direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×C3⋊C16, C3⋊C80, C6.C40, C15⋊5C16, C30.5C8, C40.4S3, C12.2C20, C120.7C2, C24.3C10, C60.14C4, C20.8Dic3, C8.2(C5×S3), C10.3(C3⋊C8), C4.2(C5×Dic3), C2.(C5×C3⋊C8), SmallGroup(240,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C5×C3⋊C16 |
Generators and relations for C5×C3⋊C16
G = < a,b,c | a5=b3=c16=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×C3⋊C16
►Regular action on 240 points
Generators in S240
(1 144 234 192 169)(2 129 235 177 170)(3 130 236 178 171)(4 131 237 179 172)(5 132 238 180 173)(6 133 239 181 174)(7 134 240 182 175)(8 135 225 183 176)(9 136 226 184 161)(10 137 227 185 162)(11 138 228 186 163)(12 139 229 187 164)(13 140 230 188 165)(14 141 231 189 166)(15 142 232 190 167)(16 143 233 191 168)(17 92 41 105 70)(18 93 42 106 71)(19 94 43 107 72)(20 95 44 108 73)(21 96 45 109 74)(22 81 46 110 75)(23 82 47 111 76)(24 83 48 112 77)(25 84 33 97 78)(26 85 34 98 79)(27 86 35 99 80)(28 87 36 100 65)(29 88 37 101 66)(30 89 38 102 67)(31 90 39 103 68)(32 91 40 104 69)(49 209 198 156 121)(50 210 199 157 122)(51 211 200 158 123)(52 212 201 159 124)(53 213 202 160 125)(54 214 203 145 126)(55 215 204 146 127)(56 216 205 147 128)(57 217 206 148 113)(58 218 207 149 114)(59 219 208 150 115)(60 220 193 151 116)(61 221 194 152 117)(62 222 195 153 118)(63 223 196 154 119)(64 224 197 155 120)
(1 212 45)(2 46 213)(3 214 47)(4 48 215)(5 216 33)(6 34 217)(7 218 35)(8 36 219)(9 220 37)(10 38 221)(11 222 39)(12 40 223)(13 224 41)(14 42 209)(15 210 43)(16 44 211)(17 188 120)(18 121 189)(19 190 122)(20 123 191)(21 192 124)(22 125 177)(23 178 126)(24 127 179)(25 180 128)(26 113 181)(27 182 114)(28 115 183)(29 184 116)(30 117 185)(31 186 118)(32 119 187)(49 166 93)(50 94 167)(51 168 95)(52 96 169)(53 170 81)(54 82 171)(55 172 83)(56 84 173)(57 174 85)(58 86 175)(59 176 87)(60 88 161)(61 162 89)(62 90 163)(63 164 91)(64 92 165)(65 150 225)(66 226 151)(67 152 227)(68 228 153)(69 154 229)(70 230 155)(71 156 231)(72 232 157)(73 158 233)(74 234 159)(75 160 235)(76 236 145)(77 146 237)(78 238 147)(79 148 239)(80 240 149)(97 132 205)(98 206 133)(99 134 207)(100 208 135)(101 136 193)(102 194 137)(103 138 195)(104 196 139)(105 140 197)(106 198 141)(107 142 199)(108 200 143)(109 144 201)(110 202 129)(111 130 203)(112 204 131)
(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)(225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
G:=sub<Sym(240)| (1,144,234,192,169)(2,129,235,177,170)(3,130,236,178,171)(4,131,237,179,172)(5,132,238,180,173)(6,133,239,181,174)(7,134,240,182,175)(8,135,225,183,176)(9,136,226,184,161)(10,137,227,185,162)(11,138,228,186,163)(12,139,229,187,164)(13,140,230,188,165)(14,141,231,189,166)(15,142,232,190,167)(16,143,233,191,168)(17,92,41,105,70)(18,93,42,106,71)(19,94,43,107,72)(20,95,44,108,73)(21,96,45,109,74)(22,81,46,110,75)(23,82,47,111,76)(24,83,48,112,77)(25,84,33,97,78)(26,85,34,98,79)(27,86,35,99,80)(28,87,36,100,65)(29,88,37,101,66)(30,89,38,102,67)(31,90,39,103,68)(32,91,40,104,69)(49,209,198,156,121)(50,210,199,157,122)(51,211,200,158,123)(52,212,201,159,124)(53,213,202,160,125)(54,214,203,145,126)(55,215,204,146,127)(56,216,205,147,128)(57,217,206,148,113)(58,218,207,149,114)(59,219,208,150,115)(60,220