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G = C108order 108 = 22·33

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C108, also denoted Z108, SmallGroup(108,2)

Series: Derived Chief Lower central Upper central

C1 — C108
C1C3C9C18C54 — C108
C1 — C108
C1 — C108

Generators and relations for C108
 G = < a | a108=1 >


Smallest permutation representation of C108
Regular action on 108 points
Generators in S108
(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)

G:=sub<Sym(108)| (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)>;

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) );

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)])

C108 is a maximal subgroup of   C27⋊C8  Dic54  D108  Q8.C54

108 conjugacy classes

class 1  2 3A3B4A4B6A6B9A···9F12A12B12C12D18A···18F27A···27R36A···36L54A···54R108A···108AJ
order123344669···91212121218···1827···2736···3654···54108···108
size111111111···111111···11···11···11···11···1

108 irreducible representations

dim111111111111
type++
imageC1C2C3C4C6C9C12C18C27C36C54C108
kernelC108C54C36C27C18C12C9C6C4C3C2C1
# reps1122264618121836

Matrix representation of C108 in GL2(𝔽109) generated by

1010
097
G:=sub<GL(2,GF(109))| [101,0,0,97] >;

C108 in GAP, Magma, Sage, TeX

C_{108}
% in TeX

G:=Group("C108");
// GroupNames label

G:=SmallGroup(108,2);
// by ID

G=gap.SmallGroup(108,2);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-3,30,66,78]);
// Polycyclic

G:=Group<a|a^108=1>;
// generators/relations

Export

Subgroup lattice of C108 in TeX

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