direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4⋊D15, C30⋊7D8, D4⋊3D30, C60.14D4, D60⋊22C22, C60.74C23, (C6×D4)⋊1D5, C15⋊16(C2×D8), C6⋊3(D4⋊D5), (C5×D4)⋊18D6, (D4×C30)⋊1C2, (C2×D4)⋊1D15, (D4×C10)⋊1S3, C10⋊3(D4⋊S3), (C3×D4)⋊18D10, (C2×D60)⋊11C2, (C2×C4).47D30, (C2×C30).145D4, C30.378(C2×D4), (C2×C20).144D6, C4.5(C15⋊7D4), C15⋊3C8⋊29C22, (C2×C12).143D10, (D4×C15)⋊20C22, C12.41(C5⋊D4), C20.39(C3⋊D4), (C2×C60).70C22, C4.11(C22×D15), C20.112(C22×S3), C12.112(C22×D5), C22.21(C15⋊7D4), C3⋊4(C2×D4⋊D5), C5⋊4(C2×D4⋊S3), (C2×C15⋊3C8)⋊4C2, C2.8(C2×C15⋊7D4), C6.103(C2×C5⋊D4), C10.103(C2×C3⋊D4), (C2×C6).77(C5⋊D4), (C2×C10).77(C3⋊D4), SmallGroup(480,896)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4⋊D15
G = < a,b,c,d,e | a2=b4=c2=d15=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >
Subgroups: 980 in 152 conjugacy classes, 55 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C6, C8, C2×C4, D4, D4, C23, D5, C10, C10, C10, C12, D6, C2×C6, C2×C6, C15, C2×C8, D8, C2×D4, C2×D4, C20, D10, C2×C10, C2×C10, C3⋊C8, D12, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, D15, C30, C30, C30, C2×D8, C5⋊2C8, D20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, C2×C3⋊C8, D4⋊S3, C2×D12, C6×D4, C60, D30, C2×C30, C2×C30, C2×C5⋊2C8, D4⋊D5, C2×D20, D4×C10, C2×D4⋊S3, C15⋊3C8, D60, D60, C2×C60, D4×C15, D4×C15, C22×D15, C22×C30, C2×D4⋊D5, C2×C15⋊3C8, D4⋊D15, C2×D60, D4×C30, C2×D4⋊D15
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, D8, C2×D4, D10, C3⋊D4, C22×S3, D15, C2×D8, C5⋊D4, C22×D5, D4⋊S3, C2×C3⋊D4, D30, D4⋊D5, C2×C5⋊D4, C2×D4⋊S3, C15⋊7D4, C22×D15, C2×D4⋊D5, D4⋊D15, C2×C15⋊7D4, C2×D4⋊D15
►On 240 points
Generators in S240
(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 127)(8 128)(9 129)(10 130)(11 131)(12 132)(13 133)(14 134)(15 135)(16 144)(17 145)(18 146)(19 147)(20 148)(21 149)(22 150)(23 136)(24 137)(25 138)(26 139)(27 140)(28 141)(29 142)(30 143)(31 164)(32 165)(33 151)(34 152)(35 153)(36 154)(37 155)(38 156)(39 157)(40 158)(41 159)(42 160)(43 161)(44 162)(45 163)(46 169)(47 170)(48 171)(49 172)(50 173)(51 174)(52 175)(53 176)(54 177)(55 178)(56 179)(57 180)(58 166)(59 167)(60 168)(61 187)(62 188)(63 189)(64 190)(65 191)(66 192)(67 193)(68 194)(69 195)(70 181)(71 182)(72 183)(73 184)(74 185)(75 186)(76 209)(77 210)(78 196)(79 197)(80 198)(81 199)(82 200)(83 201)(84 202)(85 203)(86 204)(87 205)(88 206)(89 207)(90 208)(91 222)(92 223)(93 224)(94 225)(95 211)(96 212)(97 213)(98 214)(99 215)(100 216)(101 217)(102 218)(103 219)(104 220)(105 221)(106 236)(107 237)(108 238)(109 239)(110 240)(111 226)(112 227)(113 228)(114 229)(115 230)(116 231)(117 232)(118 233)(119 234)(120 235)
