p-group, cyclic, abelian, monomial
Aliases: C128, also denoted Z128, SmallGroup(128,1)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C128 |
| C1 — C128 |
| C1 — C128 |
| C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C8 — C8 — C8 — C8 — C8 — C8 — C8 — C8 — C16 — C16 — C16 — C16 — C32 — C32 — C64 — 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 | 4A | 4B | 8A | 8B | 8C | 8D | 16A | ··· | 16H | 32A | ··· | 32P | 64A | ··· | 64AF | 128A | ··· | 128BL |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 32 | ··· | 32 | 64 | ··· | 64 | 128 | ··· | 128 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | C64 | C128 |
| kernel | C128 | C64 | C32 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 32 | 64 |
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
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