direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×C5⋊2C16, C5⋊2C48, C15⋊4C16, C30.4C8, C40.2C6, C24.4D5, C20.5C12, C10.2C24, C120.5C2, C60.12C4, C12.5Dic5, C8.2(C3×D5), C6.2(C5⋊2C8), C4.2(C3×Dic5), C2.(C3×C5⋊2C8), SmallGroup(240,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C3×C5⋊2C16 |
Generators and relations for C3×C5⋊2C16
G = < a,b,c | a3=b5=c16=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×C5⋊2C16
►Regular action on 240 points
Generators in S240
(1 118 187)(2 119 188)(3 120 189)(4 121 190)(5 122 191)(6 123 192)(7 124 177)(8 125 178)(9 126 179)(10 127 180)(11 128 181)(12 113 182)(13 114 183)(14 115 184)(15 116 185)(16 117 186)(17 45 81)(18 46 82)(19 47 83)(20 48 84)(21 33 85)(22 34 86)(23 35 87)(24 36 88)(25 37 89)(26 38 90)(27 39 91)(28 40 92)(29 41 93)(30 42 94)(31 43 95)(32 44 96)(49 105 204)(50 106 205)(51 107 206)(52 108 207)(53 109 208)(54 110 193)(55 111 194)(56 112 195)(57 97 196)(58 98 197)(59 99 198)(60 100 199)(61 101 200)(62 102 201)(63 103 202)(64 104 203)(65 239 153)(66 240 154)(67 225 155)(68 226 156)(69 227 157)(70 228 158)(71 229 159)(72 230 160)(73 231 145)(74 232 146)(75 233 147)(76 234 148)(77 235 149)(78 236 150)(79 237 151)(80 238 152)(129 220 168)(130 221 169)(131 222 170)(132 223 171)(133 224 172)(134 209 173)(135 210 174)(136 211 175)(137 212 176)(138 213 161)(139 214 162)(140 215 163)(141 216 164)(142 217 165)(143 218 166)(144 219 167)
(1 228 167 39 49)(2 50 40 168 229)(3 230 169 41 51)(4 52 42 170 231)(5 232 171 43 53)(6 54 44 172 233)(7 234 173 45 55)(8 56 46 174 235)(9 236 175 47 57)(10 58 48 176 237)(11 238 161 33 59)(12 60 34 162 239)(13 240 163 35 61)(14 62 36 164 225)(15 226 165 37 63)(16 64 38 166 227)(17 194 177 76 209)(18 210 77 178 195)(19 196 179 78 211)(20 212 79 180 197)(21 198 181 80 213)(22 214 65 182 199)(23 200 183 66 215)(24 216 67 184 201)(25 202 185 68 217)(26 218 69 186 203)(27 204 187 70 219)(28 220 71 188 205)(29 206 189 72 221)(30 222 73 190 207)(31 208 191 74 223)(32 224 75 192 193)(81 111 124 148 134)(82 135 149 125 112)(83 97 126 150 136)(84 137 151 127 98)(85 99 128 152 138)(86 139 153 113 100)(87 101 114 154 140)(88 141 155 115 102)(89 103 116 156 142)(90 143 157 117 104)(91 105 118 158 144)(92 129 159 119 106)(93 107 120 160 130)(94 131 145 121 108)(95 109 122 146 132)(96 133 147 123 110)
(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)(225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
G:=sub<Sym(240)| (1,118,187)(2,119,188)(3,120,189)(4,121,190)(5,122,191)(6,123,192)(7,124,177)(8,125,178)(9,126,179)(10,127,180)(11,128,181)(12,113,182)(13,114,183)(14,115,184)(15,116,185)(16,117,186)(17,45,81)(18,46,82)(19,47,83)(20,48,84)(21,33,85)(22,34,86)(23,35,87)(24,36,88)(25,37,89)(26,38,90)(27,39,91)(28,40,92)(29,41,93)(30,42,94)(31,43,95)(32,44,96)(49,105,204)(50,106,205)(51,107,206)(52,108,207)(53,109,208)(54,110,193)(55,111,194)(56,112,195)(57,97,196)(58,98,197)(59,99,198)(60,100,199)(61,101,200)(62,102,201)(63,103,202)(64,104,203)(65,239,153)(66,240,154)(67,225,155)(68,226,156)(69,227,157)(70,228,158)(71,229,159)(72,230,160)(73,231,145)(74,232,146)(75,233,147)(76,234,148)(77,235,149)(78,236,150)(79,237,151)(80,238,152)(129,220,168)(130,221,169)(131,222,170)(132,223,