metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.15D30, (C2×C4).25D30, (C2×Dic15)⋊9C4, (C2×C20).205D6, C22⋊C4.3D15, C30.4Q8⋊8C2, C22.6(C4×D15), (C4×Dic15)⋊15C2, (C2×C12).203D10, (C22×C10).67D6, (C22×C6).52D10, C15⋊24(C42⋊C2), C30.213(C4○D4), C2.1(D4⋊2D15), C6.88(D4⋊2D5), (C2×C30).274C23, C30.156(C22×C4), (C2×C60).171C22, Dic15.42(C2×C4), C30.38D4.1C2, C5⋊5(C23.16D6), (C22×C30).8C22, C10.88(D4⋊2S3), C3⋊4(C23.11D10), (C22×Dic15).2C2, C22.12(C22×D15), (C2×Dic15).237C22, C6.61(C2×C4×D5), C2.7(C2×C4×D15), C10.93(S3×C2×C4), (C2×C6).12(C4×D5), (C2×C30).67(C2×C4), (C2×C10).35(C4×S3), (C5×C22⋊C4).3S3, (C3×C22⋊C4).3D5, (C15×C22⋊C4).3C2, (C2×C6).270(C22×D5), (C2×C10).269(C22×S3), SmallGroup(480,842)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.15D30
G = < a,b,c,d,e | a2=b2=c2=1, d30=b, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d29 >
Subgroups: 644 in 152 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C23, C10, C10, C10, Dic3, C12, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×C12, C22×C6, C30, C30, C30, C42⋊C2, C2×Dic5, C2×C20, C22×C10, C4×Dic3, Dic3⋊C4, C6.D4, C3×C22⋊C4, C22×Dic3, Dic15, Dic15, C60, C2×C30, C2×C30, C2×C30, C4×Dic5, C10.D4, C23.D5, C5×C22⋊C4, C22×Dic5, C23.16D6, C2×Dic15, C2×Dic15, C2×C60, C22×C30, C23.11D10, C4×Dic15, C30.4Q8, C30.38D4, C15×C22⋊C4, C22×Dic15, C23.15D30
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, C4○D4, D10, C4×S3, C22×S3, D15, C42⋊C2, C4×D5, C22×D5, S3×C2×C4, D4⋊2S3, D30, C2×C4×D5, D4⋊2D5, C23.16D6, C4×D15, C22×D15, C23.11D10, C2×C4×D15, D4⋊2D15, C23.15D30
►On 240 points
Generators in S240
(1 31)(2 194)(3 33)(4 196)(5 35)(6 198)(7 37)(8 200)(9 39)(10 202)(11 41)(12 204)(13 43)(14 206)(15 45)(16 208)(17 47)(18 210)(19 49)(20 212)(21 51)(22 214)(23 53)(24 216)(25 55)(26 218)(27 57)(28 220)(29 59)(30 222)(32 224)(34 226)(36 228)(38 230)(40 232)(42 234)(44 236)(46 238)(48 240)(50 182)(52 184)(54 186)(56 188)(58 190)(60 192)(61 141)(62 92)(63 143)(64 94)(65 145)(66 96)(67 147)(68 98)(69 149)(70 100)(71 151)(72 102)(73 153)(74 104)(75 155)(76 106)(77 157)(78 108)(79 159)(80 110)(81 161)(82 112)(83 163)(84 114)(85 165)(86 116)(87 167)(88 118)(89 169)(90 120)(91 171)(93 173)(95 175)(97 177)(99 179)(101 121)(103 123)(105 125)(107 127)(109 129)(111 131)(113 133)(115 135)(117 137)(119 139)(122 152)(124 154)(126 156)(128 158)(130 160)(132 162)(134 164)(136 166)(138 168)(140 170)(142 172)(144 174)(146 176)(148 178)(150 180)(181 211)(183 213)(185 215)(187 217)(189 219)(191 221)(193 223)(195 225)(197 227)(199 229)(201 231)(203 233)(205 235)(207 237)(209 239)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(121 151)(122 152)(123 153)(124 154)(125 155)(126 156)(127 157)(128 158)(129 159)(130 160)(131 161)(132 162)(133 163)(134 164)(135 165)(136 166)(137 167)(138 168)(139 169)(140 170)(141 171)(142 172)(143 173)(144 174)(145 175)(146 176)(147 177)(148 178)(149 179)(150 180)(181 211)(182 212)(183 213)(184 214)(185 215)(186 216)(187 217)(188 218)(189 219)(190 220)(191 221)(192 222)(193 223)(194 224)(195 225)(196 226)(197 227)(198 228)(199 229)(200 230)(201 231)(202 232)(203 233)(204 234)(205 235)(206 236)(207 237)(208 238)(209 239)(210 240)