,193,151,116)(61,221,194,152,117)(62,222,195,153,118)(63,223,196,154,119)(64,224,197,155,120), (1,212,45)(2,46,213)(3,214,47)(4,48,215)(5,216,33)(6,34,217)(7,218,35)(8,36,219)(9,220,37)(10,38,221)(11,222,39)(12,40,223)(13,224,41)(14,42,209)(15,210,43)(16,44,211)(17,188,120)(18,121,189)(19,190,122)(20,123,191)(21,192,124)(22,125,177)(23,178,126)(24,127,179)(25,180,128)(26,113,181)(27,182,114)(28,115,183)(29,184,116)(30,117,185)(31,186,118)(32,119,187)(49,166,93)(50,94,167)(51,168,95)(52,96,169)(53,170,81)(54,82,171)(55,172,83)(56,84,173)(57,174,85)(58,86,175)(59,176,87)(60,88,161)(61,162,89)(62,90,163)(63,164,91)(64,92,165)(65,150,225)(66,226,151)(67,152,227)(68,228,153)(69,154,229)(70,230,155)(71,156,231)(72,232,157)(73,158,233)(74,234,159)(75,160,235)(76,236,145)(77,146,237)(78,238,147)(79,148,239)(80,240,149)(97,132,205)(98,206,133)(99,134,207)(100,208,135)(101,136,193)(102,194,137)(103,138,195)(104,196,139)(105,140,197)(106,198,141)(107,142,199)(108,200,143)(109,144,201)(110,202,129)(111,130,203)(112,204,131), (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)(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)>;
G:=Group( (1,144,234,192,169)(2,129,235,177,170)(3,130,236,178,171)(4,131,237,179,172)(5,132,238,180,173)(6,133,239,181,174)(7,134,240,182,175)(8,135,225,183,176)(9,136,226,184,161)(10,137,227,185,162)(11,138,228,186,163)(12,139,229,187,164)(13,140,230,188,165)(14,141,231,189,166)(15,142,232,190,167)(16,143,233,191,168)(17,92,41,105,70)(18,93,42,106,71)(19,94,43,107,72)(20,95,44,108,73)(21,96,45,109,74)(22,81,46,110,75)(23,82,47,111,76)(24,83,48,112,77)(25,84,33,97,78)(26,85,34,98,79)(27,86,35,99,80)(28,87,36,100,65)(29,88,37,101,66)(30,89,38,102,67)(31,90,39,103,68)(32,91,40,104,69)(49,209,198,156,121)(50,210,199,157,122)(51,211,200,158,123)(52,212,201,159,124)(53,213,202,160,125)(54,214,203,145,126)(55,215,204,146,127)(56,216,205,147,128)(57,217,206,148,113)(58,218,207,149,114)(59,219,208,150,115)(60,220,193,151,116)(61,221,194,152,117)(62,222,195,153,118)(63,223,196,154,119)(64,224,197,155,120), (1,212,45)(2,46,213)(3,214,47)(4,48,215)(5,216,33)(6,34,217)(7,218,35)(8,36,219)(9,220,37)(10,38,221)(11,222,39)(12,40,223)(13,224,41)(14,42,209)(15,210,43)(16,44,211)(17,188,120)(18,121,189)(19,190,122)(20,123,191)(21,192,124)(22,125,177)(23,178,126)(24,127,179)(25,180,128)(26,113,181)(27,182,114)(28,115,183)(29,184,116)(30,117,185)(31,186,118)(32,119,187)(49,166,93)(50,94,167)(51,168,95)(52,96,169)(53,170,81)(54,82,171)(55,172,83)(56,84,173)(57,174,85)(58,86,175)(59,176,87)(60,88,161)(61,162,89)(62,90,163)(63,164,91)(64,92,165)(65,150,225)(66,226,151)(67,152,227)(68,228,153)(69,154,229)(70,230,155)(71,156,231)(72,232,157)(73,158,233)(74,234,159)(75,160,235)(76,236,145)(77,146,237)(78,238,147)(79,148,239)(80,240,149)(97,132,205)(98,206,133)(99,134,207)(100,208,135)(101,136,193)(102,194,137)(103,138,195)(104,196,139)(105,140,197)(106,198,141)(107,142,199)(108,200,143)(109,144,201)(110,202,129)(111,130,203)(112,204,131), (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)(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240) );
