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
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊20(C4×D4), C3⋊D20⋊5C4, D10⋊10(C4×S3), D30⋊12(C2×C4), Dic3⋊2(C4×D5), C6.134(D4×D5), C3⋊4(D20⋊8C4), Dic3⋊C4⋊21D5, (C3×Dic5)⋊12D4, C30.144(C2×D4), (C2×C20).200D6, D30⋊3C4⋊26C2, C30.79(C4○D4), (C2×C12).267D10, Dic5⋊7(C3⋊D4), C30.63(C22×C4), (C22×D5).88D6, (Dic3×Dic5)⋊21C2, C10.14(C4○D12), D10⋊Dic3⋊13C2, C2.5(C12.28D10), (C2×C60).390C22, (C2×C30).134C23, C6.16(Q8⋊2D5), (C2×Dic5).216D6, (C2×Dic3).112D10, (C10×Dic3).84C22, (C6×Dic5).206C22, (C22×D15).44C22, (C2×Dic15).104C22, C5⋊3(C4×C3⋊D4), (C2×C4×D5)⋊11S3, C2.33(C4×S3×D5), C6.31(C2×C4×D5), (D5×C2×C12)⋊19C2, C10.64(S3×C2×C4), C2.3(D5×C3⋊D4), (C6×D5)⋊20(C2×C4), C22.66(C2×S3×D5), (C2×D30.C2)⋊9C2, (C2×C4).130(S3×D5), (C2×C3⋊D20).8C2, C10.36(C2×C3⋊D4), (C5×Dic3⋊C4)⋊28C2, (C5×Dic3)⋊10(C2×C4), (D5×C2×C6).104C22, (C2×C6).146(C22×D5), (C2×C10).146(C22×S3), SmallGroup(480,520)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊20(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, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, D10, D10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C3×D5, D15, C30, C4×D4, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, D30, D30, C2×C30, C4×Dic5, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C4×C3⋊D4, D30.C2, C3⋊D20, D5×C12, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, C22×D15, D20⋊8C4, Dic3×Dic5, D10⋊Dic3, C5×Dic3⋊C4, D30⋊3C4, C2×D30.C2, C2×C3⋊D20, D5×C2×C12, C15⋊20(C4×D4)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, D10, C4×S3, C3⋊D4, C22×S3, C4×D4, C4×D5, C22×D5, S3×C2×C4, C4○D12, C2×C3⋊D4, S3×D5, C2×C4×D5, D4×D5, Q8⋊2D5, C4×C3⋊D4, C2×S3×D5, D20⋊8C4, C12.28D10, C4×S3×D5, D5×C3⋊D4, C15⋊20(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 69 35 91)(2 73 36 95)(3 62 37 99)(4 66 38 103)(5 70 39 92)(6 74 40 96)(7 63 41 100)(8 67 42 104)(9 71 43 93)(10 75 44 97)(11 64 45 101)(12 68 31 105)(13 72 32 94)(14 61 33 98)(15 65 34 102)(16 84 46 110)(17 88 47 114)(18 77 48 118)(19 81 49 107)(20 85 50 111)(21 89 51 115)(22 78 52 119)(23 82 53 108)(24 86 54 112)(25 90 55 116)(26 79 56 120)(27 83 57 109)(28 87 58 113)(29 76 59 117)(30 80 60 106)(121 195 153 217)(122 184 154 221)(123 188 155 225)(124 192 156 214)(125 181 157 218)(126 185 158 222)(127 189 159 211)(128 193 160 215)(129 182 161 219)(130 186 162 223)(131 190 163 212)(132 194 164 216)(133 183 165 220)(134 187 151 224)(135 191 152 213)(136 209 174 233)(137 198 175 237)(138 202 176 226)(139 206 177 230)(140 210 178 234)(141 199 179 238)(142 203 180 227)(143 207 166 231)(144 196 167 235)(145 200 168 239)(146 204 169 228)(147 208 170 232)(148 197 171 236)(149 201 172 240)(150 205 173 229)
(1 172 16 152)(2 168 17 163)(3 179 18 159)(4 175 19 155)(5 171 20 151)(6 167 21 162)(7 178 22 158)(8 174 23 154)(9 170 24 165)(10 166 25 161)(11 177 26 157)(12 173 27 153)(13 169 28 164)(14 180 29 160)(15 176 30 156)(31 150 57 121)(32 146 58 132)(33 142 59 128)(34 138 60 124)(35 149 46 135)(36 145 47 131)(37 141 48 127)(38 137 49 123)(39 148 50 134)(40 144 51 130)(41 140 52 126)(42 136 53 122)(43 147 54 133)(44 143 55 129)(45 139 56 125)(61 227 76 215)(62 238 77 211)(63 234 78 222)(64 230 79 218)(65 226 80 214)(66 237 81 225)(67 233 82 221)(68 229 83 217)(69 240 84 213)(70 236 85 224)(71 232 86 220)(72 228 87 216)(73 239 88 212)(74 235 89 223)(75 231 90 219)(91 201 110 191)(92 197 111 187)(93 208 112 183)(94 204 113 194)(95 200 114 190)(96 196 115 186)(97 207 116 182)(98 203 117 193)(99 199 118 189)(100 210 119 185)(101 206 120 181)(102 202 106 192)(103 198 107 188)(104 209 108 184)(105 205 109 195)