(1 223)(2 224)(3 225)(4 226)(5 227)(6 228)(7 229)(8 230)(9 231)(10 232)(11 233)(12 234)(13 235)(14 236)(15 237)(16 238)(17 239)(18 240)(19 181)(20 182)(21 183)(22 184)(23 185)(24 186)(25 187)(26 188)(27 189)(28 190)(29 191)(30 192)(31 193)(32 194)(33 195)(34 196)(35 197)(36 198)(37 199)(38 200)(39 201)(40 202)(41 203)(42 204)(43 205)(44 206)(45 207)(46 208)(47 209)(48 210)(49 211)(50 212)(51 213)(52 214)(53 215)(54 216)(55 217)(56 218)(57 219)(58 220)(59 221)(60 222)(61 171)(62 172)(63 173)(64 174)(65 175)(66 176)(67 177)(68 178)(69 179)(70 180)(71 121)(72 122)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 129)(80 130)(81 131)(82 132)(83 133)(84 134)(85 135)(86 136)(87 137)(88 138)(89 139)(90 140)(91 141)(92 142)(93 143)(94 144)(95 145)(96 146)(97 147)(98 148)(99 149)(100 150)(101 151)(102 152)(103 153)(104 154)(105 155)(106 156)(107 157)(108 158)(109 159)(110 160)(111 161)(112 162)(113 163)(114 164)(115 165)(116 166)(117 167)(118 168)(119 169)(120 170)
(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)
(1 125 193 105)(2 154 194 74)(3 123 195 103)(4 152 196 72)(5 121 197 101)(6 150 198 70)(7 179 199 99)(8 148 200 68)(9 177 201 97)(10 146 202 66)(11 175 203 95)(12 144 204 64)(13 173 205 93)(14 142 206 62)(15 171 207 91)(16 140 208 120)(17 169 209 89)(18 138 210 118)(19 167 211 87)(20 136 212 116)(21 165 213 85)(22 134 214 114)(23 163 215 83)(24 132 216 112)(25 161 217 81)(26 130 218 110)(27 159 219 79)(28 128 220 108)(29 157 221 77)(30 126 222 106)(31 155 223 75)(32 124 224 104)(33 153 225 73)(34 122 226 102)(35 151 227 71)(36 180 228 100)(37 149 229 69)(38 178 230 98)(39 147 231 67)(40 176 232 96)(41 145 233 65)(42 174 234 94)(43 143 235 63)(44 172 236 92)(45 141 237 61)(46 170 238 90)(47 139 239 119)(48 168 240 88)(49 137 181 117)(50 166 182 86)(51 135 183 115)(52 164 184 84)(53 133 185 113)(54 162 186 82)(55 131 187 111)(56 160 188 80)(57 129 189 109)(58 158 190 78)(59 127 191 107)(60 156 192 76)
G:=sub<Sym(240)| (1,31)(2,194)(3,33)(4,196)(5,35)(6,198)(7,37)(8,200)(9,39)(10,202)(11,41)(12,204)(13,43)(14,206)(15,45)(16,208)(17,47)(18,210)(19,49)(20,212)(21,51)(22,214)(23,53)(24,216)(25,55)(26,218)(27,57)(28,220)(29,59)(30,222)(32,224)(34,226)(36,228)(38,230)(40,232)(42,234)(44,236)(46,238)(48,240)(50,182)(52,184)(54,186)(56,188)(58,190)(60,192)(61,141)(62,92)(63,143)(64,94)(65,145)(66,96)(67,147)(68,98)(69,149)(70,100)(71,151)(72,102)(73,153)(74,104)(75,155)(76,106)(77,157)(78,108)(79,159)(80,110)(81,161)(82,112)(83,163)(84,114)(85,165)(86,116)(87,167)(88,118)(89,169)(90,120)(91,171)(93,173)(95,175)(97,177)(99,179)(101,121)(103,123)(105,125)(107,127)(109,129)(111,131)