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## G = C15⋊20(C4×D4)  order 480 = 25·3·5

### 16th semidirect product of C15 and C4×D4 acting via C4×D4/C2×C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C15⋊20(C4×D4)
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×C3⋊D20 — C15⋊20(C4×D4)
 Lower central C15 — C30 — C15⋊20(C4×D4)
 Upper central C1 — C22 — C2×C4

Generators and relations for C1520(C4×D4)
G = < a,b,c,d | a15=b4=c4=d2=1, bab-1=dad=a4, cac-1=a11, bc=cb, bd=db, dcd=c-1 >

Subgroups: 988 in 188 conjugacy classes, 62 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×8], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×4], C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×3], D6 [×4], C2×C6, C2×C6 [×4], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], Dic5, C20 [×4], D10 [×2], D10 [×6], C2×C10, C4×S3 [×2], C2×Dic3 [×2], C2×Dic3, C3⋊D4 [×4], C2×C12, C2×C12 [×3], C22×S3, C22×C6, C3×D5 [×2], D15 [×2], C30 [×3], C4×D4, C4×D5 [×4], D20 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C5×Dic3 [×2], C5×Dic3, C3×Dic5 [×2], Dic15, C60, C6×D5 [×2], C6×D5 [×2], D30 [×2], D30 [×2], C2×C30, C4×Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C4×C3⋊D4, D30.C2 [×2], C3⋊D20 [×4], D5×C12 [×2], C6×Dic5, C10×Dic3 [×2], C2×Dic15, C2×C60, D5×C2×C6, C22×D15, D208C4, Dic3×Dic5, D10⋊Dic3, C5×Dic3⋊C4, D303C4, C2×D30.C2, C2×C3⋊D20, D5×C2×C12, C1520(C4×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, D6 [×3], C22×C4, C2×D4, C4○D4, D10 [×3], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C4×D4, C4×D5 [×2], C22×D5, S3×C2×C4, C4○D12, C2×C3⋊D4, S3×D5, C2×C4×D5, D4×D5, Q82D5, C4×C3⋊D4, C2×S3×D5, D208C4, C12.28D10, C4×S3×D5, D5×C3⋊D4, C1520(C4×D4)

