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## G = C2×Dic5order 40 = 23·5

### Direct product of C2 and Dic5

Aliases: C2×Dic5, C102C4, C22.D5, C2.2D10, C10.4C22, C53(C2×C4), (C2×C10).C2, SmallGroup(40,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×Dic5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5
 Lower central C5 — C2×Dic5
 Upper central C1 — C22

Generators and relations for C2×Dic5
G = < a,b,c | a2=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

Character table of C2×Dic5

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 10A 10B 10C 10D 10E 10F size 1 1 1 1 5 5 5 5 2 2 2 2 2 2 2 2 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 -1 -1 1 i -i -i i 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 linear of order 4 ρ7 1 -1 -1 1 -i i i -i 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 linear of order 4 ρ9 2 2 2 2 0 0 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ10 2 2 -2 -2 0 0 0 0 -1-√5/2 -1+√5/2 1+√5/2 1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 1-√5/2 orthogonal lifted from D10 ρ11 2 2 2 2 0 0 0 0 -1+√5/2 -1-√5/2 -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 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 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 -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 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 -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

Smallest permutation representation of C2×Dic5
Regular action on 40 points
Generators in S40
(1 19)(2 20)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(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 26 6 21)(2 25 7 30)(3 24 8 29)(4 23 9 28)(5 22 10 27)(11 34 16 39)(12 33 17 38)(13 32 18 37)(14 31 19 36)(15 40 20 35)

G:=sub<Sym(40)| (1,19)(2,20)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(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), (1,26,6,21)(2,25,7,30)(3,24,8,29)(4,23,9,28)(5,22,10,27)(11,34,16,39)(12,33,17,38)(13,32,18,37)(14,31,19,36)(15,40,20,35)>;

G:=Group( (1,19)(2,20)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(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), (1,26,6,21)(2,25,7,30)(3,24,8,29)(4,23,9,28)(5,22,10,27)(11,34,16,39)(12,33,17,38)(13,32,18,37)(14,31,19,36)(15,40,20,35) );

G=PermutationGroup([(1,19),(2,20),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(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)], [(1,26,6,21),(2,25,7,30),(3,24,8,29),(4,23,9,28),(5,22,10,27),(11,34,16,39),(12,33,17,38),(13,32,18,37),(14,31,19,36),(15,40,20,35)])

Matrix representation of C2×Dic5 in GL4(𝔽41) generated by

 1 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 1 0 0 0 0 40 1 0 0 33 7
,
 9 0 0 0 0 40 0 0 0 0 0 35 0 0 34 0
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,40,33,0,0,1,7],[9,0,0,0,0,40,0,0,0,0,0,34,0,0,35,0] >;

C2×Dic5 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_5
% in TeX

G:=Group("C2xDic5");
// GroupNames label

G:=SmallGroup(40,7);
// by ID

G=gap.SmallGroup(40,7);
# by ID

G:=PCGroup([4,-2,-2,-2,-5,16,515]);
// Polycyclic

G:=Group<a,b,c|a^2=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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