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## G = C174order 174 = 2·3·29

### Cyclic group

Aliases: C174, also denoted Z174, SmallGroup(174,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C174
 Chief series C1 — C29 — C87 — C174
 Lower central C1 — C174
 Upper central C1 — C174

Generators and relations for C174
G = < a | a174=1 >

Smallest permutation representation of C174
Regular action on 174 points
Generators in S174
`(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174)`

`G:=sub<Sym(174)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174)]])`

C174 is a maximal subgroup of   Dic87

174 conjugacy classes

 class 1 2 3A 3B 6A 6B 29A ··· 29AB 58A ··· 58AB 87A ··· 87BD 174A ··· 174BD order 1 2 3 3 6 6 29 ··· 29 58 ··· 58 87 ··· 87 174 ··· 174 size 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

174 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C6 C29 C58 C87 C174 kernel C174 C87 C58 C29 C6 C3 C2 C1 # reps 1 1 2 2 28 28 56 56

Matrix representation of C174 in GL1(𝔽349) generated by

 204
`G:=sub<GL(1,GF(349))| [204] >;`

C174 in GAP, Magma, Sage, TeX

`C_{174}`
`% in TeX`

`G:=Group("C174");`
`// GroupNames label`

`G:=SmallGroup(174,4);`
`// by ID`

`G=gap.SmallGroup(174,4);`
`# by ID`

`G:=PCGroup([3,-2,-3,-29]);`
`// Polycyclic`

`G:=Group<a|a^174=1>;`
`// generators/relations`

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