metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C4×D15)⋊8C4, C20.58(C4×S3), C4⋊Dic3⋊11D5, C4⋊Dic5⋊11S3, C12.26(C4×D5), C60.134(C2×C4), D30.39(C2×C4), (C2×C20).109D6, C15⋊7(C42⋊C2), (Dic3×Dic5)⋊7C2, D30⋊4C4.3C2, C30.15(C4○D4), (C2×C12).111D10, C6.5(D4⋊2D5), (C2×C30).37C23, (C2×Dic5).93D6, C10.4(D4⋊2S3), C2.2(D20⋊S3), C2.2(D12⋊D5), (C2×C60).193C22, C30.118(C22×C4), C4.15(D30.C2), C6.23(Q8⋊2D5), Dic15.47(C2×C4), (C2×Dic3).86D10, C10.24(Q8⋊3S3), (C6×Dic5).22C22, (C10×Dic3).21C22, (C22×D15).94C22, (C2×Dic15).182C22, C6.45(C2×C4×D5), C10.78(S3×C2×C4), C5⋊3(C4⋊C4⋊7S3), C3⋊2(C4⋊C4⋊7D5), (C3×C4⋊Dic5)⋊8C2, (C5×C4⋊Dic3)⋊8C2, (C2×C4×D15).13C2, C22.31(C2×S3×D5), (C2×C4).203(S3×D5), C2.10(C2×D30.C2), (C2×C6).49(C22×D5), (C2×C10).49(C22×S3), SmallGroup(480,423)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C4×D15)⋊8C4
G = < a,b,c,d | a4=b15=c2=d4=1, ab=ba, ac=ca, dad-1=a-1, cbc=b-1, dbd-1=b11, dcd-1=a2b10c >
Subgroups: 748 in 152 conjugacy classes, 60 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, D5, 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, C42⋊C2, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, D30, D30, C2×C30, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C4⋊C4⋊7S3, C6×Dic5, C10×Dic3, C4×D15, C2×Dic15, C2×C60, C22×D15, C4⋊C4⋊7D5, Dic3×Dic5, D30⋊4C4, C3×C4⋊Dic5, C5×C4⋊Dic3, C2×C4×D15, (C4×D15)⋊8C4
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, D4⋊2S3, Q8⋊3S3, S3×D5, C2×C4×D5, D4⋊2D5, Q8⋊2D5, C4⋊C4⋊7S3, D30.C2, C2×S3×D5, C4⋊C4⋊7D5, D20⋊S3, D12⋊D5, C2×D30.C2, (C4×D15)⋊8C4
►On 240 points
Generators in S240
(1 90 24 73)(2 76 25 74)(3 77 26 75)(4 78 27 61)(5 79 28 62)(6 80 29 63)(7 81 30 64)(8 82 16 65)(9 83 17 66)(10 84 18 67)(11 85 19 68)(12 86 20 69)(13 87 21 70)(14 88 22 71)(15 89 23 72)(31 113 55 92)(32 114 56 93)(33 115 57 94)(34 116 58 95)(35 117 59 96)(36 118 60 97)(37 119 46 98)(38 120 47 99)(39 106 48 100)(40 107 49 101)(41 108 50 102)(42 109 51 103)(43 110 52 104)(44 111 53 105)(45 112 54 91)(121 185 138 209)(122 186 139 210)(123 187 140 196)(124 188 141 197)(125 189 142 198)(126 190 143 199)(127 191 144 200)(128 192 145 201)(129 193 146 202)(130 194 147 203)(131 195 148 204)(132 181 149 205)(133 182 150 206)(134 183 136 207)(135 184 137 208)(151 211 173 237)(152 212 174 238)(153 213 175 239)(154 214 176 240)(155 215 177 226)(156 216 178 227)(157 217 179 228)(158 218 180 229)(159 219 166 230)(160 220 167 231)(161 221 168 232)(162 222 169 233)(163 223 170 234)(164 224 171 235)(165 225 172 236)