(1 49 21 32)(2 50 22 33)(3 51 23 34)(4 52 24 35)(5 53 25 36)(6 54 26 37)(7 55 27 38)(8 56 28 39)(9 57 29 40)(10 58 30 41)(11 59 16 42)(12 60 17 43)(13 46 18 44)(14 47 19 45)(15 48 20 31)(61 100 79 112)(62 101 80 113)(63 102 81 114)(64 103 82 115)(65 104 83 116)(66 105 84 117)(67 91 85 118)(68 92 86 119)(69 93 87 120)(70 94 88 106)(71 95 89 107)(72 96 90 108)(73 97 76 109)(74 98 77 110)(75 99 78 111)(121 172 149 165)(122 173 150 151)(123 174 136 152)(124 175 137 153)(125 176 138 154)(126 177 139 155)(127 178 140 156)(128 179 141 157)(129 180 142 158)(130 166 143 159)(131 167 144 160)(132 168 145 161)(133 169 146 162)(134 170 147 163)(135 171 148 164)(181 225 206 236)(182 211 207 237)(183 212 208 238)(184 213 209 239)(185 214 210 240)(186 215 196 226)(187 216 197 227)(188 217 198 228)(189 218 199 229)(190 219 200 230)(191 220 201 231)(192 221 202 232)(193 222 203 233)(194 223 204 234)(195 224 205 235)
(1 239)(2 240)(3 226)(4 227)(5 228)(6 229)(7 230)(8 231)(9 232)(10 233)(11 234)(12 235)(13 236)(14 237)(15 238)(16 223)(17 224)(18 225)(19 211)(20 212)(21 213)(22 214)(23 215)(24 216)(25 217)(26 218)(27 219)(28 220)(29 221)(30 222)(31 183)(32 184)(33 185)(34 186)(35 187)(36 188)(37 189)(38 190)(39 191)(40 192)(41 193)(42 194)(43 195)(44 181)(45 182)(46 206)(47 207)(48 208)(49 209)(50 210)(51 196)(52 197)(53 198)(54 199)(55 200)(56 201)(57 202)(58 203)(59 204)(60 205)(61 153)(62 154)(63 155)(64 156)(65 157)(66 158)(67 159)(68 160)(69 161)(70 162)(71 163)(72 164)(73 165)(74 151)(75 152)(76 172)(77 173)(78 174)(79 175)(80 176)(81 177)(82 178)(83 179)(84 180)(85 166)(86 167)(87 168)(88 169)(89 170)(90 171)(91 143)(92 144)(93 145)(94 146)(95 147)(96 148)(97 149)(98 150)(99 136)(100 137)(101 138)(102 139)(103 140)(104 141)(105 142)(106 133)(107 134)(108 135)(109 121)(110 122)(111 123)(112 124)(113 125)(114 126)(115 127)(116 128)(117 129)(118 130)(119 131)(120 132)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 25)(17 24)(18 23)(19 22)(20 21)(26 30)(27 29)(31 49)(32 48)(33 47)(34 46)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(41 54)(42 53)(43 52)(44 51)(45 50)(61 93)(62 92)(63 91)(64 105)(65 104)(66 103)(67 102)(68 101)(69 100)(70 99)(71 98)(72 97)(73 96)(74 95)(75 94)(76 108)(77 107)(78 106)(79 120)(80 119)(81 118)(82 117)(83 116)(84 115)(85 114)(86 113)(87 112)(88 111)(89 110)(90 109)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 146)(137 145)(138 144)(139 143)(140 142)(147 150)(148 149)(151 170)(152 169)(153 168)(154 167)(155 166)(156 180)(157 179)(158 178)(159 177)(160 176)(161 175)(162 174)(163 173)(164 172)(165 171)(181 215)(182 214)(183 213)(184 212)(185 211)(186 225)(187 224)(188 223)(189 222)(190 221)(191 220)(192 219)(193 218)(194 217)(195 216)(196 236)(197 235)(198 234)(199 233)(200 232)(201 231)(202 230)(203 229)(204 228)(205 227)(206 226)(207 240)(208 239)(209 238)(210 237)
G:=sub<Sym(240)| (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,144)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,136)(24,137)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,164)(32,165)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,157)(40,158)(41,159)(42,160)(43,161)(44,162)(45,163)(46,169)(47,170)(48,171)(49,172)(50,173)(51,174)(52,175)(53,176)(54,177)(55,178)(56,179)(57,180)(58,166)(59,167)(60,168)(61,187)(62,188)(63,189)(64,190)(65,191)(66,192)(67,193)(68,194)(69,195)(70,181)(71,182)(72,183)(73,184)(74,185)(75,186)(76,209)(77,210)(78,196)(79,197)(80,198)(81,199)(82,200)(83,201)(84,202)(85,203)(86,204)(87,205)