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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

Aliases: C15×D16, C803C6, C2409C2, C483C10, C161C30, D81C30, C30.60D8, C60.193D4, C120.103C22, (C5×D8)⋊5C6, (C3×D8)⋊5C10, C8.2(C2×C30), C4.1(D4×C15), C2.3(C15×D8), C6.15(C5×D8), (C15×D8)⋊13C2, C40.24(C2×C6), C20.36(C3×D4), C10.15(C3×D8), C12.36(C5×D4), C24.19(C2×C10), SmallGroup(480,214)

Series: Derived Chief Lower central Upper central

C1C8 — C15×D16
C1C2C4C8C40C120C15×D8 — C15×D16
C1C2C4C8 — C15×D16
C1C30C60C120 — C15×D16

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

8C2
8C2
4C22
4C22
8C6
8C6
8C10
8C10
2D4
2D4
4C2×C6
4C2×C6
4C2×C10
4C2×C10
8C30
8C30
2C3×D4
2C3×D4
2C5×D4
2C5×D4
4C2×C30
4C2×C30
2D4×C15
2D4×C15

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 2A2B2C3A3B 4 5A5B5C5D6A6B6C6D6E6F8A8B10A10B10C10D10E···10L12A12B15A···15H16A16B16C16D20A20B20C20D24A24B24C24D30A···30H30I···30X40A···40H48A···48H60A···60H80A···80P120A···120P240A···240AF
order12223345555666666881010101010···10121215···1516161616202020202424242430···3030···3040···4048···4860···6080···80120···120240···240
size118811211111188882211118···8221···12222222222221···18···82···22···22···22···22···22···2

165 irreducible representations

dim111111111111222222222222
type++++++
imageC1C2C2C3C5C6C6C10C10C15C30C30D4D8C3×D4D16C5×D4C3×D8C5×D8C3×D16D4×C15C5×D16C15×D8C15×D16
kernelC15×D16C240C15×D8C5×D16C3×D16C80C5×D8C48C3×D8D16C16D8C60C30C20C15C12C10C6C5C4C3C2C1
# reps1122424488816122444888161632

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

280
028
,
020
1714
,
146
1417
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

Export

Subgroup lattice of C15×D16 in TeX

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