metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C15⋊1C16, C30.1C8, C60.2C4, C12.2F5, C20.2Dic3, C5⋊(C3⋊C16), C3⋊(C5⋊C16), C6.(C5⋊C8), C10.(C3⋊C8), C4.2(C3⋊F5), C2.(C15⋊C8), C5⋊2C8.2S3, (C3×C5⋊2C8).3C2, SmallGroup(240,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C15⋊C16 |
Generators and relations for C15⋊C16
G = < a,b | a15=b16=1, bab-1=a2 >
Smallest permutation representation of C15⋊C16
►Regular action on 240 points
Generators in S240
(1 47 160 217 203 168 24 95 112 127 134 227 58 191 74)(2 145 204 25 97 135 59 75 48 218 169 96 128 228 192)(3 205 98 60 33 170 113 177 146 26 136 76 219 81 229)(4 99 34 114 147 137 220 230 206 61 171 178 27 77 82)(5 35 148 221 207 172 28 83 100 115 138 231 62 179 78)(6 149 208 29 101 139 63 79 36 222 173 84 116 232 180)(7 193 102 64 37 174 117 181 150 30 140 80 223 85 233)(8 103 38 118 151 141 224 234 194 49 175 182 31 65 86)(9 39 152 209 195 176 32 87 104 119 142 235 50 183 66)(10 153 196 17 105 143 51 67 40 210 161 88 120 236 184)(11 197 106 52 41 162 121 185 154 18 144 68 211 89 237)(12 107 42 122 155 129 212 238 198 53 163 186 19 69 90)(13 43 156 213 199 164 20 91 108 123 130 239 54 187 70)(14 157 200 21 109 131 55 71 44 214 165 92 124 240 188)(15 201 110 56 45 166 125 189 158 22 132 72 215 93 225)(16 111 46 126 159 133 216 226 202 57 167 190 23 73 94)
(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)| (1,47,160,217,203,168,24,95,112,127,134,227,58,191,74)(2,145,204,25,97,135,59,75,48,218,169,96,128,228,192)(3,205,98,60,33,170,113,177,146,26,136,76,219,81,229)(4,99,34,114,147,137,220,230,206,61,171,178,27,77,82)(5,35,148,221,207,172,28,83,100,115,138,231,62,179,78)(6,149,208,29,101,139,63,79,36,222,173,84,116,232,180)(7,193,102,64,37,174,117,181,150,30,140,80,223,85,233)(8,103,38,118,151,141,224,234,194,49,175,182,31,65,86)(9,39,152,209,195,176,32,87,104,119,142,235,50,183,66)(10,153,196,17,105,143,51,67,40,210,161,88,120,236,184)(11,197,106,52,41,162,121,185,154,18,144,68,211,89,237)(12,107,42,122,155,129,212,238,198,53,163,186,19,69,90)(13,43,156,213,199,164,20,91,108,123,130,239,54,187,70)(14,157,200,21,109,131,55,71,44,214,165,92,124,240,188)(15,201,110,56,45,166,125,189,158,22,132,72,215,93,225)(16,111,46,126,159,133,216,226,202,57,167,190,23,73,94), (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( (1,47,160,217,203,168,24,95,112,127,134,227,58,191,74)(2,145,204,25,97,135,59,75,48,218,169,96,128,228,192)(3,205,98,60,33,170,113,177,146,26,136,76,219,81,229)(4,99,34,114,147,137,220,230,206,61,171,178,27,77,82)(5,35,148,221,207,172,28,83,100,115,138,231,62,179,78)(6,149,208,29,101,139,63,79,36,222,173,84,116,232,180)(7,193,102,64,37,174,117,181,150,30,140,80,223,85,233)(8,103,38,118,151,141,224,234,194,49,175,182,31,65,86)(9,39,152,209,195,176,32,87,104,119,142,235,50,183,66)(10,153,196,17,105,143,51,67,40,210,161,88,120,236,184)(11,197,106,52,41,162,121,185