metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊F5, C20⋊1C4, D5.Q8, D5.1D4, Dic5⋊3C4, D10.5C22, C5⋊(C4⋊C4), (C2×F5).C2, C2.5(C2×F5), C10.4(C2×C4), (C4×D5).4C2, SmallGroup(80,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊F5
G = < a,b,c | a4=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b3 >
Character table of C4⋊F5
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 10 | 20A | 20B | |
| size | 1 | 1 | 5 | 5 | 2 | 10 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | -1 | i | i | -i | -i | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -i | i | -i | i | -1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | i | -i | i | -i | -1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | -i | -i | i | i | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ11 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ12 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ13 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -√-5 | √-5 | complex faithful |
| ρ14 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | √-5 | -√-5 | complex faithful |
►On 20 points - transitive group 20T18
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(2 3 5 4)(7 8 10 9)(11 16)(12 18 15 19)(13 20 14 17)
G:=sub<Sym(20)| (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (2,3,5,4)(7,8,10,9)(11,16)(12,18,15,19)(13,20,14,17)>;
G:=Group( (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (2,3,5,4)(7,8,10,9)(11,16)(12,18,15,19)(13,20,14,17) );
G=PermutationGroup([[(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(2,3,5,4),(7,8,10,9),(11,16),(12,18,15,19),(13,20,14,17)]])
G:=TransitiveGroup(20,18);
C4⋊F5 is a maximal subgroup of
C40⋊C4 D5.D8 D20⋊C4 Q8⋊F5 D10.C23 D4×F5 Q8×F5 Dic3⋊F5 C60⋊C4 C100⋊C4 D5.Dic10 Dic5⋊F5 C20⋊5F5 C20⋊F5 C20⋊2F5 C4⋊S5 A5⋊Q8
C4⋊F5 is a maximal quotient of
C40⋊C4 D5.D8 C40.C4 D10.Q8 C20⋊C8 Dic5⋊C8 D10.3Q8 Dic3⋊F5 C60⋊C4 C100⋊C4 D5.Dic10 Dic5⋊F5 C20⋊5F5 C20⋊F5 C20⋊2F5
Matrix representation of C4⋊F5 ►in GL4(𝔽3) generated by
| 1 | 2 | 0 | 0 |
| 2 | 2 | 0 | 0 |
| 2 | 1 | 0 | 2 |
| 1 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 2 | 0 | 2 |
| 2 | 2 | 1 | 1 |
| 0 | 2 | 1 | 2 |
| 1 | 0 | 1 | 2 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 2 |
| 0 | 1 | 1 | 1 |
G:=sub<GL(4,GF(3))| [1,2,2,1,2,2,1,0,0,0,0,1,0,0,2,0],[0,0,2,0,0,2,2,2,1,0,1,1,1,2,1,2],[1,0,0,0,0,0,0,1,1,0,1,1,2,1,2,1] >;
C4⋊F5 in GAP, Magma, Sage, TeX
C_4\rtimes F_5
% in TeX
G:=Group("C4:F5"); // GroupNames label
G:=SmallGroup(80,31);
// by ID
G=gap.SmallGroup(80,31);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,20,101,46,804,414]);
// Polycyclic
G:=Group<a,b,c|a^4=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C4⋊F5 in TeX
Character table of C4⋊F5 in TeX