(88,206)(89,207)(90,208)(91,222)(92,223)(93,224)(94,225)(95,211)(96,212)(97,213)(98,214)(99,215)(100,216)(101,217)(102,218)(103,219)(104,220)(105,221)(106,236)(107,237)(108,238)(109,239)(110,240)(111,226)(112,227)(113,228)(114,229)(115,230)(116,231)(117,232)(118,233)(119,234)(120,235), (1,49,21,32)(2,50,22,33)(3,51,23,34)(4,52,24,35)(5,53,25,36)(6,54,26,37)(7,55,27,38)(8,56,28,39)(9,57,29,40)(10,58,30,41)(11,59,16,42)(12,60,17,43)(13,46,18,44)(14,47,19,45)(15,48,20,31)(61,100,79,112)(62,101,80,113)(63,102,81,114)(64,103,82,115)(65,104,83,116)(66,105,84,117)(67,91,85,118)(68,92,86,119)(69,93,87,120)(70,94,88,106)(71,95,89,107)(72,96,90,108)(73,97,76,109)(74,98,77,110)(75,99,78,111)(121,172,149,165)(122,173,150,151)(123,174,136,152)(124,175,137,153)(125,176,138,154)(126,177,139,155)(127,178,140,156)(128,179,141,157)(129,180,142,158)(130,166,143,159)(131,167,144,160)(132,168,145,161)(133,169,146,162)(134,170,147,163)(135,171,148,164)(181,225,206,236)(182,211,207,237)(183,212,208,238)(184,213,209,239)(185,214,210,240)(186,215,196,226)(187,216,197,227)(188,217,198,228)(189,218,199,229)(190,219,200,230)(191,220,201,231)(192,221,202,232)(193,222,203,233)(194,223,204,234)(195,224,205,235), (1,239)(2,240)(3,226)(4,227)(5,228)(6,229)(7,230)(8,231)(9,232)(10,233)(11,234)(12,235)(13,236)(14,237)(15,238)(16,223)(17,224)(18,225)(19,211)(20,212)(21,213)(22,214)(23,215)(24,216)(25,217)(26,218)(27,219)(28,220)(29,221)(30,222)(31,183)(32,184)(33,185)(34,186)(35,187)(36,188)(37,189)(38,190)(39,191)(40,192)(41,193)(42,194)(43,195)(44,181)(45,182)(46,206)(47,207)(48,208)(49,209)(50,210)(51,196)(52,197)(53,198)(54,199)(55,200)(56,201)(57,202)(58,203)(59,204)(60,205)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,161)(70,162)(71,163)(72,164)(73,165)(74,151)(75,152)(76,172)(77,173)(78,174)(79,175)(80,176)(81,177)(82,178)(83,179)(84,180)(85,166)(86,167)(87,168)(88,169)(89,170)(90,171)(91,143)(92,144)(93,145)(94,146)(95,147)(96,148)(97,149)(98,150)(99,136)(100,137)(101,138)(102,139)(103,140)(104,141)(105,142)(106,133)(107,134)(108,135)(109,121)(110,122)(111,123)(112,124)(113,125)(114,126)(115,127)(116,128)(117,129)(118,130)(119,131)(120,132), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,25)(17,24)(18,23)(19,22)(20,21)(26,30)(27,29)(31,49)(32,48)(33,47)(34,46)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(61,93)(62,92)(63,91)(64,105)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,98)(72,97)(73,96)(74,95)(75,94)(76,108)(77,107)(78,106)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,146)(137,145)(138,144)(139,143)(140,142)(147,150)(148,149)(151,170)(152,169)(153,168)(154,167)(155,166)(156,180)(157,179)(158,178)(159,177)(160,176)(161,175)(162,174)(163,173)(164,172)(165,171)(181,215)(182,214)(183,213)(184,212)(185,211)(186,225)(187,224)(188,223)(189,222)(190,221)(191,220)(192,219)(193,218)(194,217)(195,216)(196,236)(197,235)(198,234)(199,233)(200,232)(201,231)(202,230)(203,229)(204,228)(205,227)(206,226)(207,240)(208,239)(209,238)(210,237)>;