(2 5)(3 9)(4 13)(7 10)(8 14)(12 15)(17 20)(18 24)(19 28)(22 25)(23 29)(27 30)(31 34)(32 38)(33 42)(36 39)(37 43)(41 44)(47 50)(48 54)(49 58)(52 55)(53 59)(57 60)(61 67)(62 71)(63 75)(65 68)(66 72)(70 73)(76 82)(77 86)(78 90)(80 83)(81 87)(85 88)(92 95)(93 99)(94 103)(97 100)(98 104)(102 105)(106 109)(107 113)(108 117)(111 114)(112 118)(116 119)(121 138)(122 142)(123 146)(124 150)(125 139)(126 143)(127 147)(128 136)(129 140)(130 144)(131 148)(132 137)(133 141)(134 145)(135 149)(151 168)(152 172)(153 176)(154 180)(155 169)(156 173)(157 177)(158 166)(159 170)(160 174)(161 178)(162 167)(163 171)(164 175)(165 179)(181 206)(182 210)(183 199)(184 203)(185 207)(186 196)(187 200)(188 204)(189 208)(190 197)(191 201)(192 205)(193 209)(194 198)(195 202)(211 232)(212 236)(213 240)(214 229)(215 233)(216 237)(217 226)(218 230)(219 234)(220 238)(221 227)(222 231)(223 235)(224 239)(225 228)
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), (1,69,35,91)(2,73,36,95)(3,62,37,99)(4,66,38,103)(5,70,39,92)(6,74,40,96)(7,63,41,100)(8,67,42,104)(9,71,43,93)(10,75,44,97)(11,64,45,101)(12,68,31,105)(13,72,32,94)(14,61,33,98)(15,65,34,102)(16,84,46,110)(17,88,47,114)(18,77,48,118)(19,81,49,107)(20,85,50,111)(21,89,51,115)(22,78,52,119)(23,82,53,108)(24,86,54,112)(25,90,55,116)(26,79,56,120)(27,83,57,109)(28,87,58,113)(29,76,59,117)(30,80,60,106)(121,195,153,217)(122,184,154,221)(123,188,155,225)(124,192,156,214)(125,181,157,218)(126,185,158,222)(127,189,159,211)(128,193,160,215)(129,182,161,219)(130,186,162,223)(131,190,163,212)(132,194,164,216)(133,183,165,220)(134,187,151,224)(135,191,152,213)(136,209,174,233)(137,198,175,237)(138,202,176,226)(139,206,177,230)(140,210,178,234)(141,199,179,238)(142,203,180,227)(143,207,166,231)(144,196,167,235)(145,200,168,239)(146,204,169,228)(147,208,170,232)(148,197,171,236)(149,201,172,240)(150,205,173,229), (1,172,16,152)(2,168,17,163)(3,179,18,159)(4,175,19,155)(5,171,20,151)(6,167,21,162)(7,178,22,158)(8,174,23,154)(9,170,24,165)(10,166,25,161)(11,177,26,157)(12,173,27,153)(13,169,28,164)(14,180,29,160)(15,176,30,156)(31,150,57,121)(32,146,58,132)(33,142,59,128)(34,138,60,124)(35,149,46,135)(36,145,47,131)(37,141,48,127)(38,137,49,123)(39,148,50,134)(40,144,51,130)(41,140,52,126)(42,136,53,122)(43,147,54,133)(44,143,55,129)(45,139,56,125)(61,227,76,215)(62,238,77,211)(63,234,78,222)(64,230,79,218)(65,226,80,214)(66,237,81,225)(67,233,82,221)(68,229,83,217)(69,240,84,213)(70,236,85,224)(71,232,86,220)(72,228,87,216)(73,239,88,212)(74,235,89,223)(75,231,90,219)(91,201,110,191)(92,197,111,187)(93,208,112,183)(94,204,113,194)(95,200,114,190)(96,196,115,186)(97,207,116,182)(98,203,117,193)(99,199,118,189)(100,210,119,185)(101,206,120,181)(102,202,106,192)(103,198,107,188)(104,209,108,184)(105,205,109,195), (2,5)(3,9)(4,13)(7,10)(8,14)(12,15)(17,20)(18,24)(19,28)(22,25)(23,29)(27,30)(31,34)(32,38)(33,42)(36,39)(37,43)(41,44)(47,50)(48,54)(49,58)(52,55)(53,59)(57,60)(61,67)(62,71)(63,75)(65,68)(66,72)(70,73)(76,82)(77,86)(78,90)(80,83