metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊1F5, (C4×C20)⋊1C4, (C2×D20)⋊3C4, C5⋊1(C42⋊C4), C20⋊4D4.1C2, (C22×D5).7D4, D10.D4⋊1C2, C10.1(C23⋊C4), (C2×D20).1C22, C22.8(C22⋊F5), C2.4(D10.D4), (C2×C4).49(C2×F5), (C2×C20).95(C2×C4), (C2×C10).8(C22⋊C4), SmallGroup(320,191)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊F5
G = < a,b,c,d | a4=b4=c5=d4=1, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, dbd-1=a2b, dcd-1=c3 >
Subgroups: 666 in 86 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, D5, C10, C10, C42, C22⋊C4, C2×D4, C20, F5, D10, C2×C10, C23⋊C4, C4⋊1D4, D20, C2×C20, C2×C20, C2×F5, C22×D5, C22×D5, C42⋊C4, C4×C20, C22⋊F5, C2×D20, C2×D20, D10.D4, C20⋊4D4, C42⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C23⋊C4, C2×F5, C42⋊C4, C22⋊F5, D10.D4, C42⋊F5
Character table of C42⋊F5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5 | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 20K | 20L | |
| size | 1 | 1 | 2 | 20 | 20 | 40 | 4 | 4 | 4 | 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 | 1 | -i | 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 | 1 | i | 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 | 1 | -i | -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 | 1 | i | -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 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 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 |
| ρ12 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | -2 | -2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from C42⋊C4 |
| ρ13 | 4 | 4 | 4 | 0 | 0 | 0 | -4 | -4 | 4 | 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 |
| ρ14 | 4 | -4 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | orthogonal lifted from C42⋊C4 |
| ρ15 | 4 | 4 | 4 | 0 | 0 | 0 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ16 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | √5 | -√5 | 1 | -√5 | √5 | √5 | -√5 | 1 | 1 | 1 | -√5 | √5 | orthogonal lifted from C22⋊F5 |
| ρ17 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -√5 | √5 | 1 | √5 | -√5 | -√5 | √5 | 1 | 1 | 1 | √5 | -√5 | orthogonal lifted from C22⋊F5 |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | √5 | -√5 | ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 | ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 | 2ζ4ζ54+2ζ4ζ53+ζ4 | ζ4ζ53-ζ4ζ52-ζ53-ζ52-1 | ζ43ζ54-ζ43ζ5-ζ54-ζ5-1 | -ζ43ζ54+ζ43ζ5-ζ54-ζ5-1 | -ζ4ζ53+ζ4ζ52-ζ53-ζ52-1 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ4ζ52+2ζ4ζ5+ζ4 | -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 | -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 | orthogonal faithful |
| ρ19 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ4ζ52+2ζ4ζ5+ζ4 | -√5 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ4ζ52+2ζ4ζ5+ζ4 | √5 | √5 | -√5 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ43ζ54+2ζ43ζ52+ζ43 | orthogonal lifted from D10.D4 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | √5 | -√5 | -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 | -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 | 2ζ4ζ52+2ζ4ζ5+ζ4 | -ζ4ζ53+ζ4ζ52-ζ53-ζ52-1 | -ζ43ζ54+ζ43ζ5-ζ54-ζ5-1 | ζ43ζ54-ζ43ζ5-ζ54-ζ5-1 | ζ4ζ53-ζ4ζ52-ζ53-ζ52-1 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ4ζ54+2ζ4ζ53+ζ4 | ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 | ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 | orthogonal faithful |
| ρ21 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ4ζ54+2ζ4ζ53+ζ4 | -√5 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ4ζ54+2ζ4ζ53+ζ4 | √5 | √5 | -√5 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ43ζ53+2ζ43ζ5+ζ43 | orthogonal lifted from D10.D4 |
| ρ22 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ43ζ53+2ζ43ζ5+ζ43 | √5 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ43ζ53+2ζ43ζ5+ζ43 | -√5 | -√5 | √5 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ4ζ52+2ζ4ζ5+ζ4 | orthogonal lifted from D10.D4 |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -√5 | √5 | -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 | ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 | 2ζ43ζ54+2ζ43ζ52+ζ43 | ζ43ζ54-ζ43ζ5-ζ54-ζ5-1 | -ζ4ζ53+ζ4ζ52-ζ53-ζ52-1 | ζ4ζ53-ζ4ζ52-ζ53-ζ52-1 | -ζ43ζ54+ζ43ζ5-ζ54-ζ5-1 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ43ζ53+2ζ43ζ5+ζ43 | -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 | ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 | orthogonal faithful |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -√5 | √5 | -ζ4ζ53+ζ4ζ52-ζ53-ζ52-1 | ζ43ζ54-ζ43ζ5-ζ54-ζ5-1 | 2ζ43ζ53+2ζ43ζ5+ζ43 | ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 | -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 | ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 | -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ43ζ54+2ζ43ζ52+ζ43 | -ζ43ζ54+ζ43ζ5-ζ54-ζ5-1 | ζ4ζ53-ζ4ζ52-ζ53-ζ52-1 | orthogonal faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | √5 | -√5 | ζ43ζ54-ζ43ζ5-ζ54-ζ5-1 | ζ4ζ53-ζ4ζ52-ζ53-ζ52-1 | 2ζ4ζ52+2ζ4ζ5+ζ4 | ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 | ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 | -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 | -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ4ζ54+2ζ4ζ53+ζ4 | -ζ4ζ53+ζ4ζ52-ζ53-ζ52-1 | -ζ43ζ54+ζ43ζ5-ζ54-ζ5-1 | orthogonal faithful |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -√5 | √5 | ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 | -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 | 2ζ43ζ53+2ζ43ζ5+ζ43 | -ζ43ζ54+ζ43ζ5-ζ54-ζ5-1 | ζ4ζ53-ζ4ζ52-ζ53-ζ52-1 | -ζ4ζ53+ζ4ζ52-ζ53-ζ52-1 | ζ43ζ54-ζ43ζ5-ζ54-ζ5-1 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ43ζ54+2ζ43ζ52+ζ43 | ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 | -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 | orthogonal faithful |
| ρ27 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ43ζ54+2ζ43ζ52+ζ43 | √5 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ43ζ54+2ζ43ζ52+ζ43 | -√5 | -√5 | √5 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ4ζ54+2ζ4ζ53+ζ4 | orthogonal lifted from D10.D4 |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | √5 | -√5 | -ζ43ζ54+ζ43ζ5-ζ54-ζ5-1 | -ζ4ζ53+ζ4ζ52-ζ53-ζ52-1 | 2ζ4ζ54+2ζ4ζ53+ζ4 | -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 | -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 | ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 | ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ4ζ52+2ζ4ζ5+ζ4 | ζ4ζ53-ζ4ζ52-ζ53-ζ52-1 | ζ43ζ54-ζ43ζ5-ζ54-ζ5-1 | orthogonal faithful |
| ρ29 | 4 | -4 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -√5 | √5 | ζ4ζ53-ζ4ζ52-ζ53-ζ52-1 | -ζ43ζ54+ζ43ζ5-ζ54-ζ5-1 | 2ζ43ζ54+2ζ43ζ52+ζ43 | -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 | ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 | -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 | ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ43ζ53+2ζ43ζ5+ζ43 | ζ43ζ54-ζ43ζ5-ζ54-ζ5-1 | -ζ4ζ53+ζ4ζ52-ζ53-ζ52-1 | orthogonal faithful |
►On 40 points
Generators in S40
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(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)
(1 33 11 23)(2 35 15 21)(3 32 14 24)(4 34 13 22)(5 31 12 25)(6 38 16 28)(7 40 20 26)(8 37 19 29)(9 39 18 27)(10 36 17 30)
G:=sub<Sym(40)| (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (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), (1,33,11,23)(2,35,15,21)(3,32,14,24)(4,34,13,22)(5,31,12,25)(6,38,16,28)(7,40,20,26)(8,37,19,29)(9,39,18,27)(10,36,17,30)>;
G:=Group( (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (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), (1,33,11,23)(2,35,15,21)(3,32,14,24)(4,34,13,22)(5,31,12,25)(6,38,16,28)(7,40,20,26)(8,37,19,29)(9,39,18,27)(10,36,17,30) );
G=PermutationGroup([[(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(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)], [(1,33,11,23),(2,35,15,21),(3,32,14,24),(4,34,13,22),(5,31,12,25),(6,38,16,28),(7,40,20,26),(8,37,19,29),(9,39,18,27),(10,36,17,30)]])
Matrix representation of C42⋊F5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 20 | 24 | 39 | 13 |
| 26 | 30 | 28 | 2 |
| 30 | 28 | 0 | 0 |
| 22 | 11 | 0 | 0 |
| 37 | 2 | 39 | 13 |
| 21 | 23 | 28 | 2 |
| 40 | 40 | 0 | 0 |
| 8 | 7 | 0 | 0 |
| 37 | 0 | 40 | 35 |
| 36 | 36 | 6 | 35 |
| 11 | 9 | 20 | 20 |
| 7 | 18 | 1 | 3 |
| 27 | 13 | 28 | 18 |
| 0 | 8 | 20 | 25 |
G:=sub<GL(4,GF(41))| [40,0,20,26,0,40,24,30,0,0,39,28,0,0,13,2],[30,22,37,21,28,11,2,23,0,0,39,28,0,0,13,2],[40,8,37,36,40,7,0,36,0,0,40,6,0,0,35,35],[11,7,27,0,9,18,13,8,20,1,28,20,20,3,18,25] >;
C42⋊F5 in GAP, Magma, Sage, TeX
C_4^2\rtimes F_5
% in TeX
G:=Group("C4^2:F5"); // GroupNames label
G:=SmallGroup(320,191);
// by ID
G=gap.SmallGroup(320,191);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,219,1571,297,136,1684,6278,3156]);
// 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^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations
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