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G = C88order 88 = 23·11

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C88, also denoted Z88, SmallGroup(88,2)

Series: Derived Chief Lower central Upper central

C1 — C88
C1C2C4C44 — C88
C1 — C88
C1 — C88

Generators and relations for C88
 G = < a | a88=1 >


Smallest permutation representation of C88
Regular action on 88 points
Generators in S88
(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)

G:=sub<Sym(88)| (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)>;

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) );

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)])

C88 is a maximal subgroup of   C11⋊C16  C88⋊C2  C8⋊D11  D88  Dic44

88 conjugacy classes

class 1  2 4A4B8A8B8C8D11A···11J22A···22J44A···44T88A···88AN
order1244888811···1122···2244···4488···88
size111111111···11···11···11···1

88 irreducible representations

dim11111111
type++
imageC1C2C4C8C11C22C44C88
kernelC88C44C22C11C8C4C2C1
# reps112410102040

Matrix representation of C88 in GL1(𝔽89) generated by

74
G:=sub<GL(1,GF(89))| [74] >;

C88 in GAP, Magma, Sage, TeX

C_{88}
% in TeX

G:=Group("C88");
// GroupNames label

G:=SmallGroup(88,2);
// by ID

G=gap.SmallGroup(88,2);
# by ID

G:=PCGroup([4,-2,-11,-2,-2,88,34]);
// Polycyclic

G:=Group<a|a^88=1>;
// generators/relations

Export

Subgroup lattice of C88 in TeX

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