direct product, cyclic, abelian, monomial
Aliases: C135, also denoted Z135, SmallGroup(135,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C135 |
| C1 — C135 |
| C1 — C135 |
Generators and relations for C135
G = < a | a135=1 >
Smallest permutation representation of C135
►Regular action on 135 points
Generators in S135
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135)
G:=sub<Sym(135)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135)]])
C135 is a maximal subgroup of
D135
135 conjugacy classes
| class | 1 | 3A | 3B | 5A | 5B | 5C | 5D | 9A | ··· | 9F | 15A | ··· | 15H | 27A | ··· | 27R | 45A | ··· | 45X | 135A | ··· | 135BT |
| order | 1 | 3 | 3 | 5 | 5 | 5 | 5 | 9 | ··· | 9 | 15 | ··· | 15 | 27 | ··· | 27 | 45 | ··· | 45 | 135 | ··· | 135 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||||
| image | C1 | C3 | C5 | C9 | C15 | C27 | C45 | C135 |
| kernel | C135 | C45 | C27 | C15 | C9 | C5 | C3 | C1 |
| # reps | 1 | 2 | 4 | 6 | 8 | 18 | 24 | 72 |
Matrix representation of C135 ►in GL1(𝔽271) generated by
| 163 |
G:=sub<GL(1,GF(271))| [163] >;
C135 in GAP, Magma, Sage, TeX
C_{135} % in TeX
G:=Group("C135"); // GroupNames label
G:=SmallGroup(135,1);
// by ID
G=gap.SmallGroup(135,1);
# by ID
G:=PCGroup([4,-3,-5,-3,-3,60,46]);
// Polycyclic
G:=Group<a|a^135=1>;
// generators/relations
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