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## G = C2×D5.D5order 200 = 23·52

### Direct product of C2 and D5.D5

Aliases: C2×D5.D5, D5⋊Dic5, D10.D5, C10⋊Dic5, C103F5, D5.2D10, C5⋊(C2×Dic5), C55(C2×F5), (C5×C10)⋊2C4, (C5×D5)⋊5C4, C524(C2×C4), (D5×C10).3C2, (C5×D5).3C22, SmallGroup(200,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×D5.D5
 Chief series C1 — C5 — C52 — C5×D5 — D5.D5 — C2×D5.D5
 Lower central C52 — C2×D5.D5
 Upper central C1 — C2

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

Smallest permutation representation of C2×D5.D5
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  C205F5  D10.D10  C2×D5×F5
C2×D5.D5 is a maximal quotient of   C20.14F5  C20.12F5  C205F5  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

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