direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C15×SD32, C80⋊6C6, D8.C30, C48⋊4C10, C16⋊2C30, C240⋊12C2, Q16⋊1C30, C30.61D8, C60.194D4, C120.104C22, C8.3(C2×C30), (C5×Q16)⋊5C6, (C5×D8).2C6, C6.16(C5×D8), C2.4(C15×D8), C4.2(D4×C15), C40.25(C2×C6), (C3×Q16)⋊5C10, (C15×D8).4C2, (C3×D8).2C10, C20.37(C3×D4), C10.16(C3×D8), C12.37(C5×D4), C24.20(C2×C10), (C15×Q16)⋊13C2, SmallGroup(480,215)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×SD32
G = < a,b,c | a15=b16=c2=1, ab=ba, ac=ca, cbc=b7 >
Smallest permutation representation of C15×SD32
►On 240 points
Generators in S240
(1 232 223 177 79 41 23 126 193 97 175 82 160 137 51)(2 233 224 178 80 42 24 127 194 98 176 83 145 138 52)(3 234 209 179 65 43 25 128 195 99 161 84 146 139 53)(4 235 210 180 66 44 26 113 196 100 162 85 147 140 54)(5 236 211 181 67 45 27 114 197 101 163 86 148 141 55)(6 237 212 182 68 46 28 115 198 102 164 87 149 142 56)(7 238 213 183 69 47 29 116 199 103 165 88 150 143 57)(8 239 214 184 70 48 30 117 200 104 166 89 151 144 58)(9 240 215 185 71 33 31 118 201 105 167 90 152 129 59)(10 225 216 186 72 34 32 119 202 106 168 91 153 130 60)(11 226 217 187 73 35 17 120 203 107 169 92 154 131 61)(12 227 218 188 74 36 18 121 204 108 170 93 155 132 62)(13 228 219 189 75 37 19 122 205 109 171 94 156 133 63)(14 229 220 190 76 38 20 123 206 110 172 95 157 134 64)(15 230 221 191 77 39 21 124 207 111 173 96 158 135 49)(16 231 222 192 78 40 22 125 208 112 174 81 159 136 50)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 29)(18 20)(19 27)(21 25)(22 32)(24 30)(26 28)(34 40)(35 47)(36 38)(37 45)(39 43)(42 48)(44 46)(49 53)(50 60)(52 58)(54 56)(55 63)(57 61)(62 64)(65 77)(66 68)(67 75)(69 73)(70 80)(72 78)(74 76)(81 91)(83 89)(84 96)(85 87)(86 94)(88 92)(93 95)(98 104)(99 111)(100 102)(101 109)(103 107)(106 112)(108 110)(113 115)(114 122)(116 120)(117 127)(119 125)(121 123)(124 128)(130 136)(131 143)(132 134)(133 141)(135 139)(138 144)(140 142)(145 151)(146 158)(147 149)(148 156)(150 154)(153 159)(155 157)(161 173)(162 164)(163 171)(165 169)(166 176)(168 174)(170 172)(178 184)(179 191)(180 182)(181 189)(183 187)(186 192)(188 190)(194 200)(195 207)(196 198)(197 205)(199 203)(202 208)(204 206)(209 221)(210 212)(211 219)(213 217)(214 224)(216 222)(218 220)(225 231)(226 238)(227 229)(228 236)(230 234)(233 239)(235 237)
G:=sub<Sym(240)| (1,232,223,177,79,41,23,126,193,97,175,82,160,137,51)(2,233,224,178,80,42,24,127,194,98,176,83,145,138,52)(3,234,209,179,65,43,25,128,195,99,161,84,146,139,53)(4,235,210,180,66,44,26,113,196,100,162,85,147,140,54)(5,236,211,181,67,45,27,114,197,101,163,86,148,141,55)(6,237,212,182,68,46,28,115,198,102,164,87,149,142,56)(7,238,213,183,69,47,29,116,199,103,165,88,150,143,57)(8,239,214,184,70,48,30,117,200,104,166,89,151,144,58)(9,240,215,185,71,33,31,118,201,105,167,90,152,129,59)(10,225,216,186,72,34,32,119,202,106,168,91,153,130,60)(11,226,217,187,73,35,17,120,203,107,169,92,154,131,61)(12,227,218,188,74,36,18,121,204,108,170,93,155,132,62)(13,228,219,189,75,37,19,122,205,109,171,94,156,133,63)(14,229,220,190,76,38,20,123,206,110,172,95,157,134,64)(15,230,221,191,