171)(133,224,172)(134,209,173)(135,210,174)(136,211,175)(137,212,176)(138,213,161)(139,214,162)(140,215,163)(141,216,164)(142,217,165)(143,218,166)(144,219,167), (1,228,167,39,49)(2,50,40,168,229)(3,230,169,41,51)(4,52,42,170,231)(5,232,171,43,53)(6,54,44,172,233)(7,234,173,45,55)(8,56,46,174,235)(9,236,175,47,57)(10,58,48,176,237)(11,238,161,33,59)(12,60,34,162,239)(13,240,163,35,61)(14,62,36,164,225)(15,226,165,37,63)(16,64,38,166,227)(17,194,177,76,209)(18,210,77,178,195)(19,196,179,78,211)(20,212,79,180,197)(21,198,181,80,213)(22,214,65,182,199)(23,200,183,66,215)(24,216,67,184,201)(25,202,185,68,217)(26,218,69,186,203)(27,204,187,70,219)(28,220,71,188,205)(29,206,189,72,221)(30,222,73,190,207)(31,208,191,74,223)(32,224,75,192,193)(81,111,124,148,134)(82,135,149,125,112)(83,97,126,150,136)(84,137,151,127,98)(85,99,128,152,138)(86,139,153,113,100)(87,101,114,154,140)(88,141,155,115,102)(89,103,116,156,142)(90,143,157,117,104)(91,105,118,158,144)(92,129,159,119,106)(93,107,120,160,130)(94,131,145,121,108)(95,109,122,146,132)(96,133,147,123,110), (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)(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)>;
G:=Group( (1,118,187)(2,119,188)(3,120,189)(4,121,190)(5,122,191)(6,123,192)(7,124,177)(8,125,178)(9,126,179)(10,127,180)(11,128,181)(12,113,182)(13,114,183)(14,115,184)(15,116,185)(16,117,186)(17,45,81)(18,46,82)(19,47,83)(20,48,84)(21,33,85)(22,34,86)(23,35,87)(24,36,88)(25,37,89)(26,38,90)(27,39,91)(28,40,92)(29,41,93)(30,42,94)(31,43,95)(32,44,96)(49,105,204)(50,106,205)(51,107,206)(52,108,207)(53,109,208)(54,110,193)(55,111,194)(56,112,195)(57,97,196)(58,98,197)(59,99,198)(60,100,199)(61,101,200)(62,102,201)(63,103,202)(64,104,203)(65,239,153)(66,240,154)(67,225,155)(68,226,156)(69,227,157)(70,228,158)(71,229,159)(72,230,160)(73,231,145)(74,232,146)(75,233,147)(76,234,148)(77,235,149)(78,236,150)(79,237,151)(80,238,152)(129,220,168)(130,221,169)(131,222,170)(132,223,171)(133,224,172)(134,209,173)(135,210,174)(136,211,175)(137,212,176)(138,213,161)(139,214,162)(140,215,163)(141,216,164)(142,217,165)(143,218,166)(144,219,167), (1,228,167,39,49)(2,50,40,168,229)(3,230,169,41,51)(4,52,42,170,231)(5,232,171,43,53)(6,54,44,172,233)(7,234,173,45,55)(8,56,46,174,235)(9,236,175,47,57)(10,58,48,176,237)(11,238,161,33,59)(12,60,34,162,239)(13,240,163,35,61)(14,62,36,164,225)(15,226,165,37,63)(16,64,38,166,227)(17,194,177,76,209)(18,210,77,178,195)(19,196,179,78,211)(20,212,79,180,197)(21,198,181,80,213)(22,214,65,182,199)(23,200,183,66,215)(24,216,67,184,201)(25,202,185,68,217)(26,218,69,186,203)(27,204,187,70,219)(28,220,71,188,205)(29,206,189,72,221)(30,222,73,190,207)(31,208,191,74,223)(32,224,75,192,193)(81,111,124,148,134)(82,135,149,125,112)(83,97,126,150,136)(84,137,151,127,98)(85,99,128,152,138)(86,139,153,113,100)(87,101,114,154,140)(88,141,155,115,102)(89,103,116,156,142)(90,143,157,117,104)(91,105,118,158,144)(92,129,159,119,106)(93,107,120,160,130)(94,131,145,121,108)(95,109,122,146,132)(96,133,147,123,110), (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)(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240) );
