metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C15⋊3C16, C30.3C8, C60.8C4, C40.2S3, C8.2D15, C24.3D5, C120.3C2, C4.2Dic15, C20.5Dic3, C12.2Dic5, C3⋊(C5⋊2C16), C5⋊2(C3⋊C16), C6.(C5⋊2C8), C2.(C15⋊3C8), C10.2(C3⋊C8), SmallGroup(240,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C15⋊3C16 |
Generators and relations for C15⋊3C16
G = < a,b | a15=b16=1, bab-1=a-1 >
Smallest permutation representation of C15⋊3C16
►Regular action on 240 points
Generators in S240
(1 230 158 173 62 222 203 178 92 38 140 28 125 99 66)(2 67 100 126 29 141 39 93 179 204 223 63 174 159 231)(3 232 160 175 64 224 205 180 94 40 142 30 127 101 68)(4 69 102 128 31 143 41 95 181 206 209 49 176 145 233)(5 234 146 161 50 210 207 182 96 42 144 32 113 103 70)(6 71 104 114 17 129 43 81 183 208 211 51 162 147 235)(7 236 148 163 52 212 193 184 82 44 130 18 115 105 72)(8 73 106 116 19 131 45 83 185 194 213 53 164 149 237)(9 238 150 165 54 214 195 186 84 46 132 20 117 107 74)(10 75 108 118 21 133 47 85 187 196 215 55 166 151 239)(11 240 152 167 56 216 197 188 86 48 134 22 119 109 76)(12 77 110 120 23 135 33 87 189 198 217 57 168 153 225)(13 226 154 169 58 218 199 190 88 34 136 24 121 111 78)(14 79 112 122 25 137 35 89 191 200 219 59 170 155 227)(15 228 156 171 60 220 201 192 90 36 138 26 123 97 80)(16 65 98 124 27 139 37 91 177 202 221 61 172 157 229)
(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)| (1,230,158,173,62,222,203,178,92,38,140,28,125,99,66)(2,67,100,126,29,141,39,93,179,204,223,63,174,159,231)(3,232,160,175,64,224,205,180,94,40,142,30,127,101,68)(4,69,102,128,31,143,41,95,181,206,209,49,176,145,233)(5,234,146,161,50,210,207,182,96,42,144,32,113,103,70)(6,71,104,114,17,129,43,81,183,208,211,51,162,147,235)(7,236,148,163,52,212,193,184,82,44,130,18,115,105,72)(8,73,106,116,19,131,45,83,185,194,213,53,164,149,237)(9,238,150,165,54,214,195,186,84,46,132,20,117,107,74)(10,75,108,118,21,133,47,85,187,196,215,55,166,151,239)(11,240,152,167,56,216,197,188,86,48,134,22,119,109,76)(12,77,110,120,23,135,33,87,189,198,217,57,168,153,225)(13,226,154,169,58,218,199,190,88,34,136,24,121,111,78)(14,79,112,122,25,137,35,89,191,200,219,59,170,155,227)(15,228,156,171,60,220,201,192,90,36,138,26,123,97,80)(16,65,98,124,27,139,37,91,177,202,221,61,172,157,229), (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( (1,230,158,173,62,222,203,178,92,38,140,28,125,99,66)(2,67,100,126,29,141,39,93,179,204,223,63,174,159,231)(3,232,160,175,64,224,205,180,94,40,142,30,127,101,68)(4,69,102,128,31,143,41,95,181,206,209,49,176,145,233)(5,234,146,161,50,210,207,182,96,42,144,32,113,103,70)(6,71,104,114,17,129,43,81,183,208,211,51,162,147,235)(7,236,148,163,52,212,193,184,82,44,130,18,115,105,72)(8,73,106,116,19,131,45,83,185,194,213,53,164,149,237)(9,238,150,165,54,214,195,186,84,46,132,20,117,107,74)(10,75,108,118,21,133,47,85,187,196,215,55,166,151,239)(11,240,152,167,56,216,197,188,86,48,134,22,119,109,76)(12,77,110,120,23,135,33,87,189,198,217,57,168,153,225)(13,226,154,169,58,218,199,190,88,34,136,24,121,111,78)(14,79,112,122,25,137,35,89,191,200,219,59,170,155,227)(15,228,156,171,60,220,201,192,90,36,138,26,123,97,80)(16,65,98,124,27,139,37,91,177,202,221,61,172,157,229), (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([[(1,230,158,173,62,222,203,178,92,38,140,28,125,99,66),(2,67,100,126,29,141,39,93,179,204,223,63,174,159,231),(3,232,160,175,64,224,205,180,94,40,142,30,127,101,68),(4,69,102,128,31,143,41,95,181,206,209,49,176,145,233),(5,234,146,161,50,210,207,182,96,42,144,32,113,103,70),(6,71,104,114,17,129,43,81,183,208,211,51,162,147,235),(7,236,148,163,52,212,193,184,82,44,130,18,115,105,72),(8,73,106,116,19,131,45,83,185,194,213,53,164,149,237),(9,238,150,165,54,214,195,186,84,46,132,20,117,107,74),(10,75,108,118,21,133,47,85,187,196,215,55,166,151,239),(11,240,152,167,56,216,197,188,86,48,134,22,119,109,76),(12,77,110,120,23,135,33,87,189,198,217,57,168,153,225),(13,226,154,169,58,218,199,190,88,34,136,24,121,111,78),(14,79,112,122,25,137,35,89,191,200,219,59,170,155,227),(15,228,156,171,60,220,201,192,90,36,138,26,123,97,80),(16,65,98,124,27,139,37,91,177,202,221,61,172,157,229)], [(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⋊3C16 is a maximal subgroup of
D5×C3⋊C16 S3×C5⋊2C16 C40.51D6 C40.52D6 C15⋊D16 C40.D6 C15⋊SD32 C15⋊Q32 C16×D15 C80⋊S3 C60.7C8 C15⋊7D16 D8.D15 C8.6D30 C15⋊7Q32
C15⋊3C16 is a maximal quotient of
C15⋊3C32
72 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 10A | 10B | 12A | 12B | 15A | 15B | 15C | 15D | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 60A | ··· | 60H | 120A | ··· | 120P |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 15 | ··· | 15 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | - | |||||||||
| image | C1 | C2 | C4 | C8 | C16 | S3 | D5 | Dic3 | Dic5 | C3⋊C8 | D15 | C5⋊2C8 | C3⋊C16 | Dic15 | C5⋊2C16 | C15⋊3C8 | C15⋊3C16 |
| kernel | C15⋊3C16 | C120 | C60 | C30 | C15 | C40 | C24 | C20 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C15⋊3C16 ►in GL4(𝔽241) generated by
| 240 | 189 | 0 | 0 |
| 52 | 52 | 0 | 0 |
| 0 | 0 | 1 | 54 |
| 0 | 0 | 174 | 239 |
| 45 | 126 | 0 | 0 |
| 196 | 196 | 0 | 0 |
| 0 | 0 | 210 | 165 |
| 0 | 0 | 19 | 31 |
G:=sub<GL(4,GF(241))| [240,52,0,0,189,52,0,0,0,0,1,174,0,0,54,239],[45,196,0,0,126,196,0,0,0,0,210,19,0,0,165,31] >;
C15⋊3C16 in GAP, Magma, Sage, TeX
C_{15}\rtimes_3C_{16} % in TeX
G:=Group("C15:3C16"); // GroupNames label
G:=SmallGroup(240,3);
// by ID
G=gap.SmallGroup(240,3);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,12,31,50,964,6917]);
// Polycyclic
G:=Group<a,b|a^15=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
Export