Copied to
clipboard

G = C168order 168 = 23·3·7

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C168, also denoted Z168, SmallGroup(168,6)

Series: Derived Chief Lower central Upper central

C1 — C168
C1C2C4C28C84 — C168
C1 — C168
C1 — C168

Generators and relations for C168
 G = < a | a168=1 >


Smallest permutation representation of C168
Regular action on 168 points
Generators in S168
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)

G:=sub<Sym(168)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)])

C168 is a maximal subgroup of   C21⋊C16  C56⋊S3  C8⋊D21  D168  Dic84

168 conjugacy classes

class 1  2 3A3B4A4B6A6B7A···7F8A8B8C8D12A12B12C12D14A···14F21A···21L24A···24H28A···28L42A···42L56A···56X84A···84X168A···168AV
order123344667···788881212121214···1421···2124···2428···2842···4256···5684···84168···168
size111111111···1111111111···11···11···11···11···11···11···11···1

168 irreducible representations

dim1111111111111111
type++
imageC1C2C3C4C6C7C8C12C14C21C24C28C42C56C84C168
kernelC168C84C56C42C28C24C21C14C12C8C7C6C4C3C2C1
# reps1122264461281212242448

Matrix representation of C168 in GL1(𝔽337) generated by

334
G:=sub<GL(1,GF(337))| [334] >;

C168 in GAP, Magma, Sage, TeX

C_{168}
% in TeX

G:=Group("C168");
// GroupNames label

G:=SmallGroup(168,6);
// by ID

G=gap.SmallGroup(168,6);
# by ID

G:=PCGroup([5,-2,-3,-7,-2,-2,210,58]);
// Polycyclic

G:=Group<a|a^168=1>;
// generators/relations

Export

Subgroup lattice of C168 in TeX

׿
×
𝔽