direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C15×Q16, C8.C30, C40.3C6, C120.6C2, C24.2C10, C30.56D4, Q8.2C30, C60.79C22, C4.3(C2×C30), C2.5(D4×C15), C6.16(C5×D4), (C5×Q8).4C6, C20.19(C2×C6), C10.16(C3×D4), (C3×Q8).2C10, (Q8×C15).4C2, C12.19(C2×C10), SmallGroup(240,88)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×Q16
G = < a,b,c | a15=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C15×Q16
►Regular action on 240 points
Generators in S240
(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)
(1 189 166 19 143 240 119 36)(2 190 167 20 144 226 120 37)(3 191 168 21 145 227 106 38)(4 192 169 22 146 228 107 39)(5 193 170 23 147 229 108 40)(6 194 171 24 148 230 109 41)(7 195 172 25 149 231 110 42)(8 181 173 26 150 232 111 43)(9 182 174 27 136 233 112 44)(10 183 175 28 137 234 113 45)(11 184 176 29 138 235 114 31)(12 185 177 30 139 236 115 32)(13 186 178 16 140 237 116 33)(14 187 179 17 141 238 117 34)(15 188 180 18 142 239 118 35)(46 133 224 199 84 94 161 66)(47 134 225 200 85 95 162 67)(48 135 211 201 86 96 163 68)(49 121 212 202 87 97 164 69)(50 122 213 203 88 98 165 70)(51 123 214 204 89 99 151 71)(52 124 215 205 90 100 152 72)(53 125 216 206 76 101 153 73)(54 126 217 207 77 102 154 74)(55 127 218 208 78 103 155 75)(56 128 219 209 79 104 156 61)(57 129 220 210 80 105 157 62)(58 130 221 196 81 91 158 63)(59 131 222 197 82 92 159 64)(60 132 223 198 83 93 160 65)
(1 210 143 62)(2 196 144 63)(3 197 145 64)(4 198 146 65)(5 199 147 66)(6 200 148 67)(7 201 149 68)(8 202 150 69)(9 203 136 70)(10 204 137 71)(11 205 138 72)(12 206 139 73)(13 207 140 74)(14 208 141 75)(15 209 142 61)(16 54 33 77)(17 55 34 78)(18 56 35 79)(19 57 36 80)(20 58 37 81)(21 59 38 82)(22 60 39 83)(23 46 40 84)(24 47 41 85)(25 48 42 86)(26 49 43 87)(27 50 44 88)(28 51 45 89)(29 52 31 90)(30 53 32 76)(91 167 130 120)(92 168 131 106)(93 169 132 107)(94 170 133 108)(95 171 134 109)(96 172 135 110)(97 173 121 111)(98 174 122 112)(99 175 123 113)(100 176 124 114)(101 177 125 115)(102 178 126 116)(103 179 127 117)(104 180 128 118)(105 166 129 119)(151 183 214 234)(152 184 215 235)(153 185 216 236)(154 186 217 237)(155 187 218 238)(156 188 219 239)(157 189 220 240)(158 190 221 226)(159 191 222 227)(160 192 223 228)(161 193 224 229)(162 194 225 230)(163 195 211 231)(164 181 212 232)(165 182 213 233)
G:=sub<Sym(240)| (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), (1,189,166,19,143,240,119,36)(2,190,167,20,144,226,120,37)(3,191,168,21,145,227,106,38)(4,192,169,22,146,228,107,39)(5,193,170,23,147,229,108,40)(6,194,171,24,148,230,109,41)(7,195,172,25,149,231,110,42)(8,181,173,26,150,232,111,43)(9,182,174,27,136,233,112,44)(10,183,175,28,137,234,113,45)(11,184,176,29,138,235,114,31)(12,185,177,30,139,236,115,32)(13,186,178,16,140,237,116,33)(14,187,179,17,141,238,117,34)(15,188,180,18,142,239,118,35)(46,133,224,199,84,94,161,66)(47,134,225,200,85,95,162,67)(48,135,211,201,86,96,163,68)(49,121,212,202,87,97,164,69)(50,122,213,203,88,98,165,70)(51,123,214,204,89,99,151,71)(52,124,215,205,90,100,152,72)(53,125,216,