metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C4×D15)⋊10C4, C20.75(C4×S3), C12.43(C4×D5), C60.155(C2×C4), C5⋊2(C42⋊2S3), (Dic3×C20)⋊9C2, (C4×Dic5)⋊11S3, (C12×Dic5)⋊9C2, (C4×Dic3)⋊12D5, D30.40(C2×C4), (C2×C20).335D6, C3⋊1(C42⋊D5), C6.32(C4○D20), C30.48(C4○D4), (C2×C12).339D10, (C2×C30).76C23, Dic15⋊5C4⋊42C2, C15⋊11(C42⋊C2), D30⋊4C4.15C2, C10.36(C4○D12), (C2×C60).237C22, C4.24(D30.C2), C30.122(C22×C4), Dic15.48(C2×C4), (C2×Dic5).168D6, (C2×Dic3).147D10, C2.3(D6.D10), (C6×Dic5).190C22, (C22×D15).99C22, (C2×Dic15).196C22, (C10×Dic3).171C22, C6.46(C2×C4×D5), C10.79(S3×C2×C4), (C2×C4×D15).22C2, C22.35(C2×S3×D5), (C2×C4).240(S3×D5), C2.11(C2×D30.C2), (C2×C6).88(C22×D5), (C2×C10).88(C22×S3), SmallGroup(480,462)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C4×D15)⋊10C4
G = < a,b,c,d | a4=b15=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b11, dcd-1=a2b10c >
Subgroups: 748 in 152 conjugacy classes, 60 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, C20, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C30, C42⋊C2, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, D30, D30, C2×C30, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C2×C4×D5, C42⋊2S3, C6×Dic5, C10×Dic3, C4×D15, C2×Dic15, C2×C60, C22×D15, C42⋊D5, D30⋊4C4, Dic15⋊5C4, C12×Dic5, Dic3×C20, C2×C4×D15, (C4×D15)⋊10C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, C4○D4, D10, C4×S3, C22×S3, C42⋊C2, C4×D5, C22×D5, S3×C2×C4, C4○D12, S3×D5, C2×C4×D5, C4○D20, C42⋊2S3, D30.C2, C2×S3×D5, C42⋊D5, D6.D10, C2×D30.C2, (C4×D15)⋊10C4
►On 240 points
Generators in S240
(1 86 16 73)(2 87 17 74)(3 88 18 75)(4 89 19 61)(5 90 20 62)(6 76 21 63)(7 77 22 64)(8 78 23 65)(9 79 24 66)(10 80 25 67)(11 81 26 68)(12 82 27 69)(13 83 28 70)(14 84 29 71)(15 85 30 72)(31 106 58 98)(32 107 59 99)(33 108 60 100)(34 109 46 101)(35 110 47 102)(36 111 48 103)(37 112 49 104)(38 113 50 105)(39 114 51 91)(40 115 52 92)(41 116 53 93)(42 117 54 94)(43 118 55 95)(44 119 56 96)(45 120 57 97)(121 205 136 188)(122 206 137 189)(123 207 138 190)(124 208 139 191)(125 209 140 192)(126 210 141 193)(127 196 142 194)(128 197 143 195)(129 198 144 181)(130 199 145 182)(131 200 146 183)(132 201 147 184)(133 202 148 185)(134 203 149 186)(135 204 150 187)(151 233 173 211)(152 234 174 212)(153 235 175 213)(154 236 176 214)(155 237 177 215)(156 238 178 216)(157 239 179 217)(158 240 180 218)(159 226 166 219)(160 227 167 220)(161 228 168 221)(162 229 169 222)(163 230 170 223)(164 231 171 224)(165 232 172 225)
