metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23⋊F5, D10.4D4, C5⋊2(C23⋊C4), C22⋊F5⋊2C2, (C22×C10)⋊3C4, (C2×Dic5)⋊2C4, C22.4(C2×F5), C10.9(C22⋊C4), C2.10(C22⋊F5), (C22×D5).15C22, (C2×C5⋊D4).7C2, (C2×C10).10(C2×C4), SmallGroup(160,86)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23⋊F5
G = < a,b,c,d,e | a2=b2=c2=d5=e4=1, ab=ba, ac=ca, ad=da, eae-1=abc, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >
2C2
4C2
10C2
10C2
2C22
4C22
5C22
5C22
10C4
20C4
20C22
20C4
2D5
2C10
2D5
4C10
5C23
10D4
10D4
10C2×C4
10C2×C4
2Dic5
4F5
4D10
4F5
Character table of C23⋊F5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 5 | 10A | 10B | 10C | 10D | 10E | 10F | 10G | |
| size | 1 | 1 | 2 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 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 | 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 | i | -1 | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | -i | -1 | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | -1 | -i | 1 | -i | i | i | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | -1 | -1 | i | 1 | i | -i | -i | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ12 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ13 | 4 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ14 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -√5 | √5 | 1 | 1 | -1 | √5 | -√5 | orthogonal lifted from C22⋊F5 |
| ρ15 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | √5 | -√5 | 1 | 1 | -1 | -√5 | √5 | orthogonal lifted from C22⋊F5 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2ζ54+2ζ52+1 | 2ζ52+2ζ5+1 | √5 | -√5 | 1 | 2ζ54+2ζ53+1 | 2ζ53+2ζ5+1 | complex faithful |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2ζ54+2ζ53+1 | 2ζ54+2ζ52+1 | -√5 | √5 | 1 | 2ζ53+2ζ5+1 | 2ζ52+2ζ5+1 | complex faithful |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2ζ52+2ζ5+1 | 2ζ53+2ζ5+1 | -√5 | √5 | 1 | 2ζ54+2ζ52+1 | 2ζ54+2ζ53+1 | complex faithful |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ53+1 | √5 | -√5 | 1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ52+1 | complex faithful |
►On 40 points
Generators in S40
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 6)(2 7)(3 8)(4 9)(5 10)(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 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 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)
G:=sub<Sym(40)| (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,6)(2,7)(3,8)(4,9)(5,10)(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), (1,2,3,4,5)(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), (2,3,5,4)(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)>;
G:=Group( (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,6)(2,7)(3,8)(4,9)(5,10)(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), (1,2,3,4,5)(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), (2,3,5,4)(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) );
G=PermutationGroup([[(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(1,6),(2,7),(3,8),(4,9),(5,10),(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)], [(1,2,3,4,5),(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)], [(2,3,5,4),(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)]])
C23⋊F5 is a maximal subgroup of
C5⋊C2≀C4 C22⋊C4⋊F5 (C22×C4)⋊F5 C24⋊2F5 C23⋊F5⋊5C2 (C2×D4)⋊7F5 (C2×D4)⋊8F5 D10.D12 C3⋊(C23⋊F5)
C23⋊F5 is a maximal quotient of
(C22×C4).F5 (C22×C4)⋊F5 C22⋊F5⋊C4 D10.SD16 (C2×D4)⋊F5 (D4×C10).C4 D10.Q16 (C2×Q8)⋊F5 (Q8×C10).C4 C24.F5 C24⋊2F5 D10.D12 C3⋊(C23⋊F5)
Matrix representation of C23⋊F5 ►in GL4(𝔽41) generated by
| 5 | 10 | 32 | 19 |
| 22 | 27 | 32 | 13 |
| 28 | 9 | 14 | 19 |
| 22 | 9 | 31 | 36 |
| 22 | 0 | 3 | 3 |
| 38 | 19 | 38 | 0 |
| 0 | 38 | 19 | 38 |
| 3 | 3 | 0 | 22 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 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))| [5,22,28,22,10,27,9,9,32,32,14,31,19,13,19,36],[22,38,0,3,0,19,38,3,3,38,19,0,3,0,38,22],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[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] >;
C23⋊F5 in GAP, Magma, Sage, TeX
C_2^3\rtimes F_5
% in TeX
G:=Group("C2^3:F5"); // GroupNames label
G:=SmallGroup(160,86);
// by ID
G=gap.SmallGroup(160,86);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,188,579,2309,1169]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^5=e^4=1,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