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## G = C2×C20order 40 = 23·5

### Abelian group of type [2,20]

Aliases: C2×C20, SmallGroup(40,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20
 Chief series C1 — C2 — C10 — C20 — C2×C20
 Lower central C1 — C2×C20
 Upper central C1 — C2×C20

Generators and relations for C2×C20
G = < a,b | a2=b20=1, ab=ba >

Smallest permutation representation of C2×C20
Regular action on 40 points
Generators in S40
(1 39)(2 40)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)
(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)| (1,39)(2,40)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38), (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( (1,39)(2,40)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38), (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([[(1,39),(2,40),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38)], [(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)]])

C2×C20 is a maximal subgroup of   C4.Dic5  C10.D4  C4⋊Dic5  D10⋊C4  C4○D20

40 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 10A ··· 10L 20A ··· 20P order 1 2 2 2 4 4 4 4 5 5 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 1 1 1 1 1 1 1 1 1 ··· 1 1 ··· 1

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C5 C10 C10 C20 kernel C2×C20 C20 C2×C10 C10 C2×C4 C4 C22 C2 # reps 1 2 1 4 4 8 4 16

Matrix representation of C2×C20 in GL2(𝔽41) generated by

 1 0 0 40
,
 21 0 0 5
G:=sub<GL(2,GF(41))| [1,0,0,40],[21,0,0,5] >;

C2×C20 in GAP, Magma, Sage, TeX

C_2\times C_{20}
% in TeX

G:=Group("C2xC20");
// GroupNames label

G:=SmallGroup(40,9);
// by ID

G=gap.SmallGroup(40,9);
# by ID

G:=PCGroup([4,-2,-2,-5,-2,80]);
// Polycyclic

G:=Group<a,b|a^2=b^20=1,a*b=b*a>;
// generators/relations

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