metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.F5, Dic5.4D4, C5⋊(C4.D4), C22.F5⋊1C2, C22.5(C2×F5), (C22×C10).4C4, (C22×D5).2C4, C2.11(C22⋊F5), C10.11(C22⋊C4), (C2×Dic5).24C22, (C2×C5⋊D4).8C2, (C2×C10).12(C2×C4), SmallGroup(160,88)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.F5
G = < a,b,c,d,e | a2=b2=c2=d5=1, e4=c, ab=ba, ac=ca, ad=da, eae-1=abc, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >
Character table of C23.F5
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 5 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | |
| size | 1 | 1 | 2 | 4 | 20 | 10 | 10 | 4 | 20 | 20 | 20 | 20 | 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 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 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 | i | i | -i | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | i | -i | -i | i | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | orthogonal lifted from C4.D4 |
| ρ12 | 4 | 4 | 4 | -4 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ13 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ14 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | √5 | -√5 | 1 | 1 | -1 | -√5 | √5 | orthogonal lifted from C22⋊F5 |
| ρ15 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -√5 | √5 | 1 | 1 | -1 | √5 | -√5 | orthogonal lifted from C22⋊F5 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 2ζ53+2ζ5+1 | 2ζ54+2ζ53+1 | √5 | -√5 | 1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ52+1 | complex faithful |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 2ζ52+2ζ5+1 | 2ζ53+2ζ5+1 | -√5 | √5 | 1 | 2ζ54+2ζ52+1 | 2ζ54+2ζ53+1 | complex faithful |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 2ζ54+2ζ53+1 | 2ζ54+2ζ52+1 | -√5 | √5 | 1 | 2ζ53+2ζ5+1 | 2ζ52+2ζ5+1 | complex faithful |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 2ζ54+2ζ52+1 | 2ζ52+2ζ5+1 | √5 | -√5 | 1 | 2ζ54+2ζ53+1 | 2ζ53+2ζ5+1 | complex faithful |
►On 40 points
Generators in S40
(1 7)(2 4)(3 5)(6 8)(9 35)(10 40)(11 33)(12 38)(13 39)(14 36)(15 37)(16 34)(17 27)(18 28)(19 25)(20 26)(21 31)(22 32)(23 29)(24 30)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)
(1 11 24 32 35)(2 25 12 36 17)(3 37 26 18 13)(4 19 38 14 27)(5 15 20 28 39)(6 29 16 40 21)(7 33 30 22 9)(8 23 34 10 31)
(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)
G:=sub<Sym(40)| (1,7)(2,4)(3,5)(6,8)(9,35)(10,40)(11,33)(12,38)(13,39)(14,36)(15,37)(16,34)(17,27)(18,28)(19,25)(20,26)(21,31)(22,32)(23,29)(24,30), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40), (1,11,24,32,35)(2,25,12,36,17)(3,37,26,18,13)(4,19,38,14,27)(5,15,20,28,39)(6,29,16,40,21)(7,33,30,22,9)(8,23,34,10,31), (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)>;
G:=Group( (1,7)(2,4)(3,5)(6,8)(9,35)(10,40)(11,33)(12,38)(13,39)(14,36)(15,37)(16,34)(17,27)(18,28)(19,25)(20,26)(21,31)(22,32)(23,29)(24,30), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40), (1,11,24,32,35)(2,25,12,36,17)(3,37,26,18,13)(4,19,38,14,27)(5,15,20,28,39)(6,29,16,40,21)(7,33,30,22,9)(8,23,34,10,31), (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) );
G=PermutationGroup([[(1,7),(2,4),(3,5),(6,8),(9,35),(10,40),(11,33),(12,38),(13,39),(14,36),(15,37),(16,34),(17,27),(18,28),(19,25),(20,26),(21,31),(22,32),(23,29),(24,30)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(34,38),(36,40)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40)], [(1,11,24,32,35),(2,25,12,36,17),(3,37,26,18,13),(4,19,38,14,27),(5,15,20,28,39),(6,29,16,40,21),(7,33,30,22,9),(8,23,34,10,31)], [(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)]])
C23.F5 is a maximal subgroup of
C5⋊C2≀C4 C22⋊C4⋊F5 (C22×C4)⋊F5 C24⋊2F5 (C4×D5).D4 (C2×D4).9F5 D5⋊(C4.D4) Dic5.D12 C5⋊(C12.D4)
C23.F5 is a maximal quotient of
C5⋊(C23⋊C8) C22.F5⋊C4 (C2×D4).F5 Dic5.SD16 (C2×Q8).F5 Dic5.Q16 C24.F5 Dic5.D12 C5⋊(C12.D4)
Matrix representation of C23.F5 ►in GL4(𝔽41) generated by
| 18 | 35 | 0 | 0 |
| 6 | 23 | 0 | 0 |
| 23 | 20 | 18 | 35 |
| 35 | 18 | 6 | 23 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 40 | 0 |
| 35 | 35 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 6 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 29 | 2 | 35 | 35 |
| 14 | 0 | 6 | 40 |
| 1 | 0 | 39 | 0 |
| 35 | 35 | 0 | 39 |
| 12 | 3 | 40 | 0 |
| 5 | 27 | 6 | 6 |
G:=sub<GL(4,GF(41))| [18,6,23,35,35,23,20,18,0,0,18,6,0,0,35,23],[1,0,1,35,0,1,0,35,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[6,1,29,14,40,0,2,0,0,0,35,6,0,0,35,40],[1,35,12,5,0,35,3,27,39,0,40,6,0,39,0,6] >;
C23.F5 in GAP, Magma, Sage, TeX
C_2^3.F_5
% in TeX
G:=Group("C2^3.F5"); // GroupNames label
G:=SmallGroup(160,88);
// by ID
G=gap.SmallGroup(160,88);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,188,86,579,2309,1169]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^5=1,e^4=c,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations
Export
Subgroup lattice of C23.F5 in TeX
Character table of C23.F5 in TeX