direct product, cyclic, abelian, monomial
Aliases: C168, also denoted Z168, SmallGroup(168,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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 | 3A | 3B | 4A | 4B | 6A | 6B | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 14A | ··· | 14F | 21A | ··· | 21L | 24A | ··· | 24H | 28A | ··· | 28L | 42A | ··· | 42L | 56A | ··· | 56X | 84A | ··· | 84X | 168A | ··· | 168AV |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 24 | ··· | 24 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
168 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C7 | C8 | C12 | C14 | C21 | C24 | C28 | C42 | C56 | C84 | C168 |
| kernel | C168 | C84 | C56 | C42 | C28 | C24 | C21 | C14 | C12 | C8 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 4 | 6 | 12 | 8 | 12 | 12 | 24 | 24 | 48 |
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
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