direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×C5⋊C16, C5⋊C48, C15⋊2C16, C10.C24, C30.2C8, C60.5C4, C12.5F5, C20.2C12, C6.2(C5⋊C8), C4.2(C3×F5), C5⋊2C8.2C6, C2.(C3×C5⋊C8), (C3×C5⋊2C8).4C2, SmallGroup(240,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C3×C5⋊C16 |
Generators and relations for C3×C5⋊C16
G = < a,b,c | a3=b5=c16=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C3×C5⋊C16
►Regular action on 240 points
Generators in S240
(1 104 196)(2 105 197)(3 106 198)(4 107 199)(5 108 200)(6 109 201)(7 110 202)(8 111 203)(9 112 204)(10 97 205)(11 98 206)(12 99 207)(13 100 208)(14 101 193)(15 102 194)(16 103 195)(17 147 213)(18 148 214)(19 149 215)(20 150 216)(21 151 217)(22 152 218)(23 153 219)(24 154 220)(25 155 221)(26 156 222)(27 157 223)(28 158 224)(29 159 209)(30 160 210)(31 145 211)(32 146 212)(33 161 66)(34 162 67)(35 163 68)(36 164 69)(37 165 70)(38 166 71)(39 167 72)(40 168 73)(41 169 74)(42 170 75)(43 171 76)(44 172 77)(45 173 78)(46 174 79)(47 175 80)(48 176 65)(49 130 122)(50 131 123)(51 132 124)(52 133 125)(53 134 126)(54 135 127)(55 136 128)(56 137 113)(57 138 114)(58 139 115)(59 140 116)(60 141 117)(61 142 118)(62 143 119)(63 144 120)(64 129 121)(81 234 182)(82 235 183)(83 236 184)(84 237 185)(85 238 186)(86 239 187)(87 240 188)(88 225 189)(89 226 190)(90 227 191)(91 228 192)(92 229 177)(93 230 178)(94 231 179)(95 232 180)(96 233 181)
(1 154 85 163 140)(2 164 155 141 86)(3 142 165 87 156)(4 88 143 157 166)(5 158 89 167 144)(6 168 159 129 90)(7 130 169 91 160)(8 92 131 145 170)(9 146 93 171 132)(10 172 147 133 94)(11 134 173 95 148)(12 96 135 149 174)(13 150 81 175 136)(14 176 151 137 82)(15 138 161 83 152)(16 84 139 153 162)(17 52 179 205 44)(18 206 53 45 180)(19 46 207 181 54)(20 182 47 55 208)(21 56 183 193 48)(22 194 57 33 184)(23 34 195 185 58)(24 186 35 59 196)(25 60 187 197 36)(26 198 61 37 188)(27 38 199 189 62)(28 190 39 63 200)(29 64 191 201 40)(30 202 49 41 192)(31 42 203 177 50)(32 178 43 51 204)(65 217 113 235 101)(66 236 218 102 114)(67 103 237 115 219)(68 116 104 220 238)(69 221 117 239 105)(70 240 222 106 118)(71 107 225 119 223)(72 120 108 224 226)(73 209 121 227 109)(74 228 210 110 122)(75 111 229 123 211)(76 124 112 212 230)(77 213 125 231 97)(78 232 214 98 126)(79 99 233 127 215)(80 128 100 216 234)
(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)| (1,104,196)(2,105,197)(3,106,198)(4,107,199)(5,108,200)(6,109,201)(7,110,202)(8,111,203)(9,112,204)(10,97,205)(11,98,206)(12,99,207)(13,100,208)(14,101,193)(15,102,194)(16,103,195)(17,147,213)(18,148,214)(19,149,215)(20,150,216)(21,151,217)(22,152,218)(23,153,219)(24,154,220)(25,155,221)(26,156,222)(27,157,223)(28,158,224)(29,159,209)(30,160,210)(31,145,211)(32,146,212)(33,161,66)(34,162,67)(35,163,68)(36,164,69)(37,165,70)(38,166,71)(39,167,72)(40,168,73)(41,169,74)(42,170,75)(43,171,76)(44,172,77)(45,173,78)(46,174,79)(47,175,80)(48,176,65)(49,130,122)(50,131,123)(51,132,124)(52,133,125)(53,134,126)(54,135,127)(55,136,128)(56,137,113)(57,138,114)(58,139,115)(59,140,116)(60,141,117)(61,142,118)(62,143,119)(63,144,120)(64,129,121)(81,234,182)(82,235,183)(83,236,184)(84,237,185)(85,238,186)(86,239,187)(87,240,188)(88,225,189)(89,226,190)(90,227,191)(91,228,192)(92,229,177)(93,230,178)(94,231,179)(95,232,180)(96,233,181), (1,154