direct product, abelian, monomial, 3-elementary
Aliases: C3×C75, SmallGroup(225,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C75 |
| C1 — C3×C75 |
| C1 — C3×C75 |
Generators and relations for C3×C75
G = < a,b | a3=b75=1, ab=ba >
Smallest permutation representation of C3×C75
►Regular action on 225 points
Generators in S225
(1 193 111)(2 194 112)(3 195 113)(4 196 114)(5 197 115)(6 198 116)(7 199 117)(8 200 118)(9 201 119)(10 202 120)(11 203 121)(12 204 122)(13 205 123)(14 206 124)(15 207 125)(16 208 126)(17 209 127)(18 210 128)(19 211 129)(20 212 130)(21 213 131)(22 214 132)(23 215 133)(24 216 134)(25 217 135)(26 218 136)(27 219 137)(28 220 138)(29 221 139)(30 222 140)(31 223 141)(32 224 142)(33 225 143)(34 151 144)(35 152 145)(36 153 146)(37 154 147)(38 155 148)(39 156 149)(40 157 150)(41 158 76)(42 159 77)(43 160 78)(44 161 79)(45 162 80)(46 163 81)(47 164 82)(48 165 83)(49 166 84)(50 167 85)(51 168 86)(52 169 87)(53 170 88)(54 171 89)(55 172 90)(56 173 91)(57 174 92)(58 175 93)(59 176 94)(60 177 95)(61 178 96)(62 179 97)(63 180 98)(64 181 99)(65 182 100)(66 183 101)(67 184 102)(68 185 103)(69 186 104)(70 187 105)(71 188 106)(72 189 107)(73 190 108)(74 191 109)(75 192 110)
(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)(151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225)
G:=sub<Sym(225)| (1,193,111)(2,194,112)(3,195,113)(4,196,114)(5,197,115)(6,198,116)(7,199,117)(8,200,118)(9,201,119)(10,202,120)(11,203,121)(12,204,122)(13,205,123)(14,206,124)(15,207,125)(16,208,126)(17,209,127)(18,210,128)(19,211,129)(20,212,130)(21,213,131)(22,214,132)(23,215,133)(24,216,134)(25,217,135)(26,218,136)(27,219,137)(28,220,138)(29,221,139)(30,222,140)(31,223,141)(32,224,142)(33,225,143)(34,151,144)(35,152,145)(36,153,146)(37,154,147)(38,155,148)(39,156,149)(40,157,150)(41,158,76)(42,159,77)(43,160,78)(44,161,79)(45,162,80)(46,163,81)(47,164,82)(48,165,83)(49,166,84)(50,167,85)(51,168,86)(52,169,87)(53,170,88)(54,171,89)(55,172,90)(56,173,91)(57,174,92)(58,175,93)(59,176,94)(60,177,95)(61,178,96)(62,179,97)(63,180,98)(64,181,99)(65,182,100)(66,183,101)(67,184,102)(68,185,103)(69,186,104)(70,187,105)(71,188,106)(72,189,107)(73,190,108)(74,191,109)(75,192,110), (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)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225)>;
G:=Group( (1,193,111)(2,194,112)(3,195,113)(4,196,114)(5,197,115)(6,198,116)(7,199,117)(8,200,118)(9,201,119)(10,202,120)(11,203,121)(12,204,122)(13,205,123)(14,206,124)(15,207,125)(16,208,126)(17,209,127)(18,210,128)(19,211,129)(20,212,130)(21,213,131)(22,214,132)(23,215,133)(24,216,134)(25,217,135)(26,218,136)(27,219,137)(28,220,138)(29,221,139)(30,222,140)(31,223,141)(32,224,142)(33,225,143)(34,151,144)(35,152,145)(36,153,146)(37,154,147)(38,155,148)(39,156,149)(40,157,150)(41,158,76)(42,159,77)(43,160,78)(44,161,79)(45,162,80)(46,163,81)(47,164,82)(48,165,83)(49,166,84)(50,167,85)(51,168,86)(52,169,87)(53,170,88)(54,171,89)(55,172,90)(56,173,91)(57,174,92)(58,175,93)(59,176,94)(60,177,95)(61,178,96)(62,179,97)(63,180,98)(64,181,99)(65,182,100)(66,183,101)(67,184,102)(68,185,103)(69,186,104)(70,187,105)(71,188,106)(72,189,107)(73,190,108)(74,191,109)(75,192,110), (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)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225) );
G=PermutationGroup([[(1,193,111),(2,194,112),(3,195,113),(4,196,114),(5,197,115),(6,198,116),(7,199,117),(8,200,118),(9,201,119),(10,202,120),(11,203,121),(12,204,122),(13,205,123),(14,206,124),(15,207,125),(16,208,126),(17,209,127),(18,210,128),(19,211,129),(20,212,130),(21,213,131),(22,214,132),(23,215,133),(24,216,134),(25,217,135),(26,218,136),(27,219,137),(28,220,138),(29,221,139),(30,222,140),(31,223,141),(32,224,142),(33,225,143),(34,151,144),(35,152,145),(36,153,146),(37,154,147),(38,155,148),(39,156,149),(40,157,150),(41,158,76),(42,159,77),(43,160,78),(44,161,79),(45,162,80),(46,163,81),(47,164,82),(48,165,83),(49,166,84),(50,167,85),(51,168,86),(52,169,87),(53,170,88),(54,171,89),(55,172,90),(56,173,91),(57,174,92),(58,175,93),(59,176,94),(60,177,95),(61,178,96),(62,179,97),(63,180,98),(64,181,99),(65,182,100),(66,183,101),(67,184,102),(68,185,103),(69,186,104),(70,187,105),(71,188,106),(72,189,107),(73,190,108),(74,191,109),(75,192,110)], [(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),(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225)]])
C3×C75 is a maximal subgroup of
C3⋊D75
225 conjugacy classes
| class | 1 | 3A | ··· | 3H | 5A | 5B | 5C | 5D | 15A | ··· | 15AF | 25A | ··· | 25T | 75A | ··· | 75FD |
| order | 1 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 15 | ··· | 15 | 25 | ··· | 25 | 75 | ··· | 75 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
225 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||
| image | C1 | C3 | C5 | C15 | C25 | C75 |
| kernel | C3×C75 | C75 | C3×C15 | C15 | C32 | C3 |
| # reps | 1 | 8 | 4 | 32 | 20 | 160 |
Matrix representation of C3×C75 ►in GL2(𝔽151) generated by
| 1 | 0 |
| 0 | 32 |
| 49 | 0 |
| 0 | 50 |
G:=sub<GL(2,GF(151))| [1,0,0,32],[49,0,0,50] >;
C3×C75 in GAP, Magma, Sage, TeX
C_3\times C_{75} % in TeX
G:=Group("C3xC75"); // GroupNames label
G:=SmallGroup(225,2);
// by ID
G=gap.SmallGroup(225,2);
# by ID
G:=PCGroup([4,-3,-3,-5,-5,70]);
// Polycyclic
G:=Group<a,b|a^3=b^75=1,a*b=b*a>;
// generators/relations
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