G:=Group( (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,144)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,136)(24,137)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,164)(32,165)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,157)(40,158)(41,159)(42,160)(43,161)(44,162)(45,163)(46,169)(47,170)(48,171)(49,172)(50,173)(51,174)(52,175)(53,176)(54,177)(55,178)(56,179)(57,180)(58,166)(59,167)(60,168)(61,187)(62,188)(63,189)(64,190)(65,191)(66,192)(67,193)(68,194)(69,195)(70,181)(71,182)(72,183)(73,184)(74,185)(75,186)(76,209)(77,210)(78,196)(79,197)(80,198)(81,199)(82,200)(83,201)(84,202)(85,203)(86,204)(87,205)(88,206)(89,207)(90,208)(91,222)(92,223)(93,224)(94,225)(95,211)(96,212)(97,213)(98,214)(99,215)(100,216)(101,217)(102,218)(103,219)(104,220)(105,221)(106,236)(107,237)(108,238)(109,239)(110,240)(111,226)(112,227)(113,228)(114,229)(115,230)(116,231)(117,232)(118,233)(119,234)(120,235), (1,49,21,32)(2,50,22,33)(3,51,23,34)(4,52,24,35)(5,53,25,36)(6,54,26,37)(7,55,27,38)(8,56,28,39)(9,57,29,40)(10,58,30,41)(11,59,16,42)(12,60,17,43)(13,46,18,44)(14,47,19,45)(15,48,20,31)(61,100,79,112)(62,101,80,113)(63,102,81,114)(64,103,82,115)(65,104,83,116)(66,105,84,117)(67,91,85,118)(68,92,86,119)(69,93,87,120)(70,94,88,106)(71,95,89,107)(72,96,90,108)(73,97,76,109)(74,98,77,110)(75,99,78,111)(121,172,149,165)(122,173,150,151)(123,174,136,152)(124,175,137,153)(125,176,138,154)(126,177,139,155)(127,178,140,156)(128,179,141,157)(129,180,142,158)(130,166,143,159)(131,167,144,160)(132,168,145,161)(133,169,146,162)(134,170,147,163)(135,171,148,164)(181,225,206,236)(182,211,207,237)(183,212,208,238)(184,213,209,239)(185,214,210,240)(186,215,196,226)(187,216,197,227)(188,217,198,228)(189,218,199,229)(190,219,200,230)(191,220,201,231)(192,221,202,232)(193,222,203,233)(194,223,204,234)(195,224,205,235), (1,239)(2,240)(3,226)(4,227)(5,228)(6,229)(7,230)(8,231)(9,232)(10,233)(11,234)(12,235)(13,236)(14,237)(15,238)(16,223)(17,224)(18,225)(19,211)(20,212)(21,213)(22,214)(23,215)(24,216)(25,217)(26,218)(27,219)(28,220)(29,221)(30,222)(31,183)(32,184)(33,185)(34,186)(35,187)(36,188)(37,189)(38,190)(39,191)(40,192)(41,193)(42,194)(43,195)(44,181)(45,182)(46,206)(47,207)(48,208)(49,209)(50,210)(51,196)(52,197)(53,198)(54,199)(55,200)(56,201)(57,202)(58,203)(59,204)(60,205)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,161)(70,162)(71,163)(72,164)(73,165)(74,151)(75,152)(76,172)(77,173)(78,174)(79,175)(80,176)(81,177)(82,178)(83,179)(84,180)(85,166)(86,167)(87,168)(88,169)(89,170)(90,171)(91,143)(92,144)(93,145)(94,146)(95,147)(96,148)(97,149)(98,150)(99,136)(100,137)(101,138)(102,139)(103,140)(104,141)(105,142)(106,133)(107,134)(108,135)(109,121)(110,122)(111,123)(112,124)(113,125)(114,126)(115,127)(116,128)(117,129)(118,130)(119,131)(120,132), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,25)(17,24)(18,23)(19,22)(20,21)(26,30)(27,29)(31,49)(32,48)(33,47)(34,46)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(61,93)(62,92)(63,91)(64,105)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,98)(72,97)(73,96)(74,95)(75,94)(76,108)(77,107)(78,106)