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G = C110order 110 = 2·5·11

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C110, also denoted Z110, SmallGroup(110,6)

Series: Derived Chief Lower central Upper central

C1 — C110
C1C11C55 — C110
C1 — C110
C1 — C110

Generators and relations for C110
 G = < a | a110=1 >


Smallest permutation representation of C110
Regular action on 110 points
Generators in S110
(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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110)

G:=sub<Sym(110)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110)])

C110 is a maximal subgroup of   Dic55

110 conjugacy classes

class 1  2 5A5B5C5D10A10B10C10D11A···11J22A···22J55A···55AN110A···110AN
order1255551010101011···1122···2255···55110···110
size11111111111···11···11···11···1

110 irreducible representations

dim11111111
type++
imageC1C2C5C10C11C22C55C110
kernelC110C55C22C11C10C5C2C1
# reps114410104040

Matrix representation of C110 in GL1(𝔽331) generated by

282
G:=sub<GL(1,GF(331))| [282] >;

C110 in GAP, Magma, Sage, TeX

C_{110}
% in TeX

G:=Group("C110");
// GroupNames label

G:=SmallGroup(110,6);
// by ID

G=gap.SmallGroup(110,6);
# by ID

G:=PCGroup([3,-2,-5,-11]);
// Polycyclic

G:=Group<a|a^110=1>;
// generators/relations

Export

Subgroup lattice of C110 in TeX

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