direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4.D15, D4.7D30, C60.16D4, C30⋊10SD16, C60.76C23, Dic30⋊21C22, (C6×D4).3D5, C6⋊3(D4.D5), (C5×D4).31D6, (C2×D4).4D15, (D4×C30).3C2, (D4×C10).3S3, (C2×C4).48D30, C15⋊22(C2×SD16), C10⋊3(D4.S3), (C3×D4).31D10, C30.380(C2×D4), (C2×C20).146D6, (C2×C30).147D4, C4.6(C15⋊7D4), C15⋊3C8⋊30C22, (C2×Dic30)⋊12C2, (C2×C12).145D10, C12.43(C5⋊D4), C20.41(C3⋊D4), (C2×C60).72C22, C4.13(C22×D15), C20.114(C22×S3), (D4×C15).36C22, C12.114(C22×D5), C22.22(C15⋊7D4), C3⋊4(C2×D4.D5), C5⋊4(C2×D4.S3), (C2×C15⋊3C8)⋊5C2, C6.105(C2×C5⋊D4), C2.10(C2×C15⋊7D4), C10.105(C2×C3⋊D4), (C2×C6).79(C5⋊D4), (C2×C10).79(C3⋊D4), SmallGroup(480,898)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4.D15
G = < a,b,c,d,e | a2=b4=c2=d15=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >
Subgroups: 596 in 136 conjugacy classes, 55 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C3⋊C8, Dic6, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, C30, C30, C30, C2×SD16, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C5×D4, C5×D4, C22×C10, C2×C3⋊C8, D4.S3, C2×Dic6, C6×D4, Dic15, C60, C2×C30, C2×C30, C2×C5⋊2C8, D4.D5, C2×Dic10, D4×C10, C2×D4.S3, C15⋊3C8, Dic30, Dic30, C2×Dic15, C2×C60, D4×C15, D4×C15, C22×C30, C2×D4.D5, C2×C15⋊3C8, D4.D15, C2×Dic30, D4×C30, C2×D4.D15
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, C3⋊D4, C22×S3, D15, C2×SD16, 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 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 78)(17 79)(18 80)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(25 87)(26 88)(27 89)(28 90)(29 76)(30 77)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 106)(41 107)(42 108)(43 109)(44 110)(45 111)(46 102)(47 103)(48 104)(49 105)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 97)(57 98)(58 99)(59 100)(60 101)(121 193)(122 194)(123 195)(124 181)(125 182)(126 183)(127 184)(128 185)(129 186)(130 187)(131 188)(132 189)(133 190)(134 191)(135 192)(136 200)(137 201)(138 202)(139 203)(140 204)(141 205)(142 206)(143 207)(144 208)(145 209)(146 210)(147 196)(148 197)(149 198)(150 199)(151 226)(152 227)(153 228)(154 229)(155 230)(156 231)(157 232)(158 233)(159 234)(160 235)(161 236)(162 237)(163 238)(164 239)(165 240)(166 223)(167 224)(168 225)(169 211)(170 212)(171 213)(172 214)(173 215)(174 216)(175 217)(176 218)(177 219)(178 220)(179 221)(180 222)
(1 58 27 34)(2 59 28 35)(3 60 29 36)(4 46 30 37)(5 47 16 38)(6 48 17 39)(7 49 18 40)(8 50 19 41)(9 51 20 42)(10 52 21 43)(11 53 22 44)(12 54 23 45)(13 55 24 31)(14 56 25 32)(15 57 26 33)(61 99 89 115)(62 100 90 116)(63 101 76 117)(64 102 77 118)(65 103 78 119)(66 104 79 120)(67 105 80 106)(68 91 81 107)(69 92 82 108)(70 93 83 109)(71 94 84 110)(72 95 85 111)(73 96 86 112)(74 97 87 113)(75 98 88 114)(121 152 136 166)(122 153 137 167)(123 154 138 168)(124 155 139 169)(125 156 140 170)(126 157 141 171)(127 158 142 172)(128 159 143 173)(129 160 144 174)(130 161 145 175)(131 162 146 176)(132 163 147 177)(133 164 148 178)(134 165 149 179)(135 151 150 180)(181 230 203 211)(182 231 204 212)(183 232 205 213)(184 233 206 214)(185 234 207 215)(186 235 208 216)(187 236 209 217)(188 237 210 218)(189 238 196 219)(190 239 197 220)(191 240 198 221)(192 226 199 222)(193 227 200 223)(194 228 201 224)(195 229 202 