direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×D5.D5, D5⋊Dic5, D10.D5, C10⋊Dic5, C10⋊3F5, D5.2D10, C5⋊(C2×Dic5), C5⋊5(C2×F5), (C5×C10)⋊2C4, (C5×D5)⋊5C4, C52⋊4(C2×C4), (D5×C10).3C2, (C5×D5).3C22, SmallGroup(200,46)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C2×D5.D5 |
Generators and relations for C2×D5.D5
G = < a,b,c,d,e | a2=b5=c2=d5=1, e2=b-1c, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b2, cd=dc, ece-1=bc, ede-1=d-1 >
Character table of C2×D5.D5
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | |
| size | 1 | 1 | 5 | 5 | 25 | 25 | 25 | 25 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | -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 | -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 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | -2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | -2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | -2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | -2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ17 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | -1 | -1 | -1 | -1 | -1 | 4 | 4 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from F5 |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | -1 | -1 | -1 | -1 | -1 | -4 | -4 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×F5 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | -1 | ζ54+2ζ53+1 | 2ζ52+ζ5+1 | ζ53+2ζ5+1 | 2ζ54+ζ52+1 | 1-√5 | 1+√5 | -ζ53+ζ52+ζ5 | -ζ54+ζ53+ζ5 | 1 | ζ54+ζ52-ζ5 | ζ54+ζ53-ζ52 | 0 | 0 | 0 | 0 | complex faithful |
| ρ20 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | -1 | ζ54+2ζ53+1 | 2ζ52+ζ5+1 | ζ53+2ζ5+1 | 2ζ54+ζ52+1 | -1+√5 | -1-√5 | ζ54+2ζ53+1 | 2ζ54+ζ52+1 | -1 | ζ53+2ζ5+1 | 2ζ52+ζ5+1 | 0 | 0 | 0 | 0 | complex lifted from D5.D5 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | -1 | 2ζ54+ζ52+1 | ζ53+2ζ5+1 | ζ54+2ζ53+1 | 2ζ52+ζ5+1 | 1+√5 | 1-√5 | -ζ54+ζ53+ζ5 | ζ54+ζ53-ζ52 | 1 | -ζ53+ζ52+ζ5 | ζ54+ζ52-ζ5 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | -1 | 2ζ52+ζ5+1 | ζ54+2ζ53+1 | 2ζ54+ζ52+1 | ζ53+2ζ5+1 | 1-√5 | 1+√5 | ζ54+ζ53-ζ52 | ζ54+ζ52-ζ5 | 1 | -ζ54+ζ53+ζ5 | -ζ53+ζ52+ζ5 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | -1 | ζ53+2ζ5+1 | 2ζ54+ζ52+1 | 2ζ52+ζ5+1 | ζ54+2ζ53+1 | -1-√5 | -1+√5 | ζ53+2ζ5+1 | ζ54+2ζ53+1 | -1 | 2ζ52+ζ5+1 | 2ζ54+ζ52+1 | 0 | 0 | 0 | 0 | complex lifted from D5.D5 |
| ρ24 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | -1 | 2ζ52+ζ5+1 | ζ54+2ζ53+1 | 2ζ54+ζ52+1 | ζ53+2ζ5+1 | -1+√5 | -1-√5 | 2ζ52+ζ5+1 | ζ53+2ζ5+1 | -1 | 2ζ54+ζ52+1 | ζ54+2ζ53+1 | 0 | 0 | 0 | 0 | complex lifted from D5.D5 |
| ρ25 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | -1 | 2ζ54+ζ52+1 | ζ53+2ζ5+1 | ζ54+2ζ53+1 | 2ζ52+ζ5+1 | -1-√5 | -1+√5 | 2ζ54+ζ52+1 | 2ζ52+ζ5+1 | -1 | ζ54+2ζ53+1 | ζ53+2ζ5+1 | 0 | 0 | 0 | 0 | complex lifted from D5.D5 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | -1 | ζ53+2ζ5+1 | 2ζ54+ζ52+1 | 2ζ52+ζ5+1 | ζ54+2ζ53+1 | 1+√5 | 1-√5 | ζ54+ζ52-ζ5 | -ζ53+ζ52+ζ5 | 1 | ζ54+ζ53-ζ52 | -ζ54+ζ53+ζ5 | 0 | 0 | 0 | 0 | complex faithful |
►On 40 points
Generators in S40
(1 12)(2 13)(3 14)(4 15)(5 11)(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 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)