(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 33)(2 32)(3 31)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 60)(17 59)(18 58)(19 57)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 47)(30 46)(61 97)(62 96)(63 95)(64 94)(65 93)(66 92)(67 91)(68 105)(69 104)(70 103)(71 102)(72 101)(73 100)(74 99)(75 98)(76 118)(77 117)(78 116)(79 115)(80 114)(81 113)(82 112)(83 111)(84 110)(85 109)(86 108)(87 107)(88 106)(89 120)(90 119)(121 172)(122 171)(123 170)(124 169)(125 168)(126 167)(127 166)(128 180)(129 179)(130 178)(131 177)(132 176)(133 175)(134 174)(135 173)(136 165)(137 164)(138 163)(139 162)(140 161)(141 160)(142 159)(143 158)(144 157)(145 156)(146 155)(147 154)(148 153)(149 152)(150 151)(181 239)(182 238)(183 237)(184 236)(185 235)(186 234)(187 233)(188 232)(189 231)(190 230)(191 229)(192 228)(193 227)(194 226)(195 240)(196 219)(197 218)(198 217)(199 216)(200 215)(201 214)(202 213)(203 212)(204 211)(205 225)(206 224)(207 223)(208 222)(209 221)(210 220)
(1 151 34 121)(2 162 35 132)(3 158 36 128)(4 154 37 124)(5 165 38 135)(6 161 39 131)(7 157 40 127)(8 153 41 123)(9 164 42 134)(10 160 43 130)(11 156 44 126)(12 152 45 122)(13 163 31 133)(14 159 32 129)(15 155 33 125)(16 173 46 136)(17 169 47 147)(18 180 48 143)(19 176 49 139)(20 172 50 150)(21 168 51 146)(22 179 52 142)(23 175 53 138)(24 171 54 149)(25 167 55 145)(26 178 56 141)(27 174 57 137)(28 170 58 148)(29 166 59 144)(30 177 60 140)(61 214 104 191)(62 225 105 187)(63 221 91 183)(64 217 92 194)(65 213 93 190)(66 224 94 186)(67 220 95 182)(68 216 96 193)(69 212 97 189)(70 223 98 185)(71 219 99 181)(72 215 100 192)(73 211 101 188)(74 222 102 184)(75 218 103 195)(76 228 114 200)(77 239 115 196)(78 235 116 207)(79 231 117 203)(80 227 118 199)(81 238 119 210)(82 234 120 206)(83 230 106 202)(84 226 107 198)(85 237 108 209)(86 233 109 205)(87 229 110 201)(88 240 111 197)(89 236 112 208)(90 232 113 204)
G:=sub<Sym(240)| (1,86,16,73)(2,87,17,74)(3,88,18,75)(4,89,19,61)(5,90,20,62)(6,76,21,63)(7,77,22,64)(8,78,23,65)(9,79,24,66)(10,80,25,67)(11,81,26,68)(12,82,27,69)(13,83,28,70)(14,84,29,71)(15,85,30,72)(31,106,58,98)(32,107,59,99)(33,108,60,100)(34,109,46,101)(35,110,47,102)(36,111,48,103)(37,112,49,104)(38,113,50,105)(39,114,51,91)(40,115,52,92)(41,116,53,93)(42,117,54,94)(43,118,55,95)(44,119,56,96)(45,120,57,97)(121,205,136,188)(122,206,137,189)(123,207,138,190)(124,208,139,191)(125,209,140,192)(126,210,141,193)(127,196,142,194)(128,197,143,195)(129,198,144,181)(130,199,145,182)(131,200,146,183)(132,201,147,184)(133,202,148,185)(134,203,149,186)(135,204,150,187)(151,233,173,211)(152,234,174,212)(153,235,175,213)(154,236,176,214)(155,237,177,215)(156,238,178,216)(157,239,179,217)(158,240,180,218)(159,226,166,219)(160,227,167,220)(161,228,168,221)(162,229,169,222)(163,230,170,223)(164,231,171,224)(165,232,172,225), (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), (1,33)(2,32)(3,31)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,60)(17,59)(18,58)(19,57)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(61,97)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,118)(77,117)(78,116)(79,115)(80,114)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,120)(90,119)(121,172)(122,171)(123,170)(124,169)(