direct product, abelian, monomial, 3-elementary
Aliases: C3×C78, SmallGroup(234,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C78 |
| C1 — C3×C78 |
| C1 — C3×C78 |
Generators and relations for C3×C78
G = < a,b | a3=b78=1, ab=ba >
Smallest permutation representation of C3×C78
►Regular action on 234 points
Generators in S234
(1 176 124)(2 177 125)(3 178 126)(4 179 127)(5 180 128)(6 181 129)(7 182 130)(8 183 131)(9 184 132)(10 185 133)(11 186 134)(12 187 135)(13 188 136)(14 189 137)(15 190 138)(16 191 139)(17 192 140)(18 193 141)(19 194 142)(20 195 143)(21 196 144)(22 197 145)(23 198 146)(24 199 147)(25 200 148)(26 201 149)(27 202 150)(28 203 151)(29 204 152)(30 205 153)(31 206 154)(32 207 155)(33 208 156)(34 209 79)(35 210 80)(36 211 81)(37 212 82)(38 213 83)(39 214 84)(40 215 85)(41 216 86)(42 217 87)(43 218 88)(44 219 89)(45 220 90)(46 221 91)(47 222 92)(48 223 93)(49 224 94)(50 225 95)(51 226 96)(52 227 97)(53 228 98)(54 229 99)(55 230 100)(56 231 101)(57 232 102)(58 233 103)(59 234 104)(60 157 105)(61 158 106)(62 159 107)(63 160 108)(64 161 109)(65 162 110)(66 163 111)(67 164 112)(68 165 113)(69 166 114)(70 167 115)(71 168 116)(72 169 117)(73 170 118)(74 171 119)(75 172 120)(76 173 121)(77 174 122)(78 175 123)
(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)(157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234)
G:=sub<Sym(234)| (1,176,124)(2,177,125)(3,178,126)(4,179,127)(5,180,128)(6,181,129)(7,182,130)(8,183,131)(9,184,132)(10,185,133)(11,186,134)(12,187,135)(13,188,136)(14,189,137)(15,190,138)(16,191,139)(17,192,140)(18,193,141)(19,194,142)(20,195,143)(21,196,144)(22,197,145)(23,198,146)(24,199,147)(25,200,148)(26,201,149)(27,202,150)(28,203,151)(29,204,152)(30,205,153)(31,206,154)(32,207,155)(33,208,156)(34,209,79)(35,210,80)(36,211,81)(37,212,82)(38,213,83)(39,214,84)(40,215,85)(41,216,86)(42,217,87)(43,218,88)(44,219,89)(45,220,90)(46,221,91)(47,222,92)(48,223,93)(49,224,94)(50,225,95)(51,226,96)(52,227,97)(53,228,98)(54,229,99)(55,230,100)(56,231,101)(57,232,102)(58,233,103)(59,234,104)(60,157,105)(61,158,106)(62,159,107)(63,160,108)(64,161,109)(65,162,110)(66,163,111)(67,164,112)(68,165,113)(69,166,114)(70,167,115)(71,168,116)(72,169,117)(73,170,118)(74,171,119)(75,172,120)(76,173,121)(77,174,122)(78,175,123), (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)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)>;
G:=Group( (1,176,124)(2,177,125)(3,178,126)(4,179,127)(5,180,128)(6,181,129)(7,182,130)(8,183,131)(9,184,132)(10,185,133)(11,186,134)(12,187,135)(13,188,136)(14,189,137)(15,190,138)(16,191,139)(17,192,140)(18,193,141)(19,194,142)(20,195,143)(21,196,144)(22,197,145)(23,198,146)(24,199,147)(25,200,148)(26,201,149)(27,202,150)(28,203,151)(29,204,152)(30,205,153)(31,206,154)(32,207,155)(33,208,156)(34,209,79)(35,210,80)(36,211,81)(37,212,82)(38,213,83)(39,214,84)(40,215,85)(41,216,86)(42,217,87)(43,218,88)(44,219,89)(45,220,90)(46,221,91)(47,222,92)(48,223,93)(49,224,94)(50,225,95)(51,226,96)(52,227,97)(53,228,98)(54,229,99)(55,230,100)(56,231,101)(57,232,102)(58,233,103)(59,234,104)(60,157,105)(61,158,106)(62,159,107)(63,160,108)(64,161,109)(65,162,110)(66,163,111)(67,164,112)(68,165,113)(69,166,114)(70,167,115)(71,168,116)(72,169,117)(73,170,118)(74,171,119)(75,172,120)(76,173,121)(77,174,122)(78,175,123), (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)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234) );
G=PermutationGroup([[(1,176,124),(2,177,125),(3,178,126),(4,179,127),(5,180,128),(6,181,129),(7,182,130),(8,183,131),(9,184,132),(10,185,133),(11,186,134),(12,187,135),(13,188,136),(14,189,137),(15,190,138),(16,191,139),(17,192,140),(18,193,141),(19,194,142),(20,195,143),(21,196,144),(22,197,145),(23,198,146),(24,199,147),(25,200,148),(26,201,149),(27,202,150),(28,203,151),(29,204,152),(30,205,153),(31,206,154),(32,207,155),(33,208,156),(34,209,79),(35,210,80),(36,211,81),(37,212,82),(38,213,83),(39,214,84),(40,215,85),(41,216,86),(42,217,87),(43,218,88),(44,219,89),(45,220,90),(46,221,91),(47,222,92),(48,223,93),(49,224,94),(50,225,95),(51,226,96),(52,227,97),(53,228,98),(54,229,99),(55,230,100),(56,231,101),(57,232,102),(58,233,103),(59,234,104),(60,157,105),(61,158,106),(62,159,107),(63,160,108),(64,161,109),(65,162,110),(66,163,111),(67,164,112),(68,165,113),(69,166,114),(70,167,115),(71,168,116),(72,169,117),(73,170,118),(74,171,119),(75,172,120),(76,173,121),(77,174,122),(78,175,123)], [(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),(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)]])
C3×C78 is a maximal subgroup of
C3⋊Dic39
234 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 13A | ··· | 13L | 26A | ··· | 26L | 39A | ··· | 39CR | 78A | ··· | 78CR |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
234 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C13 | C26 | C39 | C78 |
| kernel | C3×C78 | C3×C39 | C78 | C39 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 8 | 12 | 12 | 96 | 96 |
Matrix representation of C3×C78 ►in GL2(𝔽79) generated by
| 1 | 0 |
| 0 | 55 |
| 70 | 0 |
| 0 | 11 |
G:=sub<GL(2,GF(79))| [1,0,0,55],[70,0,0,11] >;
C3×C78 in GAP, Magma, Sage, TeX
C_3\times C_{78} % in TeX
G:=Group("C3xC78"); // GroupNames label
G:=SmallGroup(234,16);
// by ID
G=gap.SmallGroup(234,16);
# by ID
G:=PCGroup([4,-2,-3,-3,-13]);
// Polycyclic
G:=Group<a,b|a^3=b^78=1,a*b=b*a>;
// generators/relations
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