metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C4.F5, C20.1C4, C5⋊1M4(2), D10.3C4, Dic5.5C22, C5⋊C8⋊1C2, C2.4(C2×F5), C10.2(C2×C4), (C4×D5).3C2, SmallGroup(80,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.F5
G = < a,b,c | a4=b5=1, c4=a2, ab=ba, cac-1=a-1, cbc-1=b3 >
Character table of C4.F5
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5 | 8A | 8B | 8C | 8D | 10 | 20A | 20B | |
| size | 1 | 1 | 10 | 2 | 5 | 5 | 4 | 10 | 10 | 10 | 10 | 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 | -1 | 1 | i | -i | -i | i | 1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -i | -i | i | i | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | i | i | -i | -i | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -i | i | i | -i | 1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | -2 | 0 | 0 | -2i | 2i | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | complex lifted from M4(2) |
| ρ10 | 2 | -2 | 0 | 0 | 2i | -2i | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | complex lifted from M4(2) |
| ρ11 | 4 | 4 | 0 | -4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ12 | 4 | 4 | 0 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ13 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | √-5 | -√-5 | complex faithful |
| ρ14 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -√-5 | √-5 | complex faithful |
►On 40 points
(1 7 5 3)(2 4 6 8)(9 30 13 26)(10 27 14 31)(11 32 15 28)(12 29 16 25)(17 38 21 34)(18 35 22 39)(19 40 23 36)(20 37 24 33)
(1 24 32 9 35)(2 10 17 36 25)(3 37 11 26 18)(4 27 38 19 12)(5 20 28 13 39)(6 14 21 40 29)(7 33 15 30 22)(8 31 34 23 16)
(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)| (1,7,5,3)(2,4,6,8)(9,30,13,26)(10,27,14,31)(11,32,15,28)(12,29,16,25)(17,38,21,34)(18,35,22,39)(19,40,23,36)(20,37,24,33), (1,24,32,9,35)(2,10,17,36,25)(3,37,11,26,18)(4,27,38,19,12)(5,20,28,13,39)(6,14,21,40,29)(7,33,15,30,22)(8,31,34,23,16), (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( (1,7,5,3)(2,4,6,8)(9,30,13,26)(10,27,14,31)(11,32,15,28)(12,29,16,25)(17,38,21,34)(18,35,22,39)(19,40,23,36)(20,37,24,33), (1,24,32,9,35)(2,10,17,36,25)(3,37,11,26,18)(4,27,38,19,12)(5,20,28,13,39)(6,14,21,40,29)(7,33,15,30,22)(8,31,34,23,16), (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([[(1,7,5,3),(2,4,6,8),(9,30,13,26),(10,27,14,31),(11,32,15,28),(12,29,16,25),(17,38,21,34),(18,35,22,39),(19,40,23,36),(20,37,24,33)], [(1,24,32,9,35),(2,10,17,36,25),(3,37,11,26,18),(4,27,38,19,12),(5,20,28,13,39),(6,14,21,40,29),(7,33,15,30,22),(8,31,34,23,16)], [(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)]])
C4.F5 is a maximal subgroup of
C40.C4 D10.Q8 D4⋊F5 Q8⋊2F5 D5⋊M4(2) D4.F5 Q8.F5 Dic3.F5 C12.F5 C100.C4 Dic5.F5 C52⋊4M4(2) C20.12F5 C52⋊7M4(2) C52⋊8M4(2) C4.S5 C4.3S5
C4.F5 is a maximal quotient of
C20⋊C8 C10.C42 D10⋊C8 Dic3.F5 C12.F5 C100.C4 Dic5.F5 C52⋊4M4(2) C20.12F5 C52⋊7M4(2) C52⋊8M4(2)
Matrix representation of C4.F5 ►in GL4(𝔽3) generated by
| 2 | 2 | 0 | 2 |
| 0 | 1 | 2 | 1 |
| 2 | 2 | 0 | 1 |
| 2 | 0 | 1 | 0 |
| 1 | 0 | 0 | 2 |
| 1 | 0 | 0 | 1 |
| 1 | 2 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 1 | 1 |
| 2 | 0 | 2 | 0 |
| 1 | 1 | 0 | 1 |
G:=sub<GL(4,GF(3))| [2,0,2,2,2,1,2,0,0,2,0,1,2,1,1,0],[1,1,1,1,0,0,2,1,0,0,0,1,2,1,0,1],[0,0,2,1,0,0,0,1,2,1,2,0,0,1,0,1] >;
C4.F5 in GAP, Magma, Sage, TeX
C_4.F_5
% in TeX
G:=Group("C4.F5"); // GroupNames label
G:=SmallGroup(80,29);
// by ID
G=gap.SmallGroup(80,29);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,20,101,46,42,804,414]);
// Polycyclic
G:=Group<a,b,c|a^4=b^5=1,c^4=a^2,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