metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.F5, C5⋊2M4(2), Dic5.3C4, Dic5.7C22, C5⋊C8⋊2C2, C2.6(C2×F5), C10.6(C2×C4), (C2×C10).2C4, (C2×Dic5).5C2, SmallGroup(80,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.F5
G = < a,b,c,d | a2=b2=c5=1, d4=b, dad-1=ab=ba, ac=ca, bc=cb, bd=db, dcd-1=c3 >
Character table of C22.F5
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 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 | -2i | 2i | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | complex lifted from M4(2) |
| ρ10 | 2 | -2 | 0 | 2i | -2i | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | complex lifted from M4(2) |
| ρ11 | 4 | 4 | -4 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | orthogonal lifted from C2×F5 |
| ρ12 | 4 | 4 | 4 | 0 | 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 | -√5 | 1 | √5 | symplectic faithful, Schur index 2 |
| ρ14 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | √5 | 1 | -√5 | symplectic faithful, Schur index 2 |
►On 40 points
(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 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)| (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), (1,24,32,9,35)(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( (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), (1,24,32,9,35)(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([[(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)], [(1,24,32,9,35),(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)]])
C22.F5 is a maximal subgroup of
Dic5.D4 C23.F5 D5⋊M4(2) D4.F5 D6.F5 C15⋊8M4(2) C25⋊M4(2) D10.F5 C52⋊4M4(2) C102.C4 C52⋊13M4(2) C52⋊14M4(2) C22.S5
C22.F5 is a maximal quotient of
C10.C42 Dic5⋊C8 C23.2F5 D6.F5 C15⋊8M4(2) C25⋊M4(2) D10.F5 C52⋊4M4(2) C102.C4 C52⋊13M4(2) C52⋊14M4(2)
Matrix representation of C22.F5 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 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 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 |
| 0 | 0 | 6 | 40 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 23 | 6 | 0 | 0 |
| 21 | 18 | 0 | 0 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[6,1,0,0,40,0,0,0,0,0,35,6,0,0,35,40],[0,0,23,21,0,0,6,18,1,0,0,0,0,1,0,0] >;
C22.F5 in GAP, Magma, Sage, TeX
C_2^2.F_5
% in TeX
G:=Group("C2^2.F5"); // GroupNames label
G:=SmallGroup(80,33);
// by ID
G=gap.SmallGroup(80,33);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,20,101,42,804,414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^5=1,d^4=b,d*a*d^-1=a*b=b*a,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations
Export
Subgroup lattice of C22.F5 in TeX
Character table of C22.F5 in TeX