,85,163,140)(2,164,155,141,86)(3,142,165,87,156)(4,88,143,157,166)(5,158,89,167,144)(6,168,159,129,90)(7,130,169,91,160)(8,92,131,145,170)(9,146,93,171,132)(10,172,147,133,94)(11,134,173,95,148)(12,96,135,149,174)(13,150,81,175,136)(14,176,151,137,82)(15,138,161,83,152)(16,84,139,153,162)(17,52,179,205,44)(18,206,53,45,180)(19,46,207,181,54)(20,182,47,55,208)(21,56,183,193,48)(22,194,57,33,184)(23,34,195,185,58)(24,186,35,59,196)(25,60,187,197,36)(26,198,61,37,188)(27,38,199,189,62)(28,190,39,63,200)(29,64,191,201,40)(30,202,49,41,192)(31,42,203,177,50)(32,178,43,51,204)(65,217,113,235,101)(66,236,218,102,114)(67,103,237,115,219)(68,116,104,220,238)(69,221,117,239,105)(70,240,222,106,118)(71,107,225,119,223)(72,120,108,224,226)(73,209,121,227,109)(74,228,210,110,122)(75,111,229,123,211)(76,124,112,212,230)(77,213,125,231,97)(78,232,214,98,126)(79,99,233,127,215)(80,128,100,216,234), (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( (1,104,196)(2,105,197)(3,106,198)(4,107,199)(5,108,200)(6,109,201)(7,110,202)(8,111,203)(9,112,204)(10,97,205)(11,98,206)(12,99,207)(13,100,208)(14,101,193)(15,102,194)(16,103,195)(17,147,213)(18,148,214)(19,149,215)(20,150,216)(21,151,217)(22,152,218)(23,153,219)(24,154,220)(25,155,221)(26,156,222)(27,157,223)(28,158,224)(29,159,209)(30,160,210)(31,145,211)(32,146,212)(33,161,66)(34,162,67)(35,163,68)(36,164,69)(37,165,70)(38,166,71)(39,167,72)(40,168,73)(41,169,74)(42,170,75)(43,171,76)(44,172,77)(45,173,78)(46,174,79)(47,175,80)(48,176,65)(49,130,122)(50,131,123)(51,132,124)(52,133,125)(53,134,126)(54,135,127)(55,136,128)(56,137,113)(57,138,114)(58,139,115)(59,140,116)(60,141,117)(61,142,118)(62,143,119)(63,144,120)(64,129,121)(81,234,182)(82,235,183)(83,236,184)(84,237,185)(85,238,186)(86,239,187)(87,240,188)(88,225,189)(89,226,190)(90,227,191)(91,228,192)(92,229,177)(93,230,178)(94,231,179)(95,232,180)(96,233,181), (1,154,85,163,140)(2,164,155,141,86)(3,142,165,87,156)(4,88,143,157,166)(5,158,89,167,144)(6,168,159,129,90)(7,130,169,91,160)(8,92,131,145,170)(9,146,93,171,132)(10,172,147,133,94)(11,134,173,95,148)(12,96,135,149,174)(13,150,81,175,136)(14,176,151,137,82)(15,138,161,83,152)(16,84,139,153,162)(17,52,179,205,44)(18,206,53,45,180)(19,46,207,181,54)(20,182,47,55,208)(21,56,183,193,48)(22,194,57,33,184)(23,34,195,185,58)(24,186,35,59,196)(25,60,187,197,36)(26,198,61,37,188)(27,38,199,189,62)(28,190,39,63,200)(29,64,191,201,40)(30,202,49,41,192)(31,42,203,177,50)(32,178,43,51,204)(65,217,113,235,101)(66,236,218,102,114)(67,103,237,115,219)(68,116,104,220,238)(69,221,117,239,105)(70,240,222,106,118)(71,107,225,119,223)(72,120,108,224,226)(73,209,121,227,109)(74,228,210,110,122)(75,111,229,123,211)(76,124,112,212,230)(77,213,125,231,97)(78,232,214,98,126)(79,99,233,127,215)(80,128,100,216,234), (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([[(1,104,196),(2,105,197),(3,106,198),(4,107,199),(5,108,200),(6,109,201),(7,110,202),(8,111,203),(9,112,204),(10,97,205),(11,98,206),(12,99,207),(13,100,208),(14,101,193),(15,102,194),(16,103,195),(17,147,213),(18,148,214),(19,149,215),(20,150,216),(21,151,217),(22,152,218),(23,153,219),(24,154,220),(25,155,221),(26,156,222),(27,157,223),(28,158,224),(29,159,209),(30,160,210),(31,145,211),(32,146,212),(33,161,66),(34,162,67),(35,163,68