(1 7)(2 6)(3 10)(4 9)(5 8)(11 18)(12 17)(13 16)(14 20)(15 19)(21 28)(22 27)(23 26)(24 30)(25 29)(31 38)(32 37)(33 36)(34 40)(35 39)
(1 4 2 5 3)(6 8 10 7 9)(11 14 12 15 13)(16 18 20 17 19)(21 25 24 23 22)(26 27 28 29 30)(31 35 34 33 32)(36 37 38 39 40)
(1 40 8 35)(2 38 7 32)(3 36 6 34)(4 39 10 31)(5 37 9 33)(11 27 19 23)(12 30 18 25)(13 28 17 22)(14 26 16 24)(15 29 20 21)
G:=sub<Sym(40)| (1,12)(2,13)(3,14)(4,15)(5,11)(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), (1,2,3,4,5)(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), (1,7)(2,6)(3,10)(4,9)(5,8)(11,18)(12,17)(13,16)(14,20)(15,19)(21,28)(22,27)(23,26)(24,30)(25,29)(31,38)(32,37)(33,36)(34,40)(35,39), (1,4,2,5,3)(6,8,10,7,9)(11,14,12,15,13)(16,18,20,17,19)(21,25,24,23,22)(26,27,28,29,30)(31,35,34,33,32)(36,37,38,39,40), (1,40,8,35)(2,38,7,32)(3,36,6,34)(4,39,10,31)(5,37,9,33)(11,27,19,23)(12,30,18,25)(13,28,17,22)(14,26,16,24)(15,29,20,21)>;
G:=Group( (1,12)(2,13)(3,14)(4,15)(5,11)(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), (1,2,3,4,5)(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), (1,7)(2,6)(3,10)(4,9)(5,8)(11,18)(12,17)(13,16)(14,20)(15,19)(21,28)(22,27)(23,26)(24,30)(25,29)(31,38)(32,37)(33,36)(34,40)(35,39), (1,4,2,5,3)(6,8,10,7,9)(11,14,12,15,13)(16,18,20,17,19)(21,25,24,23,22)(26,27,28,29,30)(31,35,34,33,32)(36,37,38,39,40), (1,40,8,35)(2,38,7,32)(3,36,6,34)(4,39,10,31)(5,37,9,33)(11,27,19,23)(12,30,18,25)(13,28,17,22)(14,26,16,24)(15,29,20,21) );
G=PermutationGroup([[(1,12),(2,13),(3,14),(4,15),(5,11),(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)], [(1,2,3,4,5),(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)], [(1,7),(2,6),(3,10),(4,9),(5,8),(11,18),(12,17),(13,16),(14,20),(15,19),(21,28),(22,27),(23,26),(24,30),(25,29),(31,38),(32,37),(33,36),(34,40),(35,39)], [(1,4,2,5,3),(6,8,10,7,9),(11,14,12,15,13),(16,18,20,17,19),(21,25,24,23,22),(26,27,28,29,30),(31,35,34,33,32),(36,37,38,39,40)], [(1,40,8,35),(2,38,7,32),(3,36,6,34),(4,39,10,31),(5,37,9,33),(11,27,19,23),(12,30,18,25),(13,28,17,22),(14,26,16,24),(15,29,20,21)]])
C2×D5.D5 is a maximal subgroup of
Dic5×F5 D5.D20 D5.Dic10 C20⋊5F5 D10.D10 C2×D5×F5
C2×D5.D5 is a maximal quotient of C20.14F5 C20.12F5 C20⋊5F5 C102.C4 D10.D10
Matrix representation of C2×D5.D5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 16 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 10 |
| 0 | 18 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 |
| 0 | 0 | 37 | 0 |
| 37 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 31 |
| 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[16,0,0,0,0,18,0,0,0,0,37,0,0,0,0,10],[0,16,0,0,18,0,0,0,0,0,0,37,0,0,10,0],[37,0,0,0,0,37,0,0,0,0,10,0,0,0,0,10],[0,0,0,4,0,0,4,0,31,0,0,0,0,31,0,0] >;
C2×D5.D5 in GAP, Magma, Sage, TeX
C_2\times D_5.D_5
% in TeX
G:=Group("C2xD5.D5"); // GroupNames label
G:=SmallGroup(200,46);
// by ID
G=gap.SmallGroup(200,46);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,20,643,3004,1014]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^5=1,e^2=b^-1*c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^2,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations
Export
Subgroup lattice of C2×D5.D5 in TeX
Character table of C2×D5.D5 in TeX