225)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 31)(14 32)(15 33)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 57)(27 58)(28 59)(29 60)(30 46)(61 115)(62 116)(63 117)(64 118)(65 119)(66 120)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(73 112)(74 113)(75 114)(76 101)(77 102)(78 103)(79 104)(80 105)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(121 136)(122 137)(123 138)(124 139)(125 140)(126 141)(127 142)(128 143)(129 144)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(181 203)(182 204)(183 205)(184 206)(185 207)(186 208)(187 209)(188 210)(189 196)(190 197)(191 198)(192 199)(193 200)(194 201)(195 202)
(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)
(1 150 27 135)(2 149 28 134)(3 148 29 133)(4 147 30 132)(5 146 16 131)(6 145 17 130)(7 144 18 129)(8 143 19 128)(9 142 20 127)(10 141 21 126)(11 140 22 125)(12 139 23 124)(13 138 24 123)(14 137 25 122)(15 136 26 121)(31 168 55 154)(32 167 56 153)(33 166 57 152)(34 180 58 151)(35 179 59 165)(36 178 60 164)(37 177 46 163)(38 176 47 162)(39 175 48 161)(40 174 49 160)(41 173 50 159)(42 172 51 158)(43 171 52 157)(44 170 53 156)(45 169 54 155)(61 199 89 192)(62 198 90 191)(63 197 76 190)(64 196 77 189)(65 210 78 188)(66 209 79 187)(67 208 80 186)(68 207 81 185)(69 206 82 184)(70 205 83 183)(71 204 84 182)(72 203 85 181)(73 202 86 195)(74 201 87 194)(75 200 88 193)(91 234 107 215)(92 233 108 214)(93 232 109 213)(94 231 110 212)(95 230 111 211)(96 229 112 225)(97 228 113 224)(98 227 114 223)(99 226 115 222)(100 240 116 221)(101 239 117 220)(102 238 118 219)(103 237 119 218)(104 236 120 217)(105 235 106 216)
G:=sub<Sym(240)| (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,76)(30,77)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,106)(41,107)(42,108)(43,109)(44,110)(45,111)(46,102)(47,103)(48,104)(49,105)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,101)(121,193)(122,194)(123,195)(124,181)(125,182)(126,183)(127,184)(128,185)(129,186)(130,187)(131,188)(132,189)(133,190)(134,191)(135,192)(136,200)(137,201)(138,202)(139,203)(140,204)(141,205)(142,206)(143,207)(144,208)(145,209)(146,210)(147,196)(148,197)(149,198)(150,199)(151,226)(152,227)(153,228)(154,229)(155,230)(156,231)(157,232)(158,233)(159,234)(160,235)(161,236)(162,237)(163,238)(164,239)(165,240)(166,223)(167,224)(168,225)(169,211)(170,212)(171,213)(172,214)(173,215)(174,216)(175,217)(176,218)(177,219)(178,220)(179,221)(180,222), (1,58,27,34)(2,59,28,35)(3,60,29,36)(4,46,30,37)(5,47,16,38)(6,48,17,39)(7,49,18,40)(8,50,19,41)(9,51,20,42)(10,52,21,43)(11,53,22,44)(12,54,23,45)(13,55,24,31)(14,56,25,32)(15,57,26,33)(61,99,89,115)(62,100,90,116)(63,101,76,117)(64,102,77,118)(65,103,78,119)(66,104,79,120)(67,105,80,106)(68,91,81,107)(69,92,82,108)(70,93,83,109)(71,94,84,110)(72,95,85,111)(73,96,86,112)(74,97,87,113)(75,98,88,114)(121,152,136,166)(122,153,137,167)(123,154,138,168)(124,155,139,169)(125,156,140,170)(126,157,141,171)(127,158,142,172)(128,159,143,173)(129,160,144,174)(130,161,145,175)(131,162,146,176)(132,163,147,177)(133,164,148,178)(134,165,149,179)(135,151,150,180)(181,230,203,211)(182,231,204,212)(183,232,205,213)(184,233,206,214)(185,234,207,215)(186,235,208,216)(187,236,209,217)(188,237,210,218)(189,238,196,219)(190,239,197,220)(191,240,198,221)(192,226,199,222