)(81,87)(85,88)(92,95)(93,99)(94,103)(97,100)(98,104)(102,105)(106,109)(107,113)(108,117)(111,114)(112,118)(116,119)(121,138)(122,142)(123,146)(124,150)(125,139)(126,143)(127,147)(128,136)(129,140)(130,144)(131,148)(132,137)(133,141)(134,145)(135,149)(151,168)(152,172)(153,176)(154,180)(155,169)(156,173)(157,177)(158,166)(159,170)(160,174)(161,178)(162,167)(163,171)(164,175)(165,179)(181,206)(182,210)(183,199)(184,203)(185,207)(186,196)(187,200)(188,204)(189,208)(190,197)(191,201)(192,205)(193,209)(194,198)(195,202)(211,232)(212,236)(213,240)(214,229)(215,233)(216,237)(217,226)(218,230)(219,234)(220,238)(221,227)(222,231)(223,235)(224,239)(225,228)>;
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), (1,69,35,91)(2,73,36,95)(3,62,37,99)(4,66,38,103)(5,70,39,92)(6,74,40,96)(7,63,41,100)(8,67,42,104)(9,71,43,93)(10,75,44,97)(11,64,45,101)(12,68,31,105)(13,72,32,94)(14,61,33,98)(15,65,34,102)(16,84,46,110)(17,88,47,114)(18,77,48,118)(19,81,49,107)(20,85,50,111)(21,89,51,115)(22,78,52,119)(23,82,53,108)(24,86,54,112)(25,90,55,116)(26,79,56,120)(27,83,57,109)(28,87,58,113)(29,76,59,117)(30,80,60,106)(121,195,153,217)(122,184,154,221)(123,188,155,225)(124,192,156,214)(125,181,157,218)(126,185,158,222)(127,189,159,211)(128,193,160,215)(129,182,161,219)(130,186,162,223)(131,190,163,212)(132,194,164,216)(133,183,165,220)(134,187,151,224)(135,191,152,213)(136,209,174,233)(137,198,175,237)(138,202,176,226)(139,206,177,230)(140,210,178,234)(141,199,179,238)(142,203,180,227)(143,207,166,231)(144,196,167,235)(145,200,168,239)(146,204,169,228)(147,208,170,232)(148,197,171,236)(149,201,172,240)(150,205,173,229), (1,172,16,152)(2,168,17,163)(3,179,18,159)(4,175,19,155)(5,171,20,151)(6,167,21,162)(7,178,22,158)(8,174,23,154)(9,170,24,165)(10,166,25,161)(11,177,26,157)(12,173,27,153)(13,169,28,164)(14,180,29,160)(15,176,30,156)(31,150,57,121)(32,146,58,132)(33,142,59,128)(34,138,60,124)(35,149,46,135)(36,145,47,131)(37,141,48,127)(38,137,49,123)(39,148,50,134)(40,144,51,130)(41,140,52,126)(42,136,53,122)(43,147,54,133)(44,143,55,129)(45,139,56,125)(61,227,76,215)(62,238,77,211)(63,234,78,222)(64,230,79,218)(65,226,80,214)(66,237,81,225)(67,233,82,221)(68,229,83,217)(69,240,84,213)(70,236,85,224)(71,232,86,220)(72,228,87,216)(73,239,88,212)(74,235,89,223)(75,231,90,219)(91,201,110,191)(92,197,111,187)(93,208,112,183)(94,204,113,194)(95,200,114,190)(96,196,115,186)(97,207,116,182)(98,203,117,193)(99,199,118,189)(100,210,119,185)(101,206,120,181)(102,202,106,192)(103,198,107,188)(104,209,108,184)(105,205,109,195), (2,5)(3,9)(4,13)(7,10)(8,14)(12,15)(17,20)(18,24)(19,28)(22,25)(23,29)(27,30)(31,34)(32,38)(33,42)(36,39)(37,43)(41,44)(47,50)(48,54)(49,58)(52,55)(53,59)(57,60)(61,67)(62,71)(63,75)(65,68)(66,72)(70,73)(76,82)(77,86)(78,90)(80,83)(81,87)(85,88)(92,95)(93,99)(94,103)(97,100)(98,104)(102,105)(106,109)(107,113)(108,117)(111,114)(112,118)(116,119)(121,138)(122,142)(123,146)(124,150)(125,139)(126,143)(127,147)(128,136)(129,140)(130,144)(131,148)(132,137)(133,141)(134,145)(135,149)(151,168)(152,172)(153,176)(154,180)(155,169)(156,173)(157,177)(158,166)(159,170)(160,174)(161,178)(162,167)(163,171)(164,175)(165,179)(181,206)(182,210)(183,199)(184,203)(185,207)(186,196)(187,200)(188,204)(189,208)(190,197)(191,201)(192,205)(193,209)(194,198)(195,202)(211,232)(212,236)(213,240)(214,229)(215,233)(216,237)(217,226)(218,230)(219,234)(220,238)(221,227)(222,231)(223,235)(224,239)(225,228) );