),(36,164,69),(37,165,70),(38,166,71),(39,167,72),(40,168,73),(41,169,74),(42,170,75),(43,171,76),(44,172,77),(45,173,78),(46,174,79),(47,175,80),(48,176,65),(49,130,122),(50,131,123),(51,132,124),(52,133,125),(53,134,126),(54,135,127),(55,136,128),(56,137,113),(57,138,114),(58,139,115),(59,140,116),(60,141,117),(61,142,118),(62,143,119),(63,144,120),(64,129,121),(81,234,182),(82,235,183),(83,236,184),(84,237,185),(85,238,186),(86,239,187),(87,240,188),(88,225,189),(89,226,190),(90,227,191),(91,228,192),(92,229,177),(93,230,178),(94,231,179),(95,232,180),(96,233,181)], [(1,154,85,163,140),(2,164,155,141,86),(3,142,165,87,156),(4,88,143,157,166),(5,158,89,167,144),(6,168,159,129,90),(7,130,169,91,160),(8,92,131,145,170),(9,146,93,171,132),(10,172,147,133,94),(11,134,173,95,148),(12,96,135,149,174),(13,150,81,175,136),(14,176,151,137,82),(15,138,161,83,152),(16,84,139,153,162),(17,52,179,205,44),(18,206,53,45,180),(19,46,207,181,54),(20,182,47,55,208),(21,56,183,193,48),(22,194,57,33,184),(23,34,195,185,58),(24,186,35,59,196),(25,60,187,197,36),(26,198,61,37,188),(27,38,199,189,62),(28,190,39,63,200),(29,64,191,201,40),(30,202,49,41,192),(31,42,203,177,50),(32,178,43,51,204),(65,217,113,235,101),(66,236,218,102,114),(67,103,237,115,219),(68,116,104,220,238),(69,221,117,239,105),(70,240,222,106,118),(71,107,225,119,223),(72,120,108,224,226),(73,209,121,227,109),(74,228,210,110,122),(75,111,229,123,211),(76,124,112,212,230),(77,213,125,231,97),(78,232,214,98,126),(79,99,233,127,215),(80,128,100,216,234)], [(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)]])
C3×C5⋊C16 is a maximal subgroup of
D15⋊C16 C15⋊M5(2) D30.C8
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5 | 6A | 6B | 8A | 8B | 8C | 8D | 10 | 12A | 12B | 12C | 12D | 15A | 15B | 16A | ··· | 16H | 20A | 20B | 24A | ··· | 24H | 30A | 30B | 48A | ··· | 48P | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 16 | ··· | 16 | 20 | 20 | 24 | ··· | 24 | 30 | 30 | 48 | ··· | 48 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 5 | 5 | 5 | 5 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 5 | ··· | 5 | 4 | 4 | 5 | ··· | 5 | 4 | 4 | 5 | ··· | 5 | 4 | 4 | 4 | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | ||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C16 | C24 | C48 | F5 | C5⋊C8 | C3×F5 | C5⋊C16 | C3×C5⋊C8 | C3×C5⋊C16 |
| kernel | C3×C5⋊C16 | C3×C5⋊2C8 | C5⋊C16 | C60 | C5⋊2C8 | C30 | C20 | C15 | C10 | C5 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C3×C5⋊C16 ►in GL5(𝔽241)
| 225 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 |
| 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 1 | 240 |
| 240 | 0 | 0 | 0 | 0 |
| 0 | 118 | 69 | 213 | 53 |
| 0 | 90 | 122 | 1 | 171 |
| 0 | 119 | 240 | 70 | 143 |
| 0 | 188 | 212 | 123 | 172 |
G:=sub<GL(5,GF(241))| [225,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,240,240,240,240],[240,0,0,0,0,0,118,90,119,188,0,69,122,240,212,0,213,1,70,123,0,53,171,143,172] >;
C3×C5⋊C16 in GAP, Magma, Sage, TeX
C_3\times C_5\rtimes C_{16} % in TeX
G:=Group("C3xC5:C16"); // GroupNames label
G:=SmallGroup(240,5);
// by ID
G=gap.SmallGroup(240,5);
# by ID
G:=PCGroup([6,-2,-3,-2,-2,-2,-5,36,50,69,3461,1169]);
// Polycyclic
G:=Group<a,b,c|a^3=b^5=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
Export