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G = C2×Dic5order 40 = 23·5

Direct product of C2 and Dic5

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C5 — C2×Dic5
C1C5C10Dic5 — C2×Dic5
C5 — C2×Dic5
C1C22

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

5C4
5C4
5C2×C4

Character table of C2×Dic5

 class 12A2B2C4A4B4C4D5A5B10A10B10C10D10E10F
 size 1111555522222222
ρ11111111111111111    trivial
ρ211-1-1-11-1111-1-111-1-1    linear of order 2
ρ311-1-11-11-111-1-111-1-1    linear of order 2
ρ41111-1-1-1-111111111    linear of order 2
ρ51-1-11i-i-ii11-11-1-11-1    linear of order 4
ρ61-11-1ii-i-i111-1-1-1-11    linear of order 4
ρ71-1-11-iii-i11-11-1-11-1    linear of order 4
ρ81-11-1-i-iii111-1-1-1-11    linear of order 4
ρ922220000-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5
ρ1022-2-20000-1-5/2-1+5/21+5/21+5/2-1+5/2-1-5/21-5/21-5/2    orthogonal lifted from D10
ρ1122220000-1+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
ρ1222-2-20000-1+5/2-1-5/21-5/21-5/2-1-5/2-1+5/21+5/21+5/2    orthogonal lifted from D10
ρ132-2-220000-1+5/2-1-5/21-5/2-1+5/21+5/21-5/2-1-5/21+5/2    symplectic lifted from Dic5, Schur index 2
ρ142-22-20000-1-5/2-1+5/2-1-5/21+5/21-5/21+5/21-5/2-1+5/2    symplectic lifted from Dic5, Schur index 2
ρ152-2-220000-1-5/2-1+5/21+5/2-1-5/21-5/21+5/2-1+5/21-5/2    symplectic lifted from Dic5, Schur index 2
ρ162-22-20000-1+5/2-1-5/2-1+5/21-5/21+5/21-5/21+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

1000
04000
0010
0001
,
40000
0100
00401
00337
,
9000
04000
00035
00340
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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