,154,18,144,68,211,89,237)(12,107,42,122,155,129,212,238,198,53,163,186,19,69,90)(13,43,156,213,199,164,20,91,108,123,130,239,54,187,70)(14,157,200,21,109,131,55,71,44,214,165,92,124,240,188)(15,201,110,56,45,166,125,189,158,22,132,72,215,93,225)(16,111,46,126,159,133,216,226,202,57,167,190,23,73,94), (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([[(1,47,160,217,203,168,24,95,112,127,134,227,58,191,74),(2,145,204,25,97,135,59,75,48,218,169,96,128,228,192),(3,205,98,60,33,170,113,177,146,26,136,76,219,81,229),(4,99,34,114,147,137,220,230,206,61,171,178,27,77,82),(5,35,148,221,207,172,28,83,100,115,138,231,62,179,78),(6,149,208,29,101,139,63,79,36,222,173,84,116,232,180),(7,193,102,64,37,174,117,181,150,30,140,80,223,85,233),(8,103,38,118,151,141,224,234,194,49,175,182,31,65,86),(9,39,152,209,195,176,32,87,104,119,142,235,50,183,66),(10,153,196,17,105,143,51,67,40,210,161,88,120,236,184),(11,197,106,52,41,162,121,185,154,18,144,68,211,89,237),(12,107,42,122,155,129,212,238,198,53,163,186,19,69,90),(13,43,156,213,199,164,20,91,108,123,130,239,54,187,70),(14,157,200,21,109,131,55,71,44,214,165,92,124,240,188),(15,201,110,56,45,166,125,189,158,22,132,72,215,93,225),(16,111,46,126,159,133,216,226,202,57,167,190,23,73,94)], [(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)]])
C15⋊C16 is a maximal subgroup of
S3×C5⋊C16 D15⋊C16 C15⋊M5(2) D30.C8 C24.F5 C120.C4 C60.C8
C15⋊C16 is a maximal quotient of C15⋊C32
36 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5 | 6 | 8A | 8B | 8C | 8D | 10 | 12A | 12B | 15A | 15B | 16A | ··· | 16H | 20A | 20B | 24A | 24B | 24C | 24D | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 6 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | 15 | 15 | 16 | ··· | 16 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 2 | 1 | 1 | 4 | 2 | 5 | 5 | 5 | 5 | 4 | 2 | 2 | 4 | 4 | 15 | ··· | 15 | 4 | 4 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | - | |||||||||
| image | C1 | C2 | C4 | C8 | C16 | S3 | Dic3 | C3⋊C8 | C3⋊C16 | F5 | C5⋊C8 | C3⋊F5 | C5⋊C16 | C15⋊C8 | C15⋊C16 |
| kernel | C15⋊C16 | C3×C5⋊2C8 | C60 | C30 | C15 | C5⋊2C8 | C20 | C10 | C5 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C15⋊C16 ►in GL4(𝔽241) generated by
| 229 | 127 | 114 | 12 |
| 229 | 115 | 0 | 126 |
| 0 | 115 | 229 | 12 |
| 114 | 127 | 229 | 0 |
| 219 | 173 | 77 | 2 |
| 221 | 184 | 55 | 175 |
| 153 | 20 | 57 | 186 |
| 164 | 239 | 230 | 22 |
G:=sub<GL(4,GF(241))| [229,229,0,114,127,115,115,127,114,0,229,229,12,126,12,0],[219,221,153,164,173,184,20,239,77,55,57,230,2,175,186,22] >;
C15⋊C16 in GAP, Magma, Sage, TeX
C_{15}\rtimes C_{16} % in TeX
G:=Group("C15:C16"); // GroupNames label
G:=SmallGroup(240,6);
// by ID
G=gap.SmallGroup(240,6);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,12,31,50,964,5189,3467]);
// Polycyclic
G:=Group<a,b|a^15=b^16=1,b*a*b^-1=a^2>;
// generators/relations
Export