125,168)(126,167)(127,166)(128,180)(129,179)(130,178)(131,177)(132,176)(133,175)(134,174)(135,173)(136,165)(137,164)(138,163)(139,162)(140,161)(141,160)(142,159)(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)(181,239)(182,238)(183,237)(184,236)(185,235)(186,234)(187,233)(188,232)(189,231)(190,230)(191,229)(192,228)(193,227)(194,226)(195,240)(196,219)(197,218)(198,217)(199,216)(200,215)(201,214)(202,213)(203,212)(204,211)(205,225)(206,224)(207,223)(208,222)(209,221)(210,220), (1,151,34,121)(2,162,35,132)(3,158,36,128)(4,154,37,124)(5,165,38,135)(6,161,39,131)(7,157,40,127)(8,153,41,123)(9,164,42,134)(10,160,43,130)(11,156,44,126)(12,152,45,122)(13,163,31,133)(14,159,32,129)(15,155,33,125)(16,173,46,136)(17,169,47,147)(18,180,48,143)(19,176,49,139)(20,172,50,150)(21,168,51,146)(22,179,52,142)(23,175,53,138)(24,171,54,149)(25,167,55,145)(26,178,56,141)(27,174,57,137)(28,170,58,148)(29,166,59,144)(30,177,60,140)(61,214,104,191)(62,225,105,187)(63,221,91,183)(64,217,92,194)(65,213,93,190)(66,224,94,186)(67,220,95,182)(68,216,96,193)(69,212,97,189)(70,223,98,185)(71,219,99,181)(72,215,100,192)(73,211,101,188)(74,222,102,184)(75,218,103,195)(76,228,114,200)(77,239,115,196)(78,235,116,207)(79,231,117,203)(80,227,118,199)(81,238,119,210)(82,234,120,206)(83,230,106,202)(84,226,107,198)(85,237,108,209)(86,233,109,205)(87,229,110,201)(88,240,111,197)(89,236,112,208)(90,232,113,204)>;
G:=Group( (1,86,16,73)(2,87,17,74)(3,88,18,75)(4,89,19,61)(5,90,20,62)(6,76,21,63)(7,77,22,64)(8,78,23,65)(9,79,24,66)(10,80,25,67)(11,81,26,68)(12,82,27,69)(13,83,28,70)(14,84,29,71)(15,85,30,72)(31,106,58,98)(32,107,59,99)(33,108,60,100)(34,109,46,101)(35,110,47,102)(36,111,48,103)(37,112,49,104)(38,113,50,105)(39,114,51,91)(40,115,52,92)(41,116,53,93)(42,117,54,94)(43,118,55,95)(44,119,56,96)(45,120,57,97)(121,205,136,188)(122,206,137,189)(123,207,138,190)(124,208,139,191)(125,209,140,192)(126,210,141,193)(127,196,142,194)(128,197,143,195)(129,198,144,181)(130,199,145,182)(131,200,146,183)(132,201,147,184)(133,202,148,185)(134,203,149,186)(135,204,150,187)(151,233,173,211)(152,234,174,212)(153,235,175,213)(154,236,176,214)(155,237,177,215)(156,238,178,216)(157,239,179,217)(158,240,180,218)(159,226,166,219)(160,227,167,220)(161,228,168,221)(162,229,169,222)(163,230,170,223)(164,231,171,224)(165,232,172,225), (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), (1,33)(2,32)(3,31)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,60)(17,59)(18,58)(19,57)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(61,97)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,118)(77,117)(78,116)(79,115)(80,114)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,120)(90,119)(121,172)(122,171)(123,170)(124,169)(125,168)(126,167)(127,166)(128,180)(129,179)(130,178)(131,177)(132,176)(133,175)(134,174)(135,173)(136,165)(137,164)(138,163)(