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)], [(1,69,35,91),(2,73,36,95),(3,62,37,99),(4,66,38,103),(5,70,39,92),(6,74,40,96),(7,63,41,100),(8,67,42,104),(9,71,43,93),(10,75,44,97),(11,64,45,101),(12,68,31,105),(13,72,32,94),(14,61,33,98),(15,65,34,102),(16,84,46,110),(17,88,47,114),(18,77,48,118),(19,81,49,107),(20,85,50,111),(21,89,51,115),(22,78,52,119),(23,82,53,108),(24,86,54,112),(25,90,55,116),(26,79,56,120),(27,83,57,109),(28,87,58,113),(29,76,59,117),(30,80,60,106),(121,195,153,217),(122,184,154,221),(123,188,155,225),(124,192,156,214),(125,181,157,218),(126,185,158,222),(127,189,159,211),(128,193,160,215),(129,182,161,219),(130,186,162,223),(131,190,163,212),(132,194,164,216),(133,183,165,220),(134,187,151,224),(135,191,152,213),(136,209,174,233),(137,198,175,237),(138,202,176,226),(139,206,177,230),(140,210,178,234),(141,199,179,238),(142,203,180,227),(143,207,166,231),(144,196,167,235),(145,200,168,239),(146,204,169,228),(147,208,170,232),(148,197,171,236),(149,201,172,240),(150,205,173,229)], [(1,172,16,152),(2,168,17,163),(3,179,18,159),(4,175,19,155),(5,171,20,151),(6,167,21,162),(7,178,22,158),(8,174,23,154),(9,170,24,165),(10,166,25,161),(11,177,26,157),(12,173,27,153),(13,169,28,164),(14,180,29,160),(15,176,30,156),(31,150,57,121),(32,146,58,132),(33,142,59,128),(34,138,60,124),(35,149,46,135),(36,145,47,131),(37,141,48,127),(38,137,49,123),(39,148,50,134),(40,144,51,130),(41,140,52,126),(42,136,53,122),(43,147,54,133),(44,143,55,129),(45,139,56,125),(61,227,76,215),(62,238,77,211),(63,234,78,222),(64,230,79,218),(65,226,80,214),(66,237,81,225),(67,233,82,221),(68,229,83,217),(69,240,84,213),(70,236,85,224),(71,232,86,220),(72,228,87,216),(73,239,88,212),(74,235,89,223),(75,231,90,219),(91,201,110,191),(92,197,111,187),(93,208,112,183),(94,204,113,194),(95,200,114,190),(96,196,115,186),(97,207,116,182),(98,203,117,193),(99,199,118,189),(100,210,119,185),(101,206,120,181),(102,202,106,192),(103,198,107,188),(104,209,108,184),(105,205,109,195)], [(2,5),(3,9),(4,13),(7,10),(8,14),(12,15),(17,20),(18,24),(19,28),(22,25),(23,29),(27,30),(31,34),(32,38),(33,42),(36,39),(37,43),(41,44),(47,50),(48,54),(49,58),(52,55),(53,59),(57,60),(61,67),(62,71),(63,75),(65,68),(66,72),(70,73),(76,82),(77,86),(78,90),(80,83),(81,87),(85,88),(92,95),(93,99),(94,103),(97,100),(98,104),(102,105),(106,109),(107,113),(108,117),(111,114),(112,118),(116,119),(121,138),(122,142),(123,146),(124,150),(125,139),(126,143),(127,147),(128,136),(129,140),(130,144),(131,148),(132,137),(133,141),(134,145),(135,149),(151,168),(152,172),(153,176),(154,180),(155,169),(156,173),(157,177),(158,166),(159,170),(160,174),(161,178),(162,167),(163,171),(164,175),(165,179),(181,206),(182,210),(183,199),(184,203),(185,207),(186,196),(187,200),(188,204),(189,208),(190,197),(191,201),(192,205),(193,209),(194,198),(195,202),(211,232),(212,236),(213,240),(214,229),(215,233),(216,237),(217,226),(218,230),(219,234),(220,238),(221,227),(222,231),(223,235),(224,239),(225,228)]])
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 C15⋊20(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] >;
C15⋊20(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