206,76,101,153,73)(54,126,217,207,77,102,154,74)(55,127,218,208,78,103,155,75)(56,128,219,209,79,104,156,61)(57,129,220,210,80,105,157,62)(58,130,221,196,81,91,158,63)(59,131,222,197,82,92,159,64)(60,132,223,198,83,93,160,65), (1,210,143,62)(2,196,144,63)(3,197,145,64)(4,198,146,65)(5,199,147,66)(6,200,148,67)(7,201,149,68)(8,202,150,69)(9,203,136,70)(10,204,137,71)(11,205,138,72)(12,206,139,73)(13,207,140,74)(14,208,141,75)(15,209,142,61)(16,54,33,77)(17,55,34,78)(18,56,35,79)(19,57,36,80)(20,58,37,81)(21,59,38,82)(22,60,39,83)(23,46,40,84)(24,47,41,85)(25,48,42,86)(26,49,43,87)(27,50,44,88)(28,51,45,89)(29,52,31,90)(30,53,32,76)(91,167,130,120)(92,168,131,106)(93,169,132,107)(94,170,133,108)(95,171,134,109)(96,172,135,110)(97,173,121,111)(98,174,122,112)(99,175,123,113)(100,176,124,114)(101,177,125,115)(102,178,126,116)(103,179,127,117)(104,180,128,118)(105,166,129,119)(151,183,214,234)(152,184,215,235)(153,185,216,236)(154,186,217,237)(155,187,218,238)(156,188,219,239)(157,189,220,240)(158,190,221,226)(159,191,222,227)(160,192,223,228)(161,193,224,229)(162,194,225,230)(163,195,211,231)(164,181,212,232)(165,182,213,233)>;
G:=Group( (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), (1,189,166,19,143,240,119,36)(2,190,167,20,144,226,120,37)(3,191,168,21,145,227,106,38)(4,192,169,22,146,228,107,39)(5,193,170,23,147,229,108,40)(6,194,171,24,148,230,109,41)(7,195,172,25,149,231,110,42)(8,181,173,26,150,232,111,43)(9,182,174,27,136,233,112,44)(10,183,175,28,137,234,113,45)(11,184,176,29,138,235,114,31)(12,185,177,30,139,236,115,32)(13,186,178,16,140,237,116,33)(14,187,179,17,141,238,117,34)(15,188,180,18,142,239,118,35)(46,133,224,199,84,94,161,66)(47,134,225,200,85,95,162,67)(48,135,211,201,86,96,163,68)(49,121,212,202,87,97,164,69)(50,122,213,203,88,98,165,70)(51,123,214,204,89,99,151,71)(52,124,215,205,90,100,152,72)(53,125,216,206,76,101,153,73)(54,126,217,207,77,102,154,74)(55,127,218,208,78,103,155,75)(56,128,219,209,79,104,156,61)(57,129,220,210,80,105,157,62)(58,130,221,196,81,91,158,63)(59,131,222,197,82,92,159,64)(60,132,223,198,83,93,160,65), (1,210,143,62)(2,196,144,63)(3,197,145,64)(4,198,146,65)(5,199,147,66)(6,200,148,67)(7,201,149,68)(8,202,150,69)(9,203,136,70)(10,204,137,71)(11,205,138,72)(12,206,139,73)(13,207,140,74)(14,208,141,75)(15,209,142,61)(16,54,33,77)(17,55,34,78)(18,56,35,79)(19,57,36,80)(20,58,37,81)(21,59,38,82)(22,60,39,83)(23,46,40,84)(24,47,41,85)(25,48,42,86)(26,49,43,87)(27,50,44,88)(28,51,45,89)(29,52,31,90)(30,53,32,76)(91,167,130,120)(92,168,131,106)(93,169,132,107)(94,170,133,108)(95,171,134,109)(96,172,135,110)(97,173,121,111)(98,174,122,112)(99,175,123,113)(100,176,124,114)(101,177,125,115)(102,178,126,116)(103,179,127,117)(104,180,128,118)(105,166,129,119)(151,183,214,234)(152,184,215,235)(153,185,216,236)(154,186,217,237)(155,187,218,238)(156,188,219,239)(157,189,220,240)(158,190,221,226)(159,191,222,227)(160,192,223,228)(161,193,224,229)(162,194,225,230)(163,195,211,231)(164,181,212,232)(165,182,213,233) );