139,162)(140,161)(141,160)(142,159)(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)(181,239)(182,238)(183,237)(184,236)(185,235)(186,234)(187,233)(188,232)(189,231)(190,230)(191,229)(192,228)(193,227)(194,226)(195,240)(196,219)(197,218)(198,217)(199,216)(200,215)(201,214)(202,213)(203,212)(204,211)(205,225)(206,224)(207,223)(208,222)(209,221)(210,220), (1,151,34,121)(2,162,35,132)(3,158,36,128)(4,154,37,124)(5,165,38,135)(6,161,39,131)(7,157,40,127)(8,153,41,123)(9,164,42,134)(10,160,43,130)(11,156,44,126)(12,152,45,122)(13,163,31,133)(14,159,32,129)(15,155,33,125)(16,173,46,136)(17,169,47,147)(18,180,48,143)(19,176,49,139)(20,172,50,150)(21,168,51,146)(22,179,52,142)(23,175,53,138)(24,171,54,149)(25,167,55,145)(26,178,56,141)(27,174,57,137)(28,170,58,148)(29,166,59,144)(30,177,60,140)(61,214,104,191)(62,225,105,187)(63,221,91,183)(64,217,92,194)(65,213,93,190)(66,224,94,186)(67,220,95,182)(68,216,96,193)(69,212,97,189)(70,223,98,185)(71,219,99,181)(72,215,100,192)(73,211,101,188)(74,222,102,184)(75,218,103,195)(76,228,114,200)(77,239,115,196)(78,235,116,207)(79,231,117,203)(80,227,118,199)(81,238,119,210)(82,234,120,206)(83,230,106,202)(84,226,107,198)(85,237,108,209)(86,233,109,205)(87,229,110,201)(88,240,111,197)(89,236,112,208)(90,232,113,204) );
G=PermutationGroup([[(1,86,16,73),(2,87,17,74),(3,88,18,75),(4,89,19,61),(5,90,20,62),(6,76,21,63),(7,77,22,64),(8,78,23,65),(9,79,24,66),(10,80,25,67),(11,81,26,68),(12,82,27,69),(13,83,28,70),(14,84,29,71),(15,85,30,72),(31,106,58,98),(32,107,59,99),(33,108,60,100),(34,109,46,101),(35,110,47,102),(36,111,48,103),(37,112,49,104),(38,113,50,105),(39,114,51,91),(40,115,52,92),(41,116,53,93),(42,117,54,94),(43,118,55,95),(44,119,56,96),(45,120,57,97),(121,205,136,188),(122,206,137,189),(123,207,138,190),(124,208,139,191),(125,209,140,192),(126,210,141,193),(127,196,142,194),(128,197,143,195),(129,198,144,181),(130,199,145,182),(131,200,146,183),(132,201,147,184),(133,202,148,185),(134,203,149,186),(135,204,150,187),(151,233,173,211),(152,234,174,212),(153,235,175,213),(154,236,176,214),(155,237,177,215),(156,238,178,216),(157,239,179,217),(158,240,180,218),(159,226,166,219),(160,227,167,220),(161,228,168,221),(162,229,169,222),(163,230,170,223),(164,231,171,224),(165,232,172,225)], [(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)], [(1,33),(2,32),(3,31),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,36),(14,35),(15,34),(16,60),(17,59),(18,58),(19,57),(20,56),(21,55),(22,54),(23,53),(24,52),(25,51),(26,50),(27,49),(28,48),(29,47),(30,46),(61,97),(62,96),(63,95),(64,94),(65,93),(66,92),(67,91),(68,105),(69,104),(70,103),(71,102),(72,101),(73,100),(74,99),(75,98),(76,118),(77,117),(78,116),(79,115),(80,114),(81,113),(82,112),(83,111),(84,110),(85,109),(86,108),(87,107),(88,106),(89,120),(90,119),(121,172),(122,171),(123,170),(124,169),(125,168),