(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 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 36)(10 35)(11 34)(12 33)(13 32)(14 31)(15 45)(16 46)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(61 102)(62 101)(63 100)(64 99)(65 98)(66 97)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)(73 105)(74 104)(75 103)(76 110)(77 109)(78 108)(79 107)(80 106)(81 120)(82 119)(83 118)(84 117)(85 116)(86 115)(87 114)(88 113)(89 112)(90 111)(121 166)(122 180)(123 179)(124 178)(125 177)(126 176)(127 175)(128 174)(129 173)(130 172)(131 171)(132 170)(133 169)(134 168)(135 167)(136 161)(137 160)(138 159)(139 158)(140 157)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 165)(148 164)(149 163)(150 162)(181 234)(182 233)(183 232)(184 231)(185 230)(186 229)(187 228)(188 227)(189 226)(190 240)(191 239)(192 238)(193 237)(194 236)(195 235)(196 217)(197 216)(198 215)(199 214)(200 213)(201 212)(202 211)(203 225)(204 224)(205 223)(206 222)(207 221)(208 220)(209 219)(210 218)
(1 173 54 130)(2 169 55 126)(3 180 56 122)(4 176 57 133)(5 172 58 129)(6 168 59 125)(7 179 60 121)(8 175 46 132)(9 171 47 128)(10 167 48 124)(11 178 49 135)(12 174 50 131)(13 170 51 127)(14 166 52 123)(15 177 53 134)(16 153 37 149)(17 164 38 145)(18 160 39 141)(19 156 40 137)(20 152 41 148)(21 163 42 144)(22 159 43 140)(23 155 44 136)(24 151 45 147)(25 162 31 143)(26 158 32 139)(27 154 33 150)(28 165 34 146)(29 161 35 142)(30 157 36 138)(61 240 115 182)(62 236 116 193)(63 232 117 189)(64 228 118 185)(65 239 119 181)(66 235 120 192)(67 231 106 188)(68 227 107 184)(69 238 108 195)(70 234 109 191)(71 230 110 187)(72 226 111 183)(73 237 112 194)(74 233 113 190)(75 229 114 186)(76 222 92 199)(77 218 93 210)(78 214 94 206)(79 225 95 202)(80 221 96 198)(81 217 97 209)(82 213 98 205)(83 224 99 201)(84 220 100 197)(85 216 101 208)(86 212 102 204)(87 223 103 200)(88 219 104 196)(89 215 105 207)(90 211 91 203)
G:=sub<Sym(240)| (1,90,24,73)(2,76,25,74)(3,77,26,75)(4,78,27,61)(5,79,28,62)(6,80,29,63)(7,81,30,64)(8,82,16,65)(9,83,17,66)(10,84,18,67)(11,85,19,68)(12,86,20,69)(13,87,21,70)(14,88,22,71)(15,89,23,72)(31,113,55,92)(32,114,56,93)(33,115,57,94)(34,116,58,95)(35,117,59,96)(36,118,60,97)(37,119,46,98)(38,120,47,99)(39,106,48,100)(40,107,49,101)(41,108,50,102)(42,109,51,103)(43,110,52,104)(44,111,53,105)(45,112,54,91)(121,185,138,209)(122,186,139,210)(123,187,140,196)(124,188,141,197)(125,189,142,198)(126,190,143,199)(127,191,144,200)(128,192,145,201)(129,193,146,202)(130,194,147,203)(131,195,148,204)(132,181,149,205)(133,182,150,206)(134,183,136,207)(135,184,137,208)(151,211,173,237)(152,212,174,238)(153,213,175,239)(154,214,176,240)(155,215,177,226)(156,216,178,227)(157,217,179,228)(158,218,180,229)(159,219,166,230)(160,220,167,231)(161,221,168,232)(162,222,169,233)(163,223,170,234)(164,224,171,235)(165,225,172,236), (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), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,45)(16,46)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,105)(74,104)(75,103)(76,110)(77,109)(78,108)(79,107)(80,106)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111)(121,166)(122,180)(123,179)(124,178)(125,177)(126,176)(127,175)(128,174)(129,173)(130,172)(131,171)(132,170)(133,169)(134,168)(135,167)(136,161)(137,160)(138,159)(139,158)(140,157)