direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×F5, C5⋊C42, C20⋊2C4, Dic5⋊2C4, D10.4C22, D5.(C2×C4), C2.2(C2×F5), C10.3(C2×C4), (C4×D5).6C2, (C2×F5).2C2, SmallGroup(80,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C4×F5 |
Generators and relations for C4×F5
G = < a,b,c | a4=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Character table of C4×F5
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5 | 10 | 20A | 20B | |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 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 | trivial |
| ρ2 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | -i | i | 1 | -i | i | -i | -i | i | 1 | -1 | -1 | i | 1 | -1 | i | -i | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | i | -i | i | -i | -1 | 1 | -1 | 1 | -i | i | -i | i | 1 | -1 | -i | i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | i | -i | 1 | i | -i | i | i | -i | 1 | -1 | -1 | -i | 1 | -1 | -i | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | 1 | i | -1 | -i | -i | i | i | -i | -i | i | -1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 1 | -1 | -1 | 1 | i | -i | -i | -i | 1 | -1 | 1 | -1 | i | -i | i | i | 1 | -1 | -i | i | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | -1 | -1 | i | 1 | i | i | -i | -i | -i | -i | i | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ11 | 1 | -1 | -1 | 1 | -i | i | i | i | 1 | -1 | 1 | -1 | -i | i | -i | -i | 1 | -1 | i | -i | linear of order 4 |
| ρ12 | 1 | 1 | -1 | -1 | 1 | 1 | -i | -1 | i | i | -i | -i | i | i | -i | -1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | -1 | -1 | -i | 1 | -i | -i | i | i | i | i | -i | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | -i | i | -1 | -i | -i | i | i | -i | -1 | 1 | 1 | i | 1 | -1 | i | -i | linear of order 4 |
| ρ15 | 1 | -1 | -1 | 1 | -i | i | -i | i | -1 | 1 | -1 | 1 | i | -i | i | -i | 1 | -1 | i | -i | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | i | -i | -1 | i | i | -i | -i | i | -1 | 1 | 1 | -i | 1 | -1 | -i | i | linear of order 4 |
| ρ17 | 4 | 4 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ18 | 4 | 4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ19 | 4 | -4 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | i | -i | complex faithful |
| ρ20 | 4 | -4 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -i | i | complex faithful |
►On 20 points - transitive group 20T20
Generators in S20
(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)
(1 11 6 16)(2 13 10 19)(3 15 9 17)(4 12 8 20)(5 14 7 18)
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), (1,11,6,16)(2,13,10,19)(3,15,9,17)(4,12,8,20)(5,14,7,18)>;
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), (1,11,6,16)(2,13,10,19)(3,15,9,17)(4,12,8,20)(5,14,7,18) );
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)], [(1,11,6,16),(2,13,10,19),(3,15,9,17),(4,12,8,20),(5,14,7,18)]])
G:=TransitiveGroup(20,20);
C4×F5 is a maximal subgroup of
C8⋊F5 D4⋊F5 Q8⋊2F5 D10.C23 C52⋊3C42 GL2(𝔽5)
C4×F5 is a maximal quotient of C8⋊F5 C10.C42 D10.3Q8 C52⋊3C42
Matrix representation of C4×F5 ►in GL5(𝔽41)
| 9 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 1 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 | 0 |
G:=sub<GL(5,GF(41))| [9,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,40,40,40,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,40,0,0,0,0,0,0,40,0] >;
C4×F5 in GAP, Magma, Sage, TeX
C_4\times F_5
% in TeX
G:=Group("C4xF5"); // GroupNames label
G:=SmallGroup(80,30);
// by ID
G=gap.SmallGroup(80,30);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,20,46,804,414]);
// Polycyclic
G:=Group<a,b,c|a^4=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C4×F5 in TeX
Character table of C4×F5 in TeX