direct product, abelian, monomial, 3-elementary
Aliases: C3×C69, SmallGroup(207,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C69 |
| C1 — C3×C69 |
| C1 — C3×C69 |
Generators and relations for C3×C69
G = < a,b | a3=b69=1, ab=ba >
Smallest permutation representation of C3×C69
►Regular action on 207 points
Generators in S207
(1 182 70)(2 183 71)(3 184 72)(4 185 73)(5 186 74)(6 187 75)(7 188 76)(8 189 77)(9 190 78)(10 191 79)(11 192 80)(12 193 81)(13 194 82)(14 195 83)(15 196 84)(16 197 85)(17 198 86)(18 199 87)(19 200 88)(20 201 89)(21 202 90)(22 203 91)(23 204 92)(24 205 93)(25 206 94)(26 207 95)(27 139 96)(28 140 97)(29 141 98)(30 142 99)(31 143 100)(32 144 101)(33 145 102)(34 146 103)(35 147 104)(36 148 105)(37 149 106)(38 150 107)(39 151 108)(40 152 109)(41 153 110)(42 154 111)(43 155 112)(44 156 113)(45 157 114)(46 158 115)(47 159 116)(48 160 117)(49 161 118)(50 162 119)(51 163 120)(52 164 121)(53 165 122)(54 166 123)(55 167 124)(56 168 125)(57 169 126)(58 170 127)(59 171 128)(60 172 129)(61 173 130)(62 174 131)(63 175 132)(64 176 133)(65 177 134)(66 178 135)(67 179 136)(68 180 137)(69 181 138)
(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)(139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207)
G:=sub<Sym(207)| (1,182,70)(2,183,71)(3,184,72)(4,185,73)(5,186,74)(6,187,75)(7,188,76)(8,189,77)(9,190,78)(10,191,79)(11,192,80)(12,193,81)(13,194,82)(14,195,83)(15,196,84)(16,197,85)(17,198,86)(18,199,87)(19,200,88)(20,201,89)(21,202,90)(22,203,91)(23,204,92)(24,205,93)(25,206,94)(26,207,95)(27,139,96)(28,140,97)(29,141,98)(30,142,99)(31,143,100)(32,144,101)(33,145,102)(34,146,103)(35,147,104)(36,148,105)(37,149,106)(38,150,107)(39,151,108)(40,152,109)(41,153,110)(42,154,111)(43,155,112)(44,156,113)(45,157,114)(46,158,115)(47,159,116)(48,160,117)(49,161,118)(50,162,119)(51,163,120)(52,164,121)(53,165,122)(54,166,123)(55,167,124)(56,168,125)(57,169,126)(58,170,127)(59,171,128)(60,172,129)(61,173,130)(62,174,131)(63,175,132)(64,176,133)(65,177,134)(66,178,135)(67,179,136)(68,180,137)(69,181,138), (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)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207)>;
G:=Group( (1,182,70)(2,183,71)(3,184,72)(4,185,73)(5,186,74)(6,187,75)(7,188,76)(8,189,77)(9,190,78)(10,191,79)(11,192,80)(12,193,81)(13,194,82)(14,195,83)(15,196,84)(16,197,85)(17,198,86)(18,199,87)(19,200,88)(20,201,89)(21,202,90)(22,203,91)(23,204,92)(24,205,93)(25,206,94)(26,207,95)(27,139,96)(28,140,97)(29,141,98)(30,142,99)(31,143,100)(32,144,101)(33,145,102)(34,146,103)(35,147,104)(36,148,105)(37,149,106)(38,150,107)(39,151,108)(40,152,109)(41,153,110)(42,154,111)(43,155,112)(44,156,113)(45,157,114)(46,158,115)(47,159,116)(48,160,117)(49,161,118)(50,162,119)(51,163,120)(52,164,121)(53,165,122)(54,166,123)(55,167,124)(56,168,125)(57,169,126)(58,170,127)(59,171,128)(60,172,129)(61,173,130)(62,174,131)(63,175,132)(64,176,133)(65,177,134)(66,178,135)(67,179,136)(68,180,137)(69,181,138), (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)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207) );
G=PermutationGroup([[(1,182,70),(2,183,71),(3,184,72),(4,185,73),(5,186,74),(6,187,75),(7,188,76),(8,189,77),(9,190,78),(10,191,79),(11,192,80),(12,193,81),(13,194,82),(14,195,83),(15,196,84),(16,197,85),(17,198,86),(18,199,87),(19,200,88),(20,201,89),(21,202,90),(22,203,91),(23,204,92),(24,205,93),(25,206,94),(26,207,95),(27,139,96),(28,140,97),(29,141,98),(30,142,99),(31,143,100),(32,144,101),(33,145,102),(34,146,103),(35,147,104),(36,148,105),(37,149,106),(38,150,107),(39,151,108),(40,152,109),(41,153,110),(42,154,111),(43,155,112),(44,156,113),(45,157,114),(46,158,115),(47,159,116),(48,160,117),(49,161,118),(50,162,119),(51,163,120),(52,164,121),(53,165,122),(54,166,123),(55,167,124),(56,168,125),(57,169,126),(58,170,127),(59,171,128),(60,172,129),(61,173,130),(62,174,131),(63,175,132),(64,176,133),(65,177,134),(66,178,135),(67,179,136),(68,180,137),(69,181,138)], [(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),(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207)]])
C3×C69 is a maximal subgroup of
C3⋊D69
207 conjugacy classes
| class | 1 | 3A | ··· | 3H | 23A | ··· | 23V | 69A | ··· | 69FT |
| order | 1 | 3 | ··· | 3 | 23 | ··· | 23 | 69 | ··· | 69 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
207 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C23 | C69 |
| kernel | C3×C69 | C69 | C32 | C3 |
| # reps | 1 | 8 | 22 | 176 |
Matrix representation of C3×C69 ►in GL2(𝔽139) generated by
| 42 | 0 |
| 0 | 42 |
| 45 | 0 |
| 0 | 136 |
G:=sub<GL(2,GF(139))| [42,0,0,42],[45,0,0,136] >;
C3×C69 in GAP, Magma, Sage, TeX
C_3\times C_{69} % in TeX
G:=Group("C3xC69"); // GroupNames label
G:=SmallGroup(207,2);
// by ID
G=gap.SmallGroup(207,2);
# by ID
G:=PCGroup([3,-3,-3,-23]);
// Polycyclic
G:=Group<a,b|a^3=b^69=1,a*b=b*a>;
// generators/relations
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