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## G = C22×C10order 40 = 23·5

### Abelian group of type [2,2,10]

Aliases: C22×C10, SmallGroup(40,14)

Series: Derived Chief Lower central Upper central

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

C22×C10 is a maximal subgroup of   C23.D5

40 conjugacy classes

 class 1 2A ··· 2G 5A 5B 5C 5D 10A ··· 10AB order 1 2 ··· 2 5 5 5 5 10 ··· 10 size 1 1 ··· 1 1 1 1 1 1 ··· 1

40 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C5 C10 kernel C22×C10 C2×C10 C23 C22 # reps 1 7 4 28

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

 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
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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