77,39,21,124,207,111,173,96,158,135,49)(16,231,222,192,78,40,22,125,208,112,174,81,159,136,50), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,29)(18,20)(19,27)(21,25)(22,32)(24,30)(26,28)(34,40)(35,47)(36,38)(37,45)(39,43)(42,48)(44,46)(49,53)(50,60)(52,58)(54,56)(55,63)(57,61)(62,64)(65,77)(66,68)(67,75)(69,73)(70,80)(72,78)(74,76)(81,91)(83,89)(84,96)(85,87)(86,94)(88,92)(93,95)(98,104)(99,111)(100,102)(101,109)(103,107)(106,112)(108,110)(113,115)(114,122)(116,120)(117,127)(119,125)(121,123)(124,128)(130,136)(131,143)(132,134)(133,141)(135,139)(138,144)(140,142)(145,151)(146,158)(147,149)(148,156)(150,154)(153,159)(155,157)(161,173)(162,164)(163,171)(165,169)(166,176)(168,174)(170,172)(178,184)(179,191)(180,182)(181,189)(183,187)(186,192)(188,190)(194,200)(195,207)(196,198)(197,205)(199,203)(202,208)(204,206)(209,221)(210,212)(211,219)(213,217)(214,224)(216,222)(218,220)(225,231)(226,238)(227,229)(228,236)(230,234)(233,239)(235,237)>;
G:=Group( (1,232,223,177,79,41,23,126,193,97,175,82,160,137,51)(2,233,224,178,80,42,24,127,194,98,176,83,145,138,52)(3,234,209,179,65,43,25,128,195,99,161,84,146,139,53)(4,235,210,180,66,44,26,113,196,100,162,85,147,140,54)(5,236,211,181,67,45,27,114,197,101,163,86,148,141,55)(6,237,212,182,68,46,28,115,198,102,164,87,149,142,56)(7,238,213,183,69,47,29,116,199,103,165,88,150,143,57)(8,239,214,184,70,48,30,117,200,104,166,89,151,144,58)(9,240,215,185,71,33,31,118,201,105,167,90,152,129,59)(10,225,216,186,72,34,32,119,202,106,168,91,153,130,60)(11,226,217,187,73,35,17,120,203,107,169,92,154,131,61)(12,227,218,188,74,36,18,121,204,108,170,93,155,132,62)(13,228,219,189,75,37,19,122,205,109,171,94,156,133,63)(14,229,220,190,76,38,20,123,206,110,172,95,157,134,64)(15,230,221,191,77,39,21,124,207,111,173,96,158,135,49)(16,231,222,192,78,40,22,125,208,112,174,81,159,136,50), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,29)(18,20)(19,27)(21,25)(22,32)(24,30)(26,28)(34,40)(35,47)(36,38)(37,45)(39,43)(42,48)(44,46)(49,53)(50,60)(52,58)(54,56)(55,63)(57,61)(62,64)(65,77)(66,68)(67,75)(69,73)(70,80)(72,78)(74,76)(81,91)(83,89)(84,96)(85,87)(86,94)(88,92)(93,95)(98,104)(99,111)(100,102)(101,109)(103,107)(106,112)(108,110)(113,115)(114,122)(116,120)(117,127)(119,125)(121,123)(124,128)(130,136)(131,143)(132,134)(133,141)(135,139)(138,144)(140,142)(145,151)(146,158)(147,149)(148,156)(150,154)(153,159)(155,157)(161,173)(162,164)(163,171)(165,169)(166,176)(168,174)(170,172)(178,184)(179,191)(180,182)(181,189)(183,187)(186,192)(188,190)(194,200)(195,207)(196,198)(197,205)(199,203)(202,208)(204,206)(209,221)(210,212)(211,219)(213,217)(214,224)(216,222)(218,220)(225,231)(226,238)(227,229)(228,236)(230,234)(233,239)(235,237) );