(126,167),(127,166),(128,180),(129,179),(130,178),(131,177),(132,176),(133,175),(134,174),(135,173),(136,165),(137,164),(138,163),(139,162),(140,161),(141,160),(142,159),(143,158),(144,157),(145,156),(146,155),(147,154),(148,153),(149,152),(150,151),(181,239),(182,238),(183,237),(184,236),(185,235),(186,234),(187,233),(188,232),(189,231),(190,230),(191,229),(192,228),(193,227),(194,226),(195,240),(196,219),(197,218),(198,217),(199,216),(200,215),(201,214),(202,213),(203,212),(204,211),(205,225),(206,224),(207,223),(208,222),(209,221),(210,220)], [(1,151,34,121),(2,162,35,132),(3,158,36,128),(4,154,37,124),(5,165,38,135),(6,161,39,131),(7,157,40,127),(8,153,41,123),(9,164,42,134),(10,160,43,130),(11,156,44,126),(12,152,45,122),(13,163,31,133),(14,159,32,129),(15,155,33,125),(16,173,46,136),(17,169,47,147),(18,180,48,143),(19,176,49,139),(20,172,50,150),(21,168,51,146),(22,179,52,142),(23,175,53,138),(24,171,54,149),(25,167,55,145),(26,178,56,141),(27,174,57,137),(28,170,58,148),(29,166,59,144),(30,177,60,140),(61,214,104,191),(62,225,105,187),(63,221,91,183),(64,217,92,194),(65,213,93,190),(66,224,94,186),(67,220,95,182),(68,216,96,193),(69,212,97,189),(70,223,98,185),(71,219,99,181),(72,215,100,192),(73,211,101,188),(74,222,102,184),(75,218,103,195),(76,228,114,200),(77,239,115,196),(78,235,116,207),(79,231,117,203),(80,227,118,199),(81,238,119,210),(82,234,120,206),(83,230,106,202),(84,226,107,198),(85,237,108,209),(86,233,109,205),(87,229,110,201),(88,240,111,197),(89,236,112,208),(90,232,113,204)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C4×S3 | C4×D5 | C4○D12 | C4○D20 | S3×D5 | D30.C2 | C2×S3×D5 | D6.D10 |
| kernel | (C4×D15)⋊10C4 | D30⋊4C4 | Dic15⋊5C4 | C12×Dic5 | Dic3×C20 | C2×C4×D15 | C4×D15 | C4×Dic5 | C4×Dic3 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C20 | C12 | C10 | C6 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 8 | 8 | 16 | 2 | 4 | 2 | 8 |
Matrix representation of (C4×D15)⋊10C4 ►in GL4(𝔽61) generated by
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
| 18 | 18 | 0 | 0 |
| 43 | 60 | 0 | 0 |
| 0 | 0 | 1 | 41 |
| 0 | 0 | 52 | 59 |
| 18 | 18 | 0 | 0 |
| 60 | 43 | 0 | 0 |
| 0 | 0 | 59 | 20 |
| 0 | 0 | 9 | 2 |
| 31 | 17 | 0 | 0 |
| 44 | 30 | 0 | 0 |
| 0 | 0 | 34 | 58 |
| 0 | 0 | 40 | 27 |
G:=sub<GL(4,GF(61))| [50,0,0,0,0,50,0,0,0,0,50,0,0,0,0,50],[18,43,0,0,18,60,0,0,0,0,1,52,0,0,41,59],[18,60,0,0,18,43,0,0,0,0,59,9,0,0,20,2],[31,44,0,0,17,30,0,0,0,0,34,40,0,0,58,27] >;
(C4×D15)⋊10C4 in GAP, Magma, Sage, TeX
(C_4\times D_{15})\rtimes_{10}C_4 % in TeX
G:=Group("(C4xD15):10C4"); // GroupNames label
G:=SmallGroup(480,462);
// by ID
G=gap.SmallGroup(480,462);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,64,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^15=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11,d*c*d^-1=a^2*b^10*c>;
// generators/relations