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## G = C15×D16order 480 = 25·3·5

### Direct product of C15 and D16

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C15×D16
 Chief series C1 — C2 — C4 — C8 — C40 — C120 — C15×D8 — C15×D16
 Lower central C1 — C2 — C4 — C8 — C15×D16
 Upper central C1 — C30 — C60 — C120 — C15×D16

Generators and relations for C15×D16
G = < a,b,c | a15=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C15×D16
On 240 points
Generators in S240
(1 103 201 90 181 219 44 17 234 114 159 70 58 169 130)(2 104 202 91 182 220 45 18 235 115 160 71 59 170 131)(3 105 203 92 183 221 46 19 236 116 145 72 60 171 132)(4 106 204 93 184 222 47 20 237 117 146 73 61 172 133)(5 107 205 94 185 223 48 21 238 118 147 74 62 173 134)(6 108 206 95 186 224 33 22 239 119 148 75 63 174 135)(7 109 207 96 187 209 34 23 240 120 149 76 64 175 136)(8 110 208 81 188 210 35 24 225 121 150 77 49 176 137)(9 111 193 82 189 211 36 25 226 122 151 78 50 161 138)(10 112 194 83 190 212 37 26 227 123 152 79 51 162 139)(11 97 195 84 191 213 38 27 228 124 153 80 52 163 140)(12 98 196 85 192 214 39 28 229 125 154 65 53 164 141)(13 99 197 86 177 215 40 29 230 126 155 66 54 165 142)(14 100 198 87 178 216 41 30 231 127 156 67 55 166 143)(15 101 199 88 179 217 42 31 232 128 157 68 56 167 144)(16 102 200 89 180 218 43 32 233 113 158 69 57 168 129)
(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 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)(33 39)(34 38)(35 37)(40 48)(41 47)(42 46)(43 45)(49 51)(52 64)(53 63)(54 62)(55 61)(56 60)(57 59)(65 75)(66 74)(67 73)(68 72)(69 71)(76 80)(77 79)(81 83)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(97 109)(98 108)(99 107)(100 106)(101 105)(102 104)(110 112)(113 115)(116 128)(117 127)(118 126)(119 125)(120 124)(121 123)(129 131)(132 144)(133 143)(134 142)(135 141)(136 140)(137 139)(145 157)(146 156)(147 155)(148 154)(149 153)(150 152)(158 160)(162 176)(163 175)(164 174)(165 173)(166 172)(167 171)(168 170)(177 185)(178 184)(179 183)(180 182)(186 192)(187 191)(188 190)(194 208)(195 207)(196 206)(197 205)(198 204)(199 203)(200 202)(209 213)(210 212)(214 224)(215 223)(216 222)(217 221)(218 220)(225 227)(228 240)(229 239)(230 238)(231 237)(232 236)(233 235)

G:=sub<Sym(240)| (1,103,201,90,181,219,44,17,234,114,159,70,58,169,130)(2,104,202,91,182,220,45,18,235,115,160,71,59,170,131)(3,105,203,92,183,221,46,19,236,116,145,72,60,171,132)(4,106,204,93,184,222,47,20,237,117,146,73,61,172,133)(5,107,205,94,185,223,48,21,238,118,147,74,62,173,134)(6,108,206,95,186,224,33,22,239,119,148,75,63,174,135)(7,109,207,96,187,209,34,23,240,120,149,76,64,175,136)(8,110,208,81,188,210,35,24,225,121,150,77,49,176,137)(9,111,193,82,189,211,36,25,226,122,151,78,50,161,138)(10,112,194,83,190,212,37,26,227,123,152,79,51,162,139)(11,97,195,84,191,213,38,27,228,124,153,80,52,163,140)(12,98,196,85,192,214,39,28,229,125,154,65,53,164,141)(13,99,197,86,177,215,40,29,230,126,155,66,54,165,142)(14,100,198,87,178,216,41,30,231,127,156,67,55,166,143)(15,101,199,88,179,217,42,31,232,128,157,68,56,167,144)(16,102,200,89,180,218,43,32,233,113,158,69,57,168,129), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(33,39)(34,38)(35,37)(40,48)(41,47)(42,46)(43,45)(49,51)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(65,75)(66,74)(67,73)(68,72)(69,71)(76,80)(77,79)(81,83)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(97,109)(98,108)(99,107)(100,106)(101,105)(102,104)(110,112)(113,115)(116,128)(117,127)(118,126)(119,125)(120,124)(121,123)(129,131)(132,144)(133,143)(134,142)(135,141)(136,140)(137,139)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152)(158,160)(162,176)(163,175)(164,174)(165,173)(166,172)(167,171)(168,170)(177,185)(178,184)(179,183)(180,182)(186,192)(187,191)(188,190)(194,208)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202)(209,213)(210,212)(214,224)(215,223)(216,222)(217,221)(218,220)(225,227)(228,240)(229,239)(230,238)(231,237)(232,236)(233,235)>;