G=PermutationGroup([[(1,232,223,177,79,41,23,126,193,97,175,82,160,137,51),(2,233,224,178,80,42,24,127,194,98,176,83,145,138,52),(3,234,209,179,65,43,25,128,195,99,161,84,146,139,53),(4,235,210,180,66,44,26,113,196,100,162,85,147,140,54),(5,236,211,181,67,45,27,114,197,101,163,86,148,141,55),(6,237,212,182,68,46,28,115,198,102,164,87,149,142,56),(7,238,213,183,69,47,29,116,199,103,165,88,150,143,57),(8,239,214,184,70,48,30,117,200,104,166,89,151,144,58),(9,240,215,185,71,33,31,118,201,105,167,90,152,129,59),(10,225,216,186,72,34,32,119,202,106,168,91,153,130,60),(11,226,217,187,73,35,17,120,203,107,169,92,154,131,61),(12,227,218,188,74,36,18,121,204,108,170,93,155,132,62),(13,228,219,189,75,37,19,122,205,109,171,94,156,133,63),(14,229,220,190,76,38,20,123,206,110,172,95,157,134,64),(15,230,221,191,77,39,21,124,207,111,173,96,158,135,49),(16,231,222,192,78,40,22,125,208,112,174,81,159,136,50)], [(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)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,29),(18,20),(19,27),(21,25),(22,32),(24,30),(26,28),(34,40),(35,47),(36,38),(37,45),(39,43),(42,48),(44,46),(49,53),(50,60),(52,58),(54,56),(55,63),(57,61),(62,64),(65,77),(66,68),(67,75),(69,73),(70,80),(72,78),(74,76),(81,91),(83,89),(84,96),(85,87),(86,94),(88,92),(93,95),(98,104),(99,111),(100,102),(101,109),(103,107),(106,112),(108,110),(113,115),(114,122),(116,120),(117,127),(119,125),(121,123),(124,128),(130,136),(131,143),(132,134),(133,141),(135,139),(138,144),(140,142),(145,151),(146,158),(147,149),(148,156),(150,154),(153,159),(155,157),(161,173),(162,164),(163,171),(165,169),(166,176),(168,174),(170,172),(178,184),(179,191),(180,182),(181,189),(183,187),(186,192),(188,190),(194,200),(195,207),(196,198),(197,205),(199,203),(202,208),(204,206),(209,221),(210,212),(211,219),(213,217),(214,224),(216,222),(218,220),(225,231),(226,238),(227,229),(228,236),(230,234),(233,239),(235,237)]])
165 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 15A | ··· | 15H | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 24A | 24B | 24C | 24D | 30A | ··· | 30H | 30I | ··· | 30P | 40A | ··· | 40H | 48A | ··· | 48H | 60A | ··· | 60H | 60I | ··· | 60P | 80A | ··· | 80P | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 8 | 1 | 1 | 2 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 2 | 2 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
165 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C5 | C6 | C6 | C6 | C10 | C10 | C10 | C15 | C30 | C30 | C30 | D4 | D8 | C3×D4 | SD32 | C5×D4 | C3×D8 | C5×D8 | C3×SD32 | D4×C15 | C5×SD32 | C15×D8 | C15×SD32 |
| kernel | C15×SD32 | C240 | C15×D8 | C15×Q16 | C5×SD32 | C3×SD32 | C80 | C5×D8 | C5×Q16 | C48 | C3×D8 | C3×Q16 | SD32 | C16 | D8 | Q16 | C60 | C30 | C20 | C15 | C12 | C10 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C15×SD32 ►in GL3(𝔽241) generated by
| 225 | 0 | 0 |
| 0 | 91 | 0 |
| 0 | 0 | 91 |
| 240 | 0 | 0 |
| 0 | 103 | 41 |
| 0 | 200 | 103 |
| 240 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 240 |
G:=sub<GL(3,GF(241))| [225,0,0,0,91,0,0,0,91],[240,0,0,0,103,200,0,41,103],[240,0,0,0,1,0,0,0,240] >;
C15×SD32 in GAP, Magma, Sage, TeX
C_{15}\times {\rm SD}_{32} % in TeX
G:=Group("C15xSD32"); // GroupNames label
G:=SmallGroup(480,215);
// by ID
G=gap.SmallGroup(480,215);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,1680,869,6304,3161,242,15125,7572,124]);
// Polycyclic
G:=Group<a,b,c|a^15=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^7>;
// generators/relations
Export