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G = C22×C10order 40 = 23·5

Abelian group of type [2,2,10]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C10, SmallGroup(40,14)

Series: Derived Chief Lower central Upper central

C1 — C22×C10
C1C5C10C2×C10 — C22×C10
C1 — C22×C10
C1 — C22×C10

Generators and relations for C22×C10
 G = < a,b,c | a2=b2=c10=1, ab=ba, ac=ca, bc=cb >


Smallest permutation representation of C22×C10
Regular action on 40 points
Generators in S40
(1 27)(2 28)(3 29)(4 30)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)
(1 17)(2 18)(3 19)(4 20)(5 11)(6 12)(7 13)(8 14)(9 15)(10 16)(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)

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

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

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

40 conjugacy classes

class 1 2A···2G5A5B5C5D10A···10AB
order12···2555510···10
size11···111111···1

40 irreducible representations

dim1111
type++
imageC1C2C5C10
kernelC22×C10C2×C10C23C22
# reps17428

Matrix representation of C22×C10 in GL3(𝔽11) generated by

1000
010
001
,
1000
0100
001
,
100
010
002
G:=sub<GL(3,GF(11))| [10,0,0,0,1,0,0,0,1],[10,0,0,0,10,0,0,0,1],[1,0,0,0,1,0,0,0,2] >;

C22×C10 in GAP, Magma, Sage, TeX

C_2^2\times C_{10}
% in TeX

G:=Group("C2^2xC10");
// GroupNames label

G:=SmallGroup(40,14);
// by ID

G=gap.SmallGroup(40,14);
# by ID

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

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

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