G=PermutationGroup([[(1,118,187),(2,119,188),(3,120,189),(4,121,190),(5,122,191),(6,123,192),(7,124,177),(8,125,178),(9,126,179),(10,127,180),(11,128,181),(12,113,182),(13,114,183),(14,115,184),(15,116,185),(16,117,186),(17,45,81),(18,46,82),(19,47,83),(20,48,84),(21,33,85),(22,34,86),(23,35,87),(24,36,88),(25,37,89),(26,38,90),(27,39,91),(28,40,92),(29,41,93),(30,42,94),(31,43,95),(32,44,96),(49,105,204),(50,106,205),(51,107,206),(52,108,207),(53,109,208),(54,110,193),(55,111,194),(56,112,195),(57,97,196),(58,98,197),(59,99,198),(60,100,199),(61,101,200),(62,102,201),(63,103,202),(64,104,203),(65,239,153),(66,240,154),(67,225,155),(68,226,156),(69,227,157),(70,228,158),(71,229,159),(72,230,160),(73,231,145),(74,232,146),(75,233,147),(76,234,148),(77,235,149),(78,236,150),(79,237,151),(80,238,152),(129,220,168),(130,221,169),(131,222,170),(132,223,171),(133,224,172),(134,209,173),(135,210,174),(136,211,175),(137,212,176),(138,213,161),(139,214,162),(140,215,163),(141,216,164),(142,217,165),(143,218,166),(144,219,167)], [(1,228,167,39,49),(2,50,40,168,229),(3,230,169,41,51),(4,52,42,170,231),(5,232,171,43,53),(6,54,44,172,233),(7,234,173,45,55),(8,56,46,174,235),(9,236,175,47,57),(10,58,48,176,237),(11,238,161,33,59),(12,60,34,162,239),(13,240,163,35,61),(14,62,36,164,225),(15,226,165,37,63),(16,64,38,166,227),(17,194,177,76,209),(18,210,77,178,195),(19,196,179,78,211),(20,212,79,180,197),(21,198,181,80,213),(22,214,65,182,199),(23,200,183,66,215),(24,216,67,184,201),(25,202,185,68,217),(26,218,69,186,203),(27,204,187,70,219),(28,220,71,188,205),(29,206,189,72,221),(30,222,73,190,207),(31,208,191,74,223),(32,224,75,192,193),(81,111,124,148,134),(82,135,149,125,112),(83,97,126,150,136),(84,137,151,127,98),(85,99,128,152,138),(86,139,153,113,100),(87,101,114,154,140),(88,141,155,115,102),(89,103,116,156,142),(90,143,157,117,104),(91,105,118,158,144),(92,129,159,119,106),(93,107,120,160,130),(94,131,145,121,108),(95,109,122,146,132),(96,133,147,123,110)], [(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),(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)]])
C3×C5⋊2C16 is a maximal subgroup of
C15⋊C32 D15⋊2C16 C40.52D6 D30.5C8 C5⋊D48 D24.D5 Dic12⋊D5 C5⋊Dic24 D5×C48
96 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 48A | ··· | 48P | 60A | ··· | 60H | 120A | ··· | 120P |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
96 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 | C3 | C4 | C6 | C8 | C12 | C16 | C24 | C48 | D5 | Dic5 | C3×D5 | C5⋊2C8 | C3×Dic5 | C5⋊2C16 | C3×C5⋊2C8 | C3×C5⋊2C16 |
| kernel | C3×C5⋊2C16 | C120 | C5⋊2C16 | C60 | C40 | C30 | C20 | C15 | C10 | C5 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C3×C5⋊2C16 ►in GL3(𝔽241) generated by
| 225 | 0 | 0 |
| 0 | 15 | 0 |
| 0 | 0 | 15 |
| 1 | 0 | 0 |
| 0 | 51 | 240 |
| 0 | 1 | 0 |
| 111 | 0 | 0 |
| 0 | 160 | 102 |
| 0 | 68 | 81 |
G:=sub<GL(3,GF(241))| [225,0,0,0,15,0,0,0,15],[1,0,0,0,51,1,0,240,0],[111,0,0,0,160,68,0,102,81] >;
C3×C5⋊2C16 in GAP, Magma, Sage, TeX
C_3\times C_5\rtimes_2C_{16} % in TeX
G:=Group("C3xC5:2C16"); // GroupNames label
G:=SmallGroup(240,2);
// by ID
G=gap.SmallGroup(240,2);
# by ID
G:=PCGroup([6,-2,-3,-2,-2,-2,-5,36,50,69,6917]);
// 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^-1>;
// generators/relations
Export