(113,133)(115,135)(117,137)(119,139)(122,152)(124,154)(126,156)(128,158)(130,160)(132,162)(134,164)(136,166)(138,168)(140,170)(142,172)(144,174)(146,176)(148,178)(150,180)(181,211)(183,213)(185,215)(187,217)(189,219)(191,221)(193,223)(195,225)(197,227)(199,229)(201,231)(203,233)(205,235)(207,237)(209,239), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(181,211)(182,212)(183,213)(184,214)(185,215)(186,216)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(199,229)(200,230)(201,231)(202,232)(203,233)(204,234)(205,235)(206,236)(207,237)(208,238)(209,239)(210,240), (1,223)(2,224)(3,225)(4,226)(5,227)(6,228)(7,229)(8,230)(9,231)(10,232)(11,233)(12,234)(13,235)(14,236)(15,237)(16,238)(17,239)(18,240)(19,181)(20,182)(21,183)(22,184)(23,185)(24,186)(25,187)(26,188)(27,189)(28,190)(29,191)(30,192)(31,193)(32,194)(33,195)(34,196)(35,197)(36,198)(37,199)(38,200)(39,201)(40,202)(41,203)(42,204)(43,205)(44,206)(45,207)(46,208)(47,209)(48,210)(49,211)(50,212)(51,213)(52,214)(53,215)(54,216)(55,217)(56,218)(57,219)(58,220)(59,221)(60,222)(61,171)(62,172)(63,173)(64,174)(65,175)(66,176)(67,177)(68,178)(69,179)(70,180)(71,121)(72,122)(73,123)(74,124)(75,125)(76,126)(77,127)(78,128)(79,129)(80,130)(81,131)(82,132)(83,133)(84,134)(85,135)(86,136)(87,137)(88,138)(89,139)(90,140)(91,141)(92,142)(93,143)(94,144)(95,145)(96,146)(97,147)(98,148)(99,149)(100,150)(101,151)(102,152)(103,153)(104,154)(105,155)(106,156)(107,157)(108,158)(109,159)(110,160)(111,161)(112,162)(113,163)(114,164)(115,165)(116,166)(117,167)(118,168)(119,169)(120,170), (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), (1,125,193,105)(2,154,194,74)(3,123,195,103)(4,152,196,72)(5,121,197,101)(6,150,198,70)(7,179,199,99)(8,148,200,68)(9,177,201,97)(10,146,202,66)(11,175,203,95)(12,144,204,64)(13,173,205,93)(14,142,206,62)(15,171,207,91)(16,140,208,120)(17,169,209,89)(18,138,210,118)(19,167,211,87)(20,136,212,116)(21,165,213,85)(22,134,214,114)(23,163,215,83)(24,132,216,112)(25,161,217,81)(26,130,218,110)(27,159,219,79)(28,128,220,108)(29,157,221,77)(30,126,222,106)(31,155,223,75)(32,124,224,104)(33,153,225,73)(34,122,226,102)(35,151,227,71)(36,180,228,100)(37,149,229,69)(38,178,230,98)(39,147,231,67)(40,176,232,96)(41,145,233,65)(42,174,234,94)(43,143,235,63)(44,172,236,92)(45,141,237,61)(46,170,238,90)(47,139,239,119)(48,168,240,88)(49,137,181,117)(50,166,182,86)(51,135,183,115)(52,164,184,84)(53,133,185,113)(54,162,186,82)(55,131,187,111)(56,160,188,80)(57,129,189,109)(58,158,190,78)(59,127,191,107)(60,156,192,76)>;