)(193,227,200,223)(194,228,201,224)(195,229,202,225), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,31)(14,32)(15,33)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,46)(61,115)(62,116)(63,117)(64,118)(65,119)(66,120)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,113)(75,114)(76,101)(77,102)(78,103)(79,104)(80,105)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(121,136)(122,137)(123,138)(124,139)(125,140)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(181,203)(182,204)(183,205)(184,206)(185,207)(186,208)(187,209)(188,210)(189,196)(190,197)(191,198)(192,199)(193,200)(194,201)(195,202), (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), (1,150,27,135)(2,149,28,134)(3,148,29,133)(4,147,30,132)(5,146,16,131)(6,145,17,130)(7,144,18,129)(8,143,19,128)(9,142,20,127)(10,141,21,126)(11,140,22,125)(12,139,23,124)(13,138,24,123)(14,137,25,122)(15,136,26,121)(31,168,55,154)(32,167,56,153)(33,166,57,152)(34,180,58,151)(35,179,59,165)(36,178,60,164)(37,177,46,163)(38,176,47,162)(39,175,48,161)(40,174,49,160)(41,173,50,159)(42,172,51,158)(43,171,52,157)(44,170,53,156)(45,169,54,155)(61,199,89,192)(62,198,90,191)(63,197,76,190)(64,196,77,189)(65,210,78,188)(66,209,79,187)(67,208,80,186)(68,207,81,185)(69,206,82,184)(70,205,83,183)(71,204,84,182)(72,203,85,181)(73,202,86,195)(74,201,87,194)(75,200,88,193)(91,234,107,215)(92,233,108,214)(93,232,109,213)(94,231,110,212)(95,230,111,211)(96,229,112,225)(97,228,113,224)(98,227,114,223)(99,226,115,222)(100,240,116,221)(101,239,117,220)(102,238,118,219)(103,237,119,218)(104,236,120,217)(105,235,106,216)>;
G:=Group( (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,76)(30,77)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,106)(41,107)(42,108)(43,109)(44,110)(45,111)(46,102)(47,103)(48,104)(49,105)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,101)(121,193)(122,194)(123,195)(124,181)(125,182)(126,183)(127,184)(128,185)(129,186)(130,187)(131,188)(132,189)(133,190)(134,191)(135,192)(136,200)(137,201)(138,202)(139,203)(140,204)(141,205)(142,206)(143,207)(144,208)(145,209)(146,210)(147,196)(148,197)(149,198)(150,199)(151,226)(152,227)(153,228)(154,229)(155,230)(156,231)(157,232)(158,233)(159,234)(160,235)(161,236)(162,237)(163,238)(164,239)(165,240)(166,223)(167,224)(168,225)(169,211)(170,212)(171,213)(172,214)(173,215)(174,216)(175,217)(176,218)(177,219)(178,220)(179,221)(180,222), (1,58,27,34)(2,59,28,35)(3,60,29,36)(4,46,30,37)(5,47,16,38)(6,48,17,39)(7,49,18,40)(8,50,19,41)(9,51,20,42)(10,52,21,43)(11,53,22,44)(12,54,23,45)(13,55,24,31)(14,56,25,32)(15,57,26,33)(61,99,89,115)(62,100,90,116)(63,101,76,117)(64,102,77,118)(65,103,78,119)(66,104,79,120)(67,105,80,106)(68,91,81,107)(69,92,82,108)(70,93,83,109)(71,94,84,110)(72,95,85,111)(73,96,86,112)(74,97,87,113)(75,98,88,114)(121,152,136,166)(122,153,137,167)(123,154,138,168)(124,155,139,169)(125,156,140,170)(126,157,141,171)(127,158,142,172)(128,159,143,173)(129,160,144,174)(130,161,145,175)(131,162,146,176)(132,163,147,177)(133,164,148,178)(134,165,149,179)(135,151,150,180)(181,230,203,211)(182,231,204,212)(183,232,205,213)(184,233,206,214)(185,234,207,215)(186,235,208,216)(187,236,209,217)(188,237,210,218)(189,238,196,219)(190,239,197,220)(191,240,198,221)(192,226,199,222)(193,227,200,223)(194