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,165)(148,164)(149,163)(150,162)(181,234)(182,233)(183,232)(184,231)(185,230)(186,229)(187,228)(188,227)(189,226)(190,240)(191,239)(192,238)(193,237)(194,236)(195,235)(196,217)(197,216)(198,215)(199,214)(200,213)(201,212)(202,211)(203,225)(204,224)(205,223)(206,222)(207,221)(208,220)(209,219)(210,218), (1,173,54,130)(2,169,55,126)(3,180,56,122)(4,176,57,133)(5,172,58,129)(6,168,59,125)(7,179,60,121)(8,175,46,132)(9,171,47,128)(10,167,48,124)(11,178,49,135)(12,174,50,131)(13,170,51,127)(14,166,52,123)(15,177,53,134)(16,153,37,149)(17,164,38,145)(18,160,39,141)(19,156,40,137)(20,152,41,148)(21,163,42,144)(22,159,43,140)(23,155,44,136)(24,151,45,147)(25,162,31,143)(26,158,32,139)(27,154,33,150)(28,165,34,146)(29,161,35,142)(30,157,36,138)(61,240,115,182)(62,236,116,193)(63,232,117,189)(64,228,118,185)(65,239,119,181)(66,235,120,192)(67,231,106,188)(68,227,107,184)(69,238,108,195)(70,234,109,191)(71,230,110,187)(72,226,111,183)(73,237,112,194)(74,233,113,190)(75,229,114,186)(76,222,92,199)(77,218,93,210)(78,214,94,206)(79,225,95,202)(80,221,96,198)(81,217,97,209)(82,213,98,205)(83,224,99,201)(84,220,100,197)(85,216,101,208)(86,212,102,204)(87,223,103,200)(88,219,104,196)(89,215,105,207)(90,211,91,203)>;
G:=Group( (1,90,24,73)(2,76,25,74)(3,77,26,75)(4,78,27,61)(5,79,28,62)(6,80,29,63)(7,81,30,64)(8,82,16,65)(9,83,17,66)(10,84,18,67)(11,85,19,68)(12,86,20,69)(13,87,21,70)(14,88,22,71)(15,89,23,72)(31,113,55,92)(32,114,56,93)(33,115,57,94)(34,116,58,95)(35,117,59,96)(36,118,60,97)(37,119,46,98)(38,120,47,99)(39,106,48,100)(40,107,49,101)(41,108,50,102)(42,109,51,103)(43,110,52,104)(44,111,53,105)(45,112,54,91)(121,185,138,209)(122,186,139,210)(123,187,140,196)(124,188,141,197)(125,189,142,198)(126,190,143,199)(127,191,144,200)(128,192,145,201)(129,193,146,202)(130,194,147,203)(131,195,148,204)(132,181,149,205)(133,182,150,206)(134,183,136,207)(135,184,137,208)(151,211,173,237)(152,212,174,238)(153,213,175,239)(154,214,176,240)(155,215,177,226)(156,216,178,227)(157,217,179,228)(158,218,180,229)(159,219,166,230)(160,220,167,231)(161,221,168,232)(162,222,169,233)(163,223,170,234)(164,224,171,235)(165,225,172,236), (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), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,45)(16,46)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,105)(74,104)(75,103)(76,110)(77,109)(78,108)(79,107)(80,106)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111)(121,166)(122,180)(123,179)(124,178)(125,177)(126,176)(127,175)(128,174)(129,173)(130,172)(131,171)(132,170)(133,169)(134,168)(135,167)(136,161)(137,160)(138,159)(139,158)(140,157)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,165)(148,164)(149,163)(150,162)(181,234)(182,233)(183,232)(184,231)(185,230)(186,229)(187,228)(188,227)(189,226)(190,240)(191,239)(192,238)(193,237)(194,236)(195,235)(196,217)(197,216)(198,215)(199,214