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,146)(137,145)(138,144)(139,143)(140,142)(147,150)(148,149)(151,170)(152,169)(153,168)(154,167)(155,166)(156,180)(157,179)(158,178)(159,177)(160,176)(161,175)(162,174)(163,173)(164,172)(165,171)(181,215)(182,214)(183,213)(184,212)(185,211)(186,225)(187,224)(188,223)(189,222)(190,221)(191,220)(192,219)(193,218)(194,217)(195,216)(196,236)(197,235)(198,234)(199,233)(200,232)(201,231)(202,230)(203,229)(204,228)(205,227)(206,226)(207,240)(208,239)(209,238)(210,237) );
G=PermutationGroup([[(1,121),(2,122),(3,123),(4,124),(5,125),(6,126),(7,127),(8,128),(9,129),(10,130),(11,131),(12,132),(13,133),(14,134),(15,135),(16,144),(17,145),(18,146),(19,147),(20,148),(21,149),(22,150),(23,136),(24,137),(25,138),(26,139),(27,140),(28,141),(29,142),(30,143),(31,164),(32,165),(33,151),(34,152),(35,153),(36,154),(37,155),(38,156),(39,157),(40,158),(41,159),(42,160),(43,161),(44,162),(45,163),(46,169),(47,170),(48,171),(49,172),(50,173),(51,174),(52,175),(53,176),(54,177),(55,178),(56,179),(57,180),(58,166),(59,167),(60,168),(61,187),(62,188),(63,189),(64,190),(65,191),(66,192),(67,193),(68,194),(69,195),(70,181),(71,182),(72,183),(73,184),(74,185),(75,186),(76,209),(77,210),(78,196),(79,197),(80,198),(81,199),(82,200),(83,201),(84,202),(85,203),(86,204),(87,205),(88,206),(89,207),(90,208),(91,222),(92,223),(93,224),(94,225),(95,211),(96,212),(97,213),(98,214),(99,215),(100,216),(101,217),(102,218),(103,219),(104,220),(105,221),(106,236),(107,237),(108,238),(109,239),(110,240),(111,226),(112,227),(113,228),(114,229),(115,230),(116,231),(117,232),(118,233),(119,234),(120,235)], [(1,49,21,32),(2,50,22,33),(3,51,23,34),(4,52,24,35),(5,53,25,36),(6,54,26,37),(7,55,27,38),(8,56,28,39),(9,57,29,40),(10,58,30,41),(11,59,16,42),(12,60,17,43),(13,46,18,44),(14,47,19,45),(15,48,20,31),(61,100,79,112),(62,101,80,113),(63,102,81,114),(64,103,82,115),(65,104,83,116),(66,105,84,117),(67,91,85,118),(68,92,86,119),(69,93,87,120),(70,94,88,106),(71,95,89,107),(72,96,90,108),(73,97,76,109),(74,98,77,110),(75,99,78,111),(121,172,149,165),(122,173,150,151),(123,174,136,152),(124,175,137,153),(125,176,138,154),(126,177,139,155),(127,178,140,156),(128,179,141,157),(129,180,142,158),(130,166,143,159),(131,167,144,160),(132,168,145,161),(133,169,146,162),(134,170,147,163),(135,171,148,164),(181,225,206,236),(182,211,207,237),(183,212,208,238),(184,213,209,239),(185,214,210,240),(186,215,196,226),(187,216,197,227),(188,217,198,228),(189,218,199,229),(190,219,200,230),(191,220,201,231),(192,221,202,232),(193,222,203,233),(194,223,204,234),(195,224,205,235)], [(1,239),(2,240),(3,226),(4,227),(5,228),(6,229),(7,230),(8,231),(9,232),(10,233),(11,234),(12,235),(13,236),(14,237),(15,238),(16,223),(17,224),(18,225),(19,211),(20,212),(21,213),(22,214),(23,215),(24,216),(25,217),(26,218),(27,219),(28,220),(29,221),(30,222),(31,183),(32,184),(33,185),(34,186),(35,187),(36,188),(37,189),(38,190),(39,191),(40,192),(41,193),(42,194),(43,195),(44,181),(45,182),(46,206),(47,207),(48,208),(49,209),(50,210),(51,196),(52,197),(53,198),(54,199),(