metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊1C2≀C4, C23⋊F5⋊1C2, C22⋊C4⋊1F5, (C23×D5)⋊5C4, C23.F5⋊1C2, C23.1(C2×F5), (C2×Dic5).9D4, (C22×D5).9D4, C22⋊D20.1C2, C10.5(C23⋊C4), C2.8(D10.D4), C22.14(C22⋊F5), (C5×C22⋊C4)⋊1C4, (C22×C10).8(C2×C4), (C2×C5⋊D4).83C22, (C2×C10).14(C22⋊C4), SmallGroup(320,202)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊C2≀C4
G = < a,b,c,d,e,f | a5=b2=c2=d2=e2=f4=1, bab=a-1, ac=ca, ad=da, ae=ea, faf-1=a3, bc=cb, bd=db, be=eb, fbf-1=bcde, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >
Subgroups: 682 in 94 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C22⋊C4, M4(2), C2×D4, C24, Dic5, C20, F5, D10, C2×C10, C2×C10, C23⋊C4, C4.D4, C22≀C2, C5⋊C8, D20, C2×Dic5, C5⋊D4, C2×C20, C2×F5, C22×D5, C22×D5, C22×C10, C2≀C4, D10⋊C4, C5×C22⋊C4, C22.F5, C22⋊F5, C2×D20, C2×C5⋊D4, C23×D5, C23⋊F5, C23.F5, C22⋊D20, C5⋊C2≀C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C23⋊C4, C2×F5, C2≀C4, C22⋊F5, D10.D4, C5⋊C2≀C4
Character table of C5⋊C2≀C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 5 | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 2 | 4 | 20 | 20 | 20 | 8 | 20 | 40 | 40 | 4 | 40 | 40 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -i | i | 1 | i | -i | 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 | 1 | -i | i | 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 | 1 | -i | i | 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 | 1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ12 | 4 | -4 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C4 |
| ρ13 | 4 | -4 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C4 |
| ρ14 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ15 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ16 | 4 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -√5 | √5 | √5 | -√5 | orthogonal lifted from C22⋊F5 |
| ρ17 | 4 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | √5 | -√5 | -√5 | √5 | orthogonal lifted from C22⋊F5 |
| ρ18 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | -1 | √5 | -√5 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ4ζ54+2ζ4ζ53+ζ4 | orthogonal lifted from D10.D4 |
| ρ19 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | -1 | √5 | -√5 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ4ζ52+2ζ4ζ5+ζ4 | orthogonal lifted from D10.D4 |
| ρ20 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | -1 | -√5 | √5 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ43ζ54+2ζ43ζ52+ζ43 | orthogonal lifted from D10.D4 |
| ρ21 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | -1 | -√5 | √5 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ43ζ53+2ζ43ζ5+ζ43 | orthogonal lifted from D10.D4 |
| ρ22 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2√5 | 2√5 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ23 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2√5 | -2√5 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 40 points
Generators in S40
(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)
(2 5)(3 4)(6 7)(8 10)(11 12)(13 15)(16 17)(18 20)(21 38)(22 37)(23 36)(24 40)(25 39)(26 33)(27 32)(28 31)(29 35)(30 34)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 11)(7 12)(8 13)(9 14)(10 15)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 22)(2 24 5 25)(3 21 4 23)(6 26 7 28)(8 30 10 29)(9 27)(11 36 17 33)(12 38 16 31)(13 40 20 34)(14 37 19 32)(15 39 18 35)
G:=sub<Sym(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), (2,5)(3,4)(6,7)(8,10)(11,12)(13,15)(16,17)(18,20)(21,38)(22,37)(23,36)(24,40)(25,39)(26,33)(27,32)(28,31)(29,35)(30,34), (1,19)(2,20)(3,16)(4,17)(5,18)(6,11)(7,12)(8,13)(9,14)(10,15)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,22)(2,24,5,25)(3,21,4,23)(6,26,7,28)(8,30,10,29)(9,27)(11,36,17,33)(12,38,16,31)(13,40,20,34)(14,37,19,32)(15,39,18,35)>;
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), (2,5)(3,4)(6,7)(8,10)(11,12)(13,15)(16,17)(18,20)(21,38)(22,37)(23,36)(24,40)(25,39)(26,33)(27,32)(28,31)(29,35)(30,34), (1,19)(2,20)(3,16)(4,17)(5,18)(6,11)(7,12)(8,13)(9,14)(10,15)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,22)(2,24,5,25)(3,21,4,23)(6,26,7,28)(8,30,10,29)(9,27)(11,36,17,33)(12,38,16,31)(13,40,20,34)(14,37,19,32)(15,39,18,35) );
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)], [(2,5),(3,4),(6,7),(8,10),(11,12),(13,15),(16,17),(18,20),(21,38),(22,37),(23,36),(24,40),(25,39),(26,33),(27,32),(28,31),(29,35),(30,34)], [(1,19),(2,20),(3,16),(4,17),(5,18),(6,11),(7,12),(8,13),(9,14),(10,15),(21,36),(22,37),(23,38),(24,39),(25,40),(26,31),(27,32),(28,33),(29,34),(30,35)], [(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,22),(2,24,5,25),(3,21,4,23),(6,26,7,28),(8,30,10,29),(9,27),(11,36,17,33),(12,38,16,31),(13,40,20,34),(14,37,19,32),(15,39,18,35)]])
Matrix representation of C5⋊C2≀C4 ►in GL8(ℤ)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -8 | 8 | 13 | 0 |
| 0 | 0 | 0 | 0 | 5 | 8 | 0 | -8 |
| 0 | 0 | 0 | 0 | -8 | 0 | 8 | 5 |
| 0 | 0 | 0 | 0 | 0 | 13 | 8 | -8 |
G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-8,5,-8,0,0,0,0,0,8,8,0,13,0,0,0,0,13,0,8,8,0,0,0,0,0,-8,5,-8] >;
C5⋊C2≀C4 in GAP, Magma, Sage, TeX
C_5\rtimes C_2\wr C_4
% in TeX
G:=Group("C5:C2wrC4"); // GroupNames label
G:=SmallGroup(320,202);
// by ID
G=gap.SmallGroup(320,202);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,219,675,570,297,1684,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^2=f^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a^3,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d*e,c*d=d*c,c*e=e*c,f*c*f^-1=c*d*e,f*d*f^-1=d*e=e*d,e*f=f*e>;
// generators/relations
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