)(200,213)(201,212)(202,211)(203,225)(204,224)(205,223)(206,222)(207,221)(208,220)(209,219)(210,218), (1,173,54,130)(2,169,55,126)(3,180,56,122)(4,176,57,133)(5,172,58,129)(6,168,59,125)(7,179,60,121)(8,175,46,132)(9,171,47,128)(10,167,48,124)(11,178,49,135)(12,174,50,131)(13,170,51,127)(14,166,52,123)(15,177,53,134)(16,153,37,149)(17,164,38,145)(18,160,39,141)(19,156,40,137)(20,152,41,148)(21,163,42,144)(22,159,43,140)(23,155,44,136)(24,151,45,147)(25,162,31,143)(26,158,32,139)(27,154,33,150)(28,165,34,146)(29,161,35,142)(30,157,36,138)(61,240,115,182)(62,236,116,193)(63,232,117,189)(64,228,118,185)(65,239,119,181)(66,235,120,192)(67,231,106,188)(68,227,107,184)(69,238,108,195)(70,234,109,191)(71,230,110,187)(72,226,111,183)(73,237,112,194)(74,233,113,190)(75,229,114,186)(76,222,92,199)(77,218,93,210)(78,214,94,206)(79,225,95,202)(80,221,96,198)(81,217,97,209)(82,213,98,205)(83,224,99,201)(84,220,100,197)(85,216,101,208)(86,212,102,204)(87,223,103,200)(88,219,104,196)(89,215,105,207)(90,211,91,203) );
G=PermutationGroup([[(1,90,24,73),(2,76,25,74),(3,77,26,75),(4,78,27,61),(5,79,28,62),(6,80,29,63),(7,81,30,64),(8,82,16,65),(9,83,17,66),(10,84,18,67),(11,85,19,68),(12,86,20,69),(13,87,21,70),(14,88,22,71),(15,89,23,72),(31,113,55,92),(32,114,56,93),(33,115,57,94),(34,116,58,95),(35,117,59,96),(36,118,60,97),(37,119,46,98),(38,120,47,99),(39,106,48,100),(40,107,49,101),(41,108,50,102),(42,109,51,103),(43,110,52,104),(44,111,53,105),(45,112,54,91),(121,185,138,209),(122,186,139,210),(123,187,140,196),(124,188,141,197),(125,189,142,198),(126,190,143,199),(127,191,144,200),(128,192,145,201),(129,193,146,202),(130,194,147,203),(131,195,148,204),(132,181,149,205),(133,182,150,206),(134,183,136,207),(135,184,137,208),(151,211,173,237),(152,212,174,238),(153,213,175,239),(154,214,176,240),(155,215,177,226),(156,216,178,227),(157,217,179,228),(158,218,180,229),(159,219,166,230),(160,220,167,231),(161,221,168,232),(162,222,169,233),(163,223,170,234),(164,224,171,235),(165,225,172,236)], [(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)], [(1,44),(2,43),(3,42),(4,41),(5,40),(6,39),(7,38),(8,37),(9,36),(10,35),(11,34),(12,33),(13,32),(14,31),(15,45),(16,46),(17,60),(18,59),(19,58),(20,57),(21,56),(22,55),(23,54),(24,53),(25,52),(26,51),(27,50),(28,49),(29,48),(30,47),(61,102),(62,101),(63,100),(64,99),(65,98),(66,97),(67,96),(68,95),(69,94),(70,93),(71,92),(72,91),(73,105),(74,104),(75,103),(76,110),(77,109),(78,108),(79,107),(80,106),(81,120),(82,119),(83,118),(84,117),(85,116),(86,115),(87,114),(88,113),(89,112),(90,111),(121,166),(122,180),(123,179),(124,178),(125,177),(126,176),(127,175),(128,174),(129,173),(130,172),(131,171),(132,170),(133,169),(134,168),(135,167),(136,161),(137,160),(138,159),(139,158),(140,157),(141,156),(142,155),(143,154),(144,153),(145,152),(146,151),(147,165),(148,164),(149,163),(150,162),(181,234),(182,233),(183,232),(184,231),