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## G = C16×D15order 480 = 25·3·5

### Direct product of C16 and D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C16×D15
 Chief series C1 — C5 — C15 — C30 — C60 — C120 — C8×D15 — C16×D15
 Lower central C15 — C16×D15
 Upper central C1 — C16

Generators and relations for C16×D15
G = < a,b,c | a16=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C16×D15
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 95 98 116 180 214 195 72 171 238 50 156 29 38 134)(2 96 99 117 181 215 196 73 172 239 51 157 30 39 135)(3 81 100 118 182 216 197 74 173 240 52 158 31 40 136)(4 82 101 119 183 217 198 75 174 225 53 159 32 41 137)(5 83 102 120 184 218 199 76 175 226 54 160 17 42 138)(6 84 103 121 185 219 200 77 176 227 55 145 18 43 139)(7 85 104 122 186 220 201 78 161 228 56 146 19 44 140)(8 86 105 123 187 221 202 79 162 229 57 147 20 45 141)(9 87 106 124 188 222 203 80 163 230 58 148 21 46 142)(10 88 107 125 189 223 204 65 164 231 59 149 22 47 143)(11 89 108 126 190 224 205 66 165 232 60 150 23 48 144)(12 90 109 127 191 209 206 67 166 233 61 151 24 33 129)(13 91 110 128 192 210 207 68 167 234 62 152 25 34 130)(14 92 111 113 177 211 208 69 168 235 63 153 26 35 131)(15 93 112 114 178 212 193 70 169 236 64 154 27 36 132)(16 94 97 115 179 213 194 71 170 237 49 155 28 37 133)
(1 134)(2 135)(3 136)(4 137)(5 138)(6 139)(7 140)(8 141)(9 142)(10 143)(11 144)(12 129)(13 130)(14 131)(15 132)(16 133)(17 102)(18 103)(19 104)(20 105)(21 106)(22 107)(23 108)(24 109)(25 110)(26 111)(27 112)(28 97)(29 98)(30 99)(31 100)(32 101)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 81)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 89)(49 179)(50 180)(51 181)(52 182)(53 183)(54 184)(55 185)(56 186)(57 187)(58 188)(59 189)(60 190)(61 191)(62 192)(63 177)(64 178)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)(121 145)(122 146)(123 147)(124 148)(125 149)(126 150)(127 151)(128 152)(161 201)(162 202)(163 203)(164 204)(165 205)(166 206)(167 207)(168 208)(169 193)(170 194)(171 195)(172 196)(173 197)(174 198)(175 199)(176 200)(209 233)(210 234)(211 235)(212 236)(213 237)(214 238)(215 239)(216 240)(217 225)(218 226)(219 227)(220 228)(221 229)(222 230)(223 231)(224 232)

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

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

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

144 conjugacy classes

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

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 C8 C16 S3 D5 D6 D10 C4×S3 D15 C4×D5 S3×C8 D30 C8×D5 S3×C16 C4×D15 D5×C16 C8×D15 C16×D15 kernel C16×D15 C15⋊3C16 C240 C8×D15 C15⋊3C8 C4×D15 Dic15 D30 D15 C80 C48 C40 C24 C20 C16 C12 C10 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 2 2 4 4 16 1 2 1 2 2 4 4 4 4 8 8 8 16 16 32

Matrix representation of C16×D15 in GL3(𝔽241) generated by

 115 0 0 0 211 0 0 0 211
,
 1 0 0 0 94 110 0 131 161
,
 240 0 0 0 157 94 0 148 84
G:=sub<GL(3,GF(241))| [115,0,0,0,211,0,0,0,211],[1,0,0,0,94,131,0,110,161],[240,0,0,0,157,148,0,94,84] >;

C16×D15 in GAP, Magma, Sage, TeX

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

G:=Group("C16xD15");
// GroupNames label

G:=SmallGroup(480,157);
// by ID

G=gap.SmallGroup(480,157);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,36,58,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^16=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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