+ζ4 | ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 | ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 | ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 | ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 | 2ζ4ζ54+2ζ4ζ52+ζ4 | 2ζ43ζ54+2ζ43ζ53+ζ43 | ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 | ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 | ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 | complex faithful |
►On 80 points
Generators in S80
(1 51 11 41)(2 52 12 42)(3 53 13 43)(4 54 14 44)(5 55 15 45)(6 56 16 46)(7 57 17 47)(8 58 18 48)(9 59 19 49)(10 60 20 50)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)
(1 26 6 21)(2 27 7 22)(3 28 8 23)(4 29 9 24)(5 30 10 25)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(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)
(2 3 5 4)(7 8 10 9)(11 16)(12 18 15 19)(13 20 14 17)(21 31 26 36)(22 33 30 39)(23 35 29 37)(24 32 28 40)(25 34 27 38)(41 61 56 71)(42 63 60 74)(43 65 59 72)(44 62 58 75)(45 64 57 73)(46 66 51 76)(47 68 55 79)(48 70 54 77)(49 67 53 80)(50 69 52 78)
G:=sub<Sym(80)| (1,51,11,41)(2,52,12,42)(3,53,13,43)(4,54,14,44)(5,55,15,45)(6,56,16,46)(7,57,17,47)(8,58,18,48)(9,59,19,49)(10,60,20,50)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70), (1,26,6,21)(2,27,7,22)(3,28,8,23)(4,29,9,24)(5,30,10,25)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (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), (2,3,5,4)(7,8,10,9)(11,16)(12,18,15,19)(13,20,14,17)(21,31,26,36)(22,33,30,39)(23,35,29,37)(24,32,28,40)(25,34,27,38)(41,61,56,71)(42,63,60,74)(43,65,59,72)(44,62,58,75)(45,64,57,73)(46,66,51,76)(47,68,55,79)(48,70,54,77)(49,67,53,80)(50,69,52,78)>;
G:=Group( (1,51,11,41)(2,52,12,42)(3,53,13,43)(4,54,14,44)(5,55,15,45)(6,56,16,46)(7,57,17,47)(8,58,18,48)(9,59,19,49)(10,60,20,50)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70), (1,26,6,21)(2,27,7,22)(3,28,8,23)(4,29,9,24)(5,30,10,25)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (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), (2,3,5,4)(7,8,10,9)(11,16)(12,18,15,19)(13,20,14,17)(21,31,26,36)(22,33,30,39)(23,35,29,37)(24,32,28,40)(25,34,27,38)(41,61,56,71)(42,63,60,74)(43,65,59,72)(44,62,58,75)(45,64,57,73)(46,66,51,76)(47,68,55,79)(48,70,54,77)(49,67,53,80)(50,69,52,78) );
G=PermutationGroup([[(1,51,11,41),(2,52,12,42),(3,53,13,43),(4,54,14,44),(5,55,15,45),(6,56,16,46),(7,57,17,47),(8,58,18,48),(9,59,19,49),(10,60,20,50),(21,71,31,61),(22,72,32,62),(23,73,33,63),(24,74,34,64),(25,75,35,65),(26,76,36,66),(27,77,37,67),(28,78,38,68),(29,79,39,69),(30,80,40,70)], [(1,26,6,21),(2,27,7,22),(3,28,8,23),(4,29,9,24),(5,30,10,25),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35),(41,66,46,61),(42,67,47,62),(43,68,48,63),(44,69,49,64),(45,70,50,65),(51,76,56,71),(52,77,57,72),(53,78,58,73),(54,79,59,74),(55,80,60,75)], [(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)], [(2,3,5,4),(7,8,10,9),(11,16),(12,18,15,19),(13,20,14,17),(21,31,26,36),(22,33,30,39),(23,35,29,37),(24,32,28,40),(25,34,27,38),(41,61,56,71),(42,63,60,74),(43,65,59,72),(44,62,58,75),(45,64,57,73),(46,66,51,76),(47,68,55,79),(48,70,54,77),(49,67,53,80),(50,69,52,78)]])
Matrix representation of C42⋊2F5 ►in GL4(𝔽41) generated by
| 26 | 9 | 7 | 29 |
| 12 | 38 | 21 | 19 |
| 22 | 34 | 19 | 2 |
| 39 | 20 | 32 | 17 |
| 3 | 6 | 7 | 40 |
| 1 | 4 | 7 | 8 |
| 33 | 34 | 37 | 40 |
| 1 | 34 | 35 | 38 |
| 40 | 40 | 40 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 40 | 40 | 40 | 40 |
G:=sub<GL(4,GF(41))| [26,12,22,39,9,38,34,20,7,21,19,32,29,19,2,17],[3,1,33,1,6,4,34,34,7,7,37,35,40,8,40,38],[40,1,0,0,40,0,1,0,40,0,0,1,40,0,0,0],[1,0,0,40,0,0,1,40,0,0,0,40,0,1,0,40] >;
C42⋊2F5 in GAP, Magma, Sage, TeX
C_4^2\rtimes_2F_5
% in TeX
G:=Group("C4^2:2F5"); // GroupNames label
G:=SmallGroup(320,192);
// by ID
G=gap.SmallGroup(320,192);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,555,675,297,136,1684,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^5=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^3>;
// generators/relations
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