,228,201,224)(195,229,202,225), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,31)(14,32)(15,33)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,46)(61,115)(62,116)(63,117)(64,118)(65,119)(66,120)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,113)(75,114)(76,101)(77,102)(78,103)(79,104)(80,105)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(121,136)(122,137)(123,138)(124,139)(125,140)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(181,203)(182,204)(183,205)(184,206)(185,207)(186,208)(187,209)(188,210)(189,196)(190,197)(191,198)(192,199)(193,200)(194,201)(195,202), (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), (1,150,27,135)(2,149,28,134)(3,148,29,133)(4,147,30,132)(5,146,16,131)(6,145,17,130)(7,144,18,129)(8,143,19,128)(9,142,20,127)(10,141,21,126)(11,140,22,125)(12,139,23,124)(13,138,24,123)(14,137,25,122)(15,136,26,121)(31,168,55,154)(32,167,56,153)(33,166,57,152)(34,180,58,151)(35,179,59,165)(36,178,60,164)(37,177,46,163)(38,176,47,162)(39,175,48,161)(40,174,49,160)(41,173,50,159)(42,172,51,158)(43,171,52,157)(44,170,53,156)(45,169,54,155)(61,199,89,192)(62,198,90,191)(63,197,76,190)(64,196,77,189)(65,210,78,188)(66,209,79,187)(67,208,80,186)(68,207,81,185)(69,206,82,184)(70,205,83,183)(71,204,84,182)(72,203,85,181)(73,202,86,195)(74,201,87,194)(75,200,88,193)(91,234,107,215)(92,233,108,214)(93,232,109,213)(94,231,110,212)(95,230,111,211)(96,229,112,225)(97,228,113,224)(98,227,114,223)(99,226,115,222)(100,240,116,221)(101,239,117,220)(102,238,118,219)(103,237,119,218)(104,236,120,217)(105,235,106,216) );
G=PermutationGroup([[(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,73),(14,74),(15,75),(16,78),(17,79),(18,80),(19,81),(20,82),(21,83),(22,84),(23,85),(24,86),(25,87),(26,88),(27,89),(28,90),(29,76),(30,77),(31,112),(32,113),(33,114),(34,115),(35,116),(36,117),(37,118),(38,119),(39,120),(40,106),(41,107),(42,108),(43,109),(44,110),(45,111),(46,102),(47,103),(48,104),(49,105),(50,91),(51,92),(52,93),(53,94),(54,95),(55,96),(56,97),(57,98),(58,99),(59,100),(60,101),(121,193),(122,194),(123,195),(124,181),(125,182),(126,183),(127,184),(128,185),(129,186),(130,187),(131,188),(132,189),(133,190),(134,191),(135,192),(136,200),(137,201),(138,202),(139,203),(140,204),(141,205),(142,206),(143,207),(144,208),(145,209),(146,210),(147,196),(148,197),(149,198),(150,199),(151,226),(152,227),(153,228),(154,229),(155,230),(156,231),(157,232),(158,233),(159,234),(160,235),(161,236),(162,237),(163,238),(164,239),(165,240),(166,223),(167,224),(168,225),(169,211),(170,212),(171,213),(172,214),(173,215),(174,216),(175,217),(176,218),(177,219),(178,220),(179,221),(180,222)], [(1,58,27,34),(2,59,28,35),(3,60,29,36),(4,46,30,37),(5,47,16,38),(6,48,17,39),(7,49,18,40),(8,50,19,41),(9,51,20,42),(10,52,21,43),(11,53,22,44),(12,54,23,45),(13,55,24,31),(14,56,25,32),(15,57,26,33),(61,99,89,115),(62,100,90,116),(63,101,76,117),(64,102,77,118),(65,103,78,119),(66,104,79,120),(67,105,80,106),(68,91,81,107),(69,92,82,108),(70,93,83,109),(71,94,84,110),(72,95,85,111),(73,96,86,112),(74,97,87,113),(75,98,88,114),(121,152,136,166),(122,153,137,167),(123,154,138,168),(124,155,139,169),(125,156,140,170),(126,157,141,171),(127,158,142,172),(128,159,143,173),(129,160,144,174),(130,161,145,175),(131,162,146,176),