Smallest permutation representation of C1520(C4×D4)
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 65 32 92)(2 69 33 96)(3 73 34 100)(4 62 35 104)(5 66 36 93)(6 70 37 97)(7 74 38 101)(8 63 39 105)(9 67 40 94)(10 71 41 98)(11 75 42 102)(12 64 43 91)(13 68 44 95)(14 72 45 99)(15 61 31 103)(16 90 57 117)(17 79 58 106)(18 83 59 110)(19 87 60 114)(20 76 46 118)(21 80 47 107)(22 84 48 111)(23 88 49 115)(24 77 50 119)(25 81 51 108)(26 85 52 112)(27 89 53 116)(28 78 54 120)(29 82 55 109)(30 86 56 113)(121 189 161 223)(122 193 162 212)(123 182 163 216)(124 186 164 220)(125 190 165 224)(126 194 151 213)(127 183 152 217)(128 187 153 221)(129 191 154 225)(130 195 155 214)(131 184 156 218)(132 188 157 222)(133 192 158 211)(134 181 159 215)(135 185 160 219)(136 209 174 231)(137 198 175 235)(138 202 176 239)(139 206 177 228)(140 210 178 232)(141 199 179 236)(142 203 180 240)(143 207 166 229)(144 196 167 233)(145 200 168 237)(146 204 169 226)(147 208 170 230)(148 197 171 234)(149 201 172 238)(150 205 173 227)
(1 168 20 156)(2 179 21 152)(3 175 22 163)(4 171 23 159)(5 167 24 155)(6 178 25 151)(7 174 26 162)(8 170 27 158)(9 166 28 154)(10 177 29 165)(11 173 30 161)(12 169 16 157)(13 180 17 153)(14 176 18 164)(15 172 19 160)(31 149 60 135)(32 145 46 131)(33 141 47 127)(34 137 48 123)(35 148 49 134)(36 144 50 130)(37 140 51 126)(38 136 52 122)(39 147 53 133)(40 143 54 129)(41 139 55 125)(42 150 56 121)(43 146 57 132)(44 142 58 128)(45 138 59 124)(61 238 87 219)(62 234 88 215)(63 230 89 211)(64 226 90 222)(65 237 76 218)(66 233 77 214)(67 229 78 225)(68 240 79 221)(69 236 80 217)(70 232 81 213)(71 228 82 224)(72 239 83 220)(73 235 84 216)(74 231 85 212)(75 227 86 223)(91 204 117 188)(92 200 118 184)(93 196 119 195)(94 207 120 191)(95 203 106 187)(96 199 107 183)(97 210 108 194)(98 206 109 190)(99 202 110 186)(100 198 111 182)(101 209 112 193)(102 205 113 189)(103 201 114 185)(104 197 115 181)(105 208 116 192)
(2 5)(3 9)(4 13)(7 10)(8 14)(12 15)(16 19)(17 23)(18 27)(21 24)(22 28)(26 29)(31 43)(33 36)(34 40)(35 44)(38 41)(39 45)(47 50)(48 54)(49 58)(52 55)(53 59)(57 60)(61 64)(62 68)(63 72)(66 69)(67 73)(71 74)(77 80)(78 84)(79 88)(82 85)(83 89)(87 90)(91 103)(93 96)(94 100)(95 104)(98 101)(99 105)(106 115)(107 119)(109 112)(110 116)(111 120)(114 117)(121 150)(122 139)(123 143)(124 147)(125 136)(126 140)(127 144)(128 148)(129 137)(130 141)(131 145)(132 149)(133 138)(134 142)(135 146)(151 178)(152 167)(153 171)(154 175)(155 179)(156 168)(157 172)(158 176)(159 180)(160 169)(161 173)(162 177)(163 166)(164 170)(165 174)(181 203)(182 207)(183 196)(184 200)(185 204)(186 208)(187 197)(188 201)(189 205)(190 209)(191 198)(192 202)(193 206)(194 210)(195 199)(211 239)(212 228)(213 232)(214 236)(215 240)(216 229)(217 233)(218 237)(219 226)(220 230)(221 234)(222 238)(223 227)(224 231)(225 235)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 20A 20B 20C 20D 20E ··· 20L 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 12 12 12 12 12 12 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 10 30 30 2 2 2 5 5 5 5 6 6 6 6 30 30 2 2 2 2 2 10 10 10 10 2 ··· 2 2 2 2 2 10 10 10 10 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

72 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 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D5 D6 D6 D6 C4○D4 D10 D10 C3⋊D4 C4×S3 C4×D5 C4○D12 S3×D5 D4×D5 Q8⋊2D5 C2×S3×D5 C12.28D10 C4×S3×D5 D5×C3⋊D4 kernel C15⋊20(C4×D4) Dic3×Dic5 D10⋊Dic3 C5×Dic3⋊C4 D30⋊3C4 C2×D30.C2 C2×C3⋊D20 D5×C2×C12 C3⋊D20 C2×C4×D5 C3×Dic5 Dic3⋊C4 C2×Dic5 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 Dic5 D10 Dic3 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 2 1 1 1 2 4 2 4 4 8 4 2 2 2 2 4 4 4

Matrix representation of C1520(C4×D4) in GL6(𝔽61)

 13 0 0 0 0 0 0 47 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 17 43 0 0 0 0 17 0
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 17 1 0 0 0 0 17 44
,
 0 60 0 0 0 0 1 0 0 0 0 0 0 0 1 59 0 0 0 0 1 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 1 60 0 0 0 0 0 0 17 1 0 0 0 0 17 44

G:=sub<GL(6,GF(61))| [13,0,0,0,0,0,0,47,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,17,0,0,0,0,43,0],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,17,0,0,0,0,1,44],[0,1,0,0,0,0,60,0,0,0,0,0,0,0,1,1,0,0,0,0,59,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,60,0,0,0,0,0,0,1,1,0,0,0,0,0,60,0,0,0,0,0,0,17,17,0,0,0,0,1,44] >;

C1520(C4×D4) in GAP, Magma, Sage, TeX

C_{15}\rtimes_{20}(C_4\times D_4)
% in TeX

G:=Group("C15:20(C4xD4)");
// GroupNames label

G:=SmallGroup(480,520);
// by ID

G=gap.SmallGroup(480,520);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,219,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^4=c^4=d^2=1,b*a*b^-1=d*a*d=a^4,c*a*c^-1=a^11,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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