G=PermutationGroup([[(1,144,234,192,169),(2,129,235,177,170),(3,130,236,178,171),(4,131,237,179,172),(5,132,238,180,173),(6,133,239,181,174),(7,134,240,182,175),(8,135,225,183,176),(9,136,226,184,161),(10,137,227,185,162),(11,138,228,186,163),(12,139,229,187,164),(13,140,230,188,165),(14,141,231,189,166),(15,142,232,190,167),(16,143,233,191,168),(17,92,41,105,70),(18,93,42,106,71),(19,94,43,107,72),(20,95,44,108,73),(21,96,45,109,74),(22,81,46,110,75),(23,82,47,111,76),(24,83,48,112,77),(25,84,33,97,78),(26,85,34,98,79),(27,86,35,99,80),(28,87,36,100,65),(29,88,37,101,66),(30,89,38,102,67),(31,90,39,103,68),(32,91,40,104,69),(49,209,198,156,121),(50,210,199,157,122),(51,211,200,158,123),(52,212,201,159,124),(53,213,202,160,125),(54,214,203,145,126),(55,215,204,146,127),(56,216,205,147,128),(57,217,206,148,113),(58,218,207,149,114),(59,219,208,150,115),(60,220,193,151,116),(61,221,194,152,117),(62,222,195,153,118),(63,223,196,154,119),(64,224,197,155,120)], [(1,212,45),(2,46,213),(3,214,47),(4,48,215),(5,216,33),(6,34,217),(7,218,35),(8,36,219),(9,220,37),(10,38,221),(11,222,39),(12,40,223),(13,224,41),(14,42,209),(15,210,43),(16,44,211),(17,188,120),(18,121,189),(19,190,122),(20,123,191),(21,192,124),(22,125,177),(23,178,126),(24,127,179),(25,180,128),(26,113,181),(27,182,114),(28,115,183),(29,184,116),(30,117,185),(31,186,118),(32,119,187),(49,166,93),(50,94,167),(51,168,95),(52,96,169),(53,170,81),(54,82,171),(55,172,83),(56,84,173),(57,174,85),(58,86,175),(59,176,87),(60,88,161),(61,162,89),(62,90,163),(63,164,91),(64,92,165),(65,150,225),(66,226,151),(67,152,227),(68,228,153),(69,154,229),(70,230,155),(71,156,231),(72,232,157),(73,158,233),(74,234,159),(75,160,235),(76,236,145),(77,146,237),(78,238,147),(79,148,239),(80,240,149),(97,132,205),(98,206,133),(99,134,207),(100,208,135),(101,136,193),(102,194,137),(103,138,195),(104,196,139),(105,140,197),(106,198,141),(107,142,199),(108,200,143),(109,144,201),(110,202,129),(111,130,203),(112,204,131)], [(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),(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)]])
C5×C3⋊C16 is a maximal subgroup of
D15⋊2C16 C40.51D6 D30.5C8 C3⋊D80 D40.S3 C24.D10 C3⋊Dic40 S3×C80
120 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 15C | 15D | 16A | ··· | 16H | 20A | ··· | 20H | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40P | 60A | ··· | 60H | 80A | ··· | 80AF | 120A | ··· | 120P |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | ··· | 16 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C4 | C5 | C8 | C10 | C16 | C20 | C40 | C80 | S3 | Dic3 | C3⋊C8 | C5×S3 | C3⋊C16 | C5×Dic3 | C5×C3⋊C8 | C5×C3⋊C16 |
| kernel | C5×C3⋊C16 | C120 | C60 | C3⋊C16 | C30 | C24 | C15 | C12 | C6 | C3 | C40 | C20 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 16 | 32 | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 16 |
Matrix representation of C5×C3⋊C16 ►in GL2(𝔽41) generated by
| 18 | 0 |
| 0 | 18 |
| 3 | 15 |
| 21 | 37 |
| 0 | 34 |
| 18 | 0 |
G:=sub<GL(2,GF(41))| [18,0,0,18],[3,21,15,37],[0,18,34,0] >;
C5×C3⋊C16 in GAP, Magma, Sage, TeX
C_5\times C_3\rtimes C_{16} % in TeX
G:=Group("C5xC3:C16"); // GroupNames label
G:=SmallGroup(240,1);
// by ID
G=gap.SmallGroup(240,1);
# by ID
G:=PCGroup([6,-2,-5,-2,-2,-2,-3,60,50,69,5765]);
// Polycyclic
G:=Group<a,b,c|a^5=b^3=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export