(132,163,147,177),(133,164,148,178),(134,165,149,179),(135,151,150,180),(181,230,203,211),(182,231,204,212),(183,232,205,213),(184,233,206,214),(185,234,207,215),(186,235,208,216),(187,236,209,217),(188,237,210,218),(189,238,196,219),(190,239,197,220),(191,240,198,221),(192,226,199,222),(193,227,200,223),(194,228,201,224),(195,229,202,225)], [(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,43),(11,44),(12,45),(13,31),(14,32),(15,33),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,56),(26,57),(27,58),(28,59),(29,60),(30,46),(61,115),(62,116),(63,117),(64,118),(65,119),(66,120),(67,106),(68,107),(69,108),(70,109),(71,110),(72,111),(73,112),(74,113),(75,114),(76,101),(77,102),(78,103),(79,104),(80,105),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(121,136),(122,137),(123,138),(124,139),(125,140),(126,141),(127,142),(128,143),(129,144),(130,145),(131,146),(132,147),(133,148),(134,149),(135,150),(181,203),(182,204),(183,205),(184,206),(185,207),(186,208),(187,209),(188,210),(189,196),(190,197),(191,198),(192,199),(193,200),(194,201),(195,202)], [(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)], [(1,150,27,135),(2,149,28,134),(3,148,29,133),(4,147,30,132),(5,146,16,131),(6,145,17,130),(7,144,18,129),(8,143,19,128),(9,142,20,127),(10,141,21,126),(11,140,22,125),(12,139,23,124),(13,138,24,123),(14,137,25,122),(15,136,26,121),(31,168,55,154),(32,167,56,153),(33,166,57,152),(34,180,58,151),(35,179,59,165),(36,178,60,164),(37,177,46,163),(38,176,47,162),(39,175,48,161),(40,174,49,160),(41,173,50,159),(42,172,51,158),(43,171,52,157),(44,170,53,156),(45,169,54,155),(61,199,89,192),(62,198,90,191),(63,197,76,190),(64,196,77,189),(65,210,78,188),(66,209,79,187),(67,208,80,186),(68,207,81,185),(69,206,82,184),(70,205,83,183),(71,204,84,182),(72,203,85,181),(73,202,86,195),(74,201,87,194),(75,200,88,193),(91,234,107,215),(92,233,108,214),(93,232,109,213),(94,231,110,212),(95,230,111,211),(96,229,112,225),(97,228,113,224),(98,227,114,223),(99,226,115,222),(100,240,116,221),(101,239,117,220),(102,238,118,219),(103,237,119,218),(104,236,120,217),(105,235,106,216)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 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 | 3 | 4 | 4 | 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 | 2 | 2 | 2 | 60 | 60 | 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 | SD16 | 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×Dic30 | 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 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 231 | 0 | 0 | 0 |
| 0 | 24 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 24 | 0 | 0 |
| 231 | 0 | 0 | 0 |
| 0 | 0 | 19 | 222 |
| 0 | 0 | 222 | 222 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240],[240,0,0,0,0,240,0,0,0,0,0,240,0,0,1,0],[240,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[231,0,0,0,0,24,0,0,0,0,1,0,0,0,0,1],[0,231,0,0,24,0,0,0,0,0,19,222,0,0,222,222] >;
C2×D4.D15 in GAP, Magma, Sage, TeX
C_2\times D_4.D_{15} % in TeX
G:=Group("C2xD4.D15"); // GroupNames label
G:=SmallGroup(480,898);
// by ID
G=gap.SmallGroup(480,898);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,675,185,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^15=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations