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G = C128order 128 = 27

Cyclic group

p-group, cyclic, abelian, monomial

Aliases: C128, also denoted Z128, SmallGroup(128,1)

Series: Derived Chief Lower central Upper central Jennings

C1 — C128
C1C2C4C8C16C32C64 — C128
C1 — C128
C1 — C128
C1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C8C8C8C8C8C8C8C8C16C16C16C16C32C32C64 — C128

Generators and relations for C128
 G = < a | a128=1 >


Smallest permutation representation of C128
Regular action on 128 points
Generators in S128
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)

G:=sub<Sym(128)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)])

128 conjugacy classes

class 1  2 4A4B8A8B8C8D16A···16H32A···32P64A···64AF128A···128BL
order1244888816···1632···3264···64128···128
size111111111···11···11···11···1

128 irreducible representations

dim11111111
type++
imageC1C2C4C8C16C32C64C128
kernelC128C64C32C16C8C4C2C1
# reps11248163264

Matrix representation of C128 in GL1(𝔽257) generated by

122
G:=sub<GL(1,GF(257))| [122] >;

C128 in GAP, Magma, Sage, TeX

C_{128}
% in TeX

G:=Group("C128");
// GroupNames label

G:=SmallGroup(128,1);
// by ID

G=gap.SmallGroup(128,1);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-2,14,36,58,80,102,124]);
// Polycyclic

G:=Group<a|a^128=1>;
// generators/relations

Export

Subgroup lattice of C128 in TeX

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