G:=Group( (1,103,201,90,181,219,44,17,234,114,159,70,58,169,130)(2,104,202,91,182,220,45,18,235,115,160,71,59,170,131)(3,105,203,92,183,221,46,19,236,116,145,72,60,171,132)(4,106,204,93,184,222,47,20,237,117,146,73,61,172,133)(5,107,205,94,185,223,48,21,238,118,147,74,62,173,134)(6,108,206,95,186,224,33,22,239,119,148,75,63,174,135)(7,109,207,96,187,209,34,23,240,120,149,76,64,175,136)(8,110,208,81,188,210,35,24,225,121,150,77,49,176,137)(9,111,193,82,189,211,36,25,226,122,151,78,50,161,138)(10,112,194,83,190,212,37,26,227,123,152,79,51,162,139)(11,97,195,84,191,213,38,27,228,124,153,80,52,163,140)(12,98,196,85,192,214,39,28,229,125,154,65,53,164,141)(13,99,197,86,177,215,40,29,230,126,155,66,54,165,142)(14,100,198,87,178,216,41,30,231,127,156,67,55,166,143)(15,101,199,88,179,217,42,31,232,128,157,68,56,167,144)(16,102,200,89,180,218,43,32,233,113,158,69,57,168,129), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(33,39)(34,38)(35,37)(40,48)(41,47)(42,46)(43,45)(49,51)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(65,75)(66,74)(67,73)(68,72)(69,71)(76,80)(77,79)(81,83)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(97,109)(98,108)(99,107)(100,106)(101,105)(102,104)(110,112)(113,115)(116,128)(117,127)(118,126)(119,125)(120,124)(121,123)(129,131)(132,144)(133,143)(134,142)(135,141)(136,140)(137,139)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152)(158,160)(162,176)(163,175)(164,174)(165,173)(166,172)(167,171)(168,170)(177,185)(178,184)(179,183)(180,182)(186,192)(187,191)(188,190)(194,208)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202)(209,213)(210,212)(214,224)(215,223)(216,222)(217,221)(218,220)(225,227)(228,240)(229,239)(230,238)(231,237)(232,236)(233,235) );