(185,230),(186,229),(187,228),(188,227),(189,226),(190,240),(191,239),(192,238),(193,237),(194,236),(195,235),(196,217),(197,216),(198,215),(199,214),(200,213),(201,212),(202,211),(203,225),(204,224),(205,223),(206,222),(207,221),(208,220),(209,219),(210,218)], [(1,173,54,130),(2,169,55,126),(3,180,56,122),(4,176,57,133),(5,172,58,129),(6,168,59,125),(7,179,60,121),(8,175,46,132),(9,171,47,128),(10,167,48,124),(11,178,49,135),(12,174,50,131),(13,170,51,127),(14,166,52,123),(15,177,53,134),(16,153,37,149),(17,164,38,145),(18,160,39,141),(19,156,40,137),(20,152,41,148),(21,163,42,144),(22,159,43,140),(23,155,44,136),(24,151,45,147),(25,162,31,143),(26,158,32,139),(27,154,33,150),(28,165,34,146),(29,161,35,142),(30,157,36,138),(61,240,115,182),(62,236,116,193),(63,232,117,189),(64,228,118,185),(65,239,119,181),(66,235,120,192),(67,231,106,188),(68,227,107,184),(69,238,108,195),(70,234,109,191),(71,230,110,187),(72,226,111,183),(73,237,112,194),(74,233,113,190),(75,229,114,186),(76,222,92,199),(77,218,93,210),(78,214,94,206),(79,225,95,202),(80,221,96,198),(81,217,97,209),(82,213,98,205),(83,224,99,201),(84,220,100,197),(85,216,101,208),(86,212,102,204),(87,223,103,200),(88,219,104,196),(89,215,105,207),(90,211,91,203)]])
66 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 | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 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 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 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 | D4⋊2S3 | Q8⋊3S3 | S3×D5 | D4⋊2D5 | Q8⋊2D5 | D30.C2 | C2×S3×D5 | D20⋊S3 | D12⋊D5 |
| kernel | (C4×D15)⋊8C4 | Dic3×Dic5 | D30⋊4C4 | C3×C4⋊Dic5 | C5×C4⋊Dic3 | C2×C4×D15 | C4×D15 | C4⋊Dic5 | C4⋊Dic3 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C20 | C12 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of (C4×D15)⋊8C4 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 50 |
| 59 | 15 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 44 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 46 | 0 | 0 | 0 | 0 |
| 49 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 18 | 0 | 0 |
| 0 | 0 | 45 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 51 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,50],[59,12,0,0,0,0,15,1,0,0,0,0,0,0,0,44,0,0,0,0,18,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,49,0,0,0,0,46,59,0,0,0,0,0,0,44,45,0,0,0,0,18,17,0,0,0,0,0,0,60,0,0,0,0,0,0,1],[11,51,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
(C4×D15)⋊8C4 in GAP, Magma, Sage, TeX
(C_4\times D_{15})\rtimes_8C_4 % in TeX
G:=Group("(C4xD15):8C4"); // GroupNames label
G:=SmallGroup(480,423);
// by ID
G=gap.SmallGroup(480,423);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,176,422,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^15=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=b^-1,d*b*d^-1=b^11,d*c*d^-1=a^2*b^10*c>;
// generators/relations