G=PermutationGroup([[(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)], [(1,189,166,19,143,240,119,36),(2,190,167,20,144,226,120,37),(3,191,168,21,145,227,106,38),(4,192,169,22,146,228,107,39),(5,193,170,23,147,229,108,40),(6,194,171,24,148,230,109,41),(7,195,172,25,149,231,110,42),(8,181,173,26,150,232,111,43),(9,182,174,27,136,233,112,44),(10,183,175,28,137,234,113,45),(11,184,176,29,138,235,114,31),(12,185,177,30,139,236,115,32),(13,186,178,16,140,237,116,33),(14,187,179,17,141,238,117,34),(15,188,180,18,142,239,118,35),(46,133,224,199,84,94,161,66),(47,134,225,200,85,95,162,67),(48,135,211,201,86,96,163,68),(49,121,212,202,87,97,164,69),(50,122,213,203,88,98,165,70),(51,123,214,204,89,99,151,71),(52,124,215,205,90,100,152,72),(53,125,216,206,76,101,153,73),(54,126,217,207,77,102,154,74),(55,127,218,208,78,103,155,75),(56,128,219,209,79,104,156,61),(57,129,220,210,80,105,157,62),(58,130,221,196,81,91,158,63),(59,131,222,197,82,92,159,64),(60,132,223,198,83,93,160,65)], [(1,210,143,62),(2,196,144,63),(3,197,145,64),(4,198,146,65),(5,199,147,66),(6,200,148,67),(7,201,149,68),(8,202,150,69),(9,203,136,70),(10,204,137,71),(11,205,138,72),(12,206,139,73),(13,207,140,74),(14,208,141,75),(15,209,142,61),(16,54,33,77),(17,55,34,78),(18,56,35,79),(19,57,36,80),(20,58,37,81),(21,59,38,82),(22,60,39,83),(23,46,40,84),(24,47,41,85),(25,48,42,86),(26,49,43,87),(27,50,44,88),(28,51,45,89),(29,52,31,90),(30,53,32,76),(91,167,130,120),(92,168,131,106),(93,169,132,107),(94,170,133,108),(95,171,134,109),(96,172,135,110),(97,173,121,111),(98,174,122,112),(99,175,123,113),(100,176,124,114),(101,177,125,115),(102,178,126,116),(103,179,127,117),(104,180,128,118),(105,166,129,119),(151,183,214,234),(152,184,215,235),(153,185,216,236),(154,186,217,237),(155,187,218,238),(156,188,219,239),(157,189,220,240),(158,190,221,226),(159,191,222,227),(160,192,223,228),(161,193,224,229),(162,194,225,230),(163,195,211,231),(164,181,212,232),(165,182,213,233)]])
C15×Q16 is a maximal subgroup of
C8.6D30 C15⋊7Q32 Q16⋊D15 D120⋊8C2
105 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 12F | 15A | ··· | 15H | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 24A | 24B | 24C | 24D | 30A | ··· | 30H | 40A | ··· | 40H | 60A | ··· | 60H | 60I | ··· | 60X | 120A | ··· | 120P |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C5 | C6 | C6 | C10 | C10 | C15 | C30 | C30 | D4 | Q16 | C3×D4 | C5×D4 | C3×Q16 | C5×Q16 | D4×C15 | C15×Q16 |
| kernel | C15×Q16 | C120 | Q8×C15 | C5×Q16 | C3×Q16 | C40 | C5×Q8 | C24 | C3×Q8 | Q16 | C8 | Q8 | C30 | C15 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C15×Q16 ►in GL2(𝔽31) generated by
| 10 | 0 |
| 0 | 10 |
| 7 | 7 |
| 7 | 16 |
| 0 | 30 |
| 1 | 0 |
G:=sub<GL(2,GF(31))| [10,0,0,10],[7,7,7,16],[0,1,30,0] >;
C15×Q16 in GAP, Magma, Sage, TeX
C_{15}\times Q_{16} % in TeX
G:=Group("C15xQ16"); // GroupNames label
G:=SmallGroup(240,88);
// by ID
G=gap.SmallGroup(240,88);
# by ID
G:=PCGroup([6,-2,-2,-3,-5,-2,-2,720,745,727,5404,2710,88]);
// Polycyclic
G:=Group<a,b,c|a^15=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export