55,200),(56,201),(57,202),(58,203),(59,204),(60,205),(61,153),(62,154),(63,155),(64,156),(65,157),(66,158),(67,159),(68,160),(69,161),(70,162),(71,163),(72,164),(73,165),(74,151),(75,152),(76,172),(77,173),(78,174),(79,175),(80,176),(81,177),(82,178),(83,179),(84,180),(85,166),(86,167),(87,168),(88,169),(89,170),(90,171),(91,143),(92,144),(93,145),(94,146),(95,147),(96,148),(97,149),(98,150),(99,136),(100,137),(101,138),(102,139),(103,140),(104,141),(105,142),(106,133),(107,134),(108,135),(109,121),(110,122),(111,123),(112,124),(113,125),(114,126),(115,127),(116,128),(117,129),(118,130),(119,131),(120,132)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,25),(17,24),(18,23),(19,22),(20,21),(26,30),(27,29),(31,49),(32,48),(33,47),(34,46),(35,60),(36,59),(37,58),(38,57),(39,56),(40,55),(41,54),(42,53),(43,52),(44,51),(45,50),(61,93),(62,92),(63,91),(64,105),(65,104),(66,103),(67,102),(68,101),(69,100),(70,99),(71,98),(72,97),(73,96),(74,95),(75,94),(76,108),(77,107),(78,106),(79,120),(80,119),(81,118),(82,117),(83,116),(84,115),(85,114),(86,113),(87,112),(88,111),(89,110),(90,109),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129),(136,146),(137,145),(138,144),(139,143),(140,142),(147,150),(148,149),(151,170),(152,169),(153,168),(154,167),(155,166),(156,180),(157,179),(158,178),(159,177),(160,176),(161,175),(162,174),(163,173),(164,172),(165,171),(181,215),(182,214),(183,213),(184,212),(185,211),(186,225),(187,224),(188,223),(189,222),(190,221),(191,220),(192,219),(193,218),(194,217),(195,216),(196,236),(197,235),(198,234),(199,233),(200,232),(201,231),(202,230),(203,229),(204,228),(205,227),(206,226),(207,240),(208,239),(209,238),(210,237)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D8 | D10 | D10 | C3⋊D4 | C3⋊D4 | D15 | C5⋊D4 | C5⋊D4 | D30 | D30 | C15⋊7D4 | C15⋊7D4 | D4⋊S3 | D4⋊D5 | D4⋊D15 |
| kernel | C2×D4⋊D15 | C2×C15⋊3C8 | D4⋊D15 | C2×D60 | D4×C30 | D4×C10 | C60 | C2×C30 | C6×D4 | C2×C20 | C5×D4 | C30 | C2×C12 | C3×D4 | C20 | C2×C10 | C2×D4 | C12 | C2×C6 | C2×C4 | D4 | C4 | C22 | C10 | C6 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | 4 | 8 |
Matrix representation of C2×D4⋊D15 ►in GL4(𝔽241) generated by
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 192 |
| 0 | 0 | 123 | 1 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 22 | 57 |
| 0 | 0 | 93 | 219 |
| 110 | 173 | 0 | 0 |
| 215 | 84 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 80 | 64 | 0 | 0 |
| 77 | 161 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 118 | 240 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,240,123,0,0,192,1],[240,0,0,0,0,240,0,0,0,0,22,93,0,0,57,219],[110,215,0,0,173,84,0,0,0,0,1,0,0,0,0,1],[80,77,0,0,64,161,0,0,0,0,1,118,0,0,0,240] >;
C2×D4⋊D15 in GAP, Magma, Sage, TeX
C_2\times D_4\rtimes D_{15} % in TeX
G:=Group("C2xD4:D15"); // GroupNames label
G:=SmallGroup(480,896);
// by ID
G=gap.SmallGroup(480,896);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,675,185,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^15=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations