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G = C136order 136 = 23·17

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C136, also denoted Z136, SmallGroup(136,2)

Series: Derived Chief Lower central Upper central

C1 — C136
C1C2C4C68 — C136
C1 — C136
C1 — C136

Generators and relations for C136
 G = < a | a136=1 >


Smallest permutation representation of C136
Regular action on 136 points
Generators in S136
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136)

G:=sub<Sym(136)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136)])

C136 is a maximal subgroup of   C174C16  C8⋊D17  C136⋊C2  D136  Dic68

136 conjugacy classes

class 1  2 4A4B8A8B8C8D17A···17P34A···34P68A···68AF136A···136BL
order1244888817···1734···3468···68136···136
size111111111···11···11···11···1

136 irreducible representations

dim11111111
type++
imageC1C2C4C8C17C34C68C136
kernelC136C68C34C17C8C4C2C1
# reps112416163264

Matrix representation of C136 in GL1(𝔽137) generated by

21
G:=sub<GL(1,GF(137))| [21] >;

C136 in GAP, Magma, Sage, TeX

C_{136}
% in TeX

G:=Group("C136");
// GroupNames label

G:=SmallGroup(136,2);
// by ID

G=gap.SmallGroup(136,2);
# by ID

G:=PCGroup([4,-2,-17,-2,-2,136,34]);
// Polycyclic

G:=Group<a|a^136=1>;
// generators/relations

Export

Subgroup lattice of C136 in TeX

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