G=PermutationGroup([(1,103,201,90,181,219,44,17,234,114,159,70,58,169,130),(2,104,202,91,182,220,45,18,235,115,160,71,59,170,131),(3,105,203,92,183,221,46,19,236,116,145,72,60,171,132),(4,106,204,93,184,222,47,20,237,117,146,73,61,172,133),(5,107,205,94,185,223,48,21,238,118,147,74,62,173,134),(6,108,206,95,186,224,33,22,239,119,148,75,63,174,135),(7,109,207,96,187,209,34,23,240,120,149,76,64,175,136),(8,110,208,81,188,210,35,24,225,121,150,77,49,176,137),(9,111,193,82,189,211,36,25,226,122,151,78,50,161,138),(10,112,194,83,190,212,37,26,227,123,152,79,51,162,139),(11,97,195,84,191,213,38,27,228,124,153,80,52,163,140),(12,98,196,85,192,214,39,28,229,125,154,65,53,164,141),(13,99,197,86,177,215,40,29,230,126,155,66,54,165,142),(14,100,198,87,178,216,41,30,231,127,156,67,55,166,143),(15,101,199,88,179,217,42,31,232,128,157,68,56,167,144),(16,102,200,89,180,218,43,32,233,113,158,69,57,168,129)], [(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)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(18,32),(19,31),(20,30),(21,29),(22,28),(23,27),(24,26),(33,39),(34,38),(35,37),(40,48),(41,47),(42,46),(43,45),(49,51),(52,64),(53,63),(54,62),(55,61),(56,60),(57,59),(65,75),(66,74),(67,73),(68,72),(69,71),(76,80),(77,79),(81,83),(84,96),(85,95),(86,94),(87,93),(88,92),(89,91),(97,109),(98,108),(99,107),(100,106),(101,105),(102,104),(110,112),(113,115),(116,128),(117,127),(118,126),(119,125),(120,124),(121,123),(129,131),(132,144),(133,143),(134,142),(135,141),(136,140),(137,139),(145,157),(146,156),(147,155),(148,154),(149,153),(150,152),(158,160),(162,176),(163,175),(164,174),(165,173),(166,172),(167,171),(168,170),(177,185),(178,184),(179,183),(180,182),(186,192),(187,191),(188,190),(194,208),(195,207),(196,206),(197,205),(198,204),(199,203),(200,202),(209,213),(210,212),(214,224),(215,223),(216,222),(217,221),(218,220),(225,227),(228,240),(229,239),(230,238),(231,237),(232,236),(233,235)])

165 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 8A 8B 10A 10B 10C 10D 10E ··· 10L 12A 12B 15A ··· 15H 16A 16B 16C 16D 20A 20B 20C 20D 24A 24B 24C 24D 30A ··· 30H 30I ··· 30X 40A ··· 40H 48A ··· 48H 60A ··· 60H 80A ··· 80P 120A ··· 120P 240A ··· 240AF order 1 2 2 2 3 3 4 5 5 5 5 6 6 6 6 6 6 8 8 10 10 10 10 10 ··· 10 12 12 15 ··· 15 16 16 16 16 20 20 20 20 24 24 24 24 30 ··· 30 30 ··· 30 40 ··· 40 48 ··· 48 60 ··· 60 80 ··· 80 120 ··· 120 240 ··· 240 size 1 1 8 8 1 1 2 1 1 1 1 1 1 8 8 8 8 2 2 1 1 1 1 8 ··· 8 2 2 1 ··· 1 2 2 2 2 2 2 2 2 2 2 2 2 1 ··· 1 8 ··· 8 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

165 irreducible representations

 dim 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 C3 C5 C6 C6 C10 C10 C15 C30 C30 D4 D8 C3×D4 D16 C5×D4 C3×D8 C5×D8 C3×D16 D4×C15 C5×D16 C15×D8 C15×D16 kernel C15×D16 C240 C15×D8 C5×D16 C3×D16 C80 C5×D8 C48 C3×D8 D16 C16 D8 C60 C30 C20 C15 C12 C10 C6 C5 C4 C3 C2 C1 # reps 1 1 2 2 4 2 4 4 8 8 8 16 1 2 2 4 4 4 8 8 8 16 16 32

Matrix representation of C15×D16 in GL2(𝔽31) generated by

 28 0 0 28
,
 0 20 17 14
,
 14 6 14 17
G:=sub<GL(2,GF(31))| [28,0,0,28],[0,17,20,14],[14,14,6,17] >;

C15×D16 in GAP, Magma, Sage, TeX

C_{15}\times D_{16}
% in TeX

G:=Group("C15xD16");
// GroupNames label

G:=SmallGroup(480,214);
// by ID

G=gap.SmallGroup(480,214);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,869,6304,3161,242,15125,7572,124]);
// 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^-1>;
// generators/relations

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