G:=Group( (1,31)(2,194)(3,33)(4,196)(5,35)(6,198)(7,37)(8,200)(9,39)(10,202)(11,41)(12,204)(13,43)(14,206)(15,45)(16,208)(17,47)(18,210)(19,49)(20,212)(21,51)(22,214)(23,53)(24,216)(25,55)(26,218)(27,57)(28,220)(29,59)(30,222)(32,224)(34,226)(36,228)(38,230)(40,232)(42,234)(44,236)(46,238)(48,240)(50,182)(52,184)(54,186)(56,188)(58,190)(60,192)(61,141)(62,92)(63,143)(64,94)(65,145)(66,96)(67,147)(68,98)(69,149)(70,100)(71,151)(72,102)(73,153)(74,104)(75,155)(76,106)(77,157)(78,108)(79,159)(80,110)(81,161)(82,112)(83,163)(84,114)(85,165)(86,116)(87,167)(88,118)(89,169)(90,120)(91,171)(93,173)(95,175)(97,177)(99,179)(101,121)(103,123)(105,125)(107,127)(109,129)(111,131)(113,133)(115,135)(117,137)(119,139)(122,152)(124,154)(126,156)(128,158)(130,160)(132,162)(134,164)(136,166)(138,168)(140,170)(142,172)(144,174)(146,176)(148,178)(150,180)(181,211)(183,213)(185,215)(187,217)(189,219)(191,221)(193,223)(195,225)(197,227)(199,229)(201,231)(203,233)(205,235)(207,237)(209,239), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(181,211)(182,212)(183,213)(184,214)(185,215)(186,216)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(199,229)(200,230)(201,231)(202,232)(203,233)(204,234)(205,235)(206,236)(207,237)(208,238)(209,239)(210,240), (1,223)(2,224)(3,225)(4,226)(5,227)(6,228)(7,229)(8,230)(9,231)(10,232)(11,233)(12,234)(13,235)(14,236)(15,237)(16,238)(17,239)(18,240)(19,181)(20,182)(21,183)(22,184)(23,185)(24,186)(25,187)(26,188)(27,189)(28,190)(29,191)(30,192)(31,193)(32,194)(33,195)(34,196)(35,197)(36,198)(37,199)(38,200)(39,201)(40,202)(41,203)(42,204)(43,205)(44,206)(45,207)(46,208)(47,209)(48,210)(49,211)(50,212)(51,213)(52,214)(53,215)(54,216)(55,217)(56,218)(57,219)(58,220)(59,221)(60,222)(61,171)(62,172)(63,173)(64,174)(65,175)(66,176)(67,177)(68,178)(69,179)(70,180)(71,121)(72,122)(73,123)(74,124)(75,125)(76,126)(77,127)(78,128)(79,129)(80,130)(81,131)(82,132)(83,133)(84,134)(85,135)(86,136)(87,137)(88,138)(89,139)(90,140)(91,141)(92,142)(93,143)(94,144)(95,145)(96,146)(97,147)(98,148)(99,149)(100,150)(101,151)(102,152)(103,153)(104,154)(105,155)(106,156)(107,157)(108,158)(109,159)(110,160)(111,161)(112,162)(113,163)(114,164)(115,165)(116,166)(117,167)(118,168)(119,169)(120,170), (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), (1,125,193,105)(2,154,194,74)(3,123,195,103)(4,152,196,72)(5,121,197,101)(6,150,198,70)(7,179,199,99)(8,148,200,68)(9,177,201,97)(10,146,202,66)(11,175,203,95)(12,144,204,64)(13,173,205,93)(14,142,206,62)(15,171,207,91)(16,140,208,120)(17,169,209,89)(18,138,210,118)(19,167,211,87)(20,136,212,116)(21,165,213,85)(22,134,214,114)(23,163,215,83)(24,132,216,112)(25,161,217,81)(26,130,218,110)(27,159,219,79)(28,128,220,108)(29,157,221,77)(30,126,222,106)(31,155,223,75)(32,124,224,104)(33,153,225,73)(34,122,226,102)(35,151,227,71)(36,180,228,100)(37,149,229,69)(38,178,230,98)(39,147,231,67)(40,176,232,96)(41,145,233,65)(42,174,234,94)(43,143,235,63)(44,172,236,92)(45,141,237,61)(46,170,238,90)(47,139,239,119)(48,168,240,88)(49,137,181,117)(50,166,182,86)(51,135,183,115)(52,164,184,84)(53,133,185,113)(54,162,186,82)(55,131,187,111)(56,160,188,80)(57,129,189,109)(58,158,190,78)(59,127,191,107)(60,156,192,76) );
G=PermutationGroup([[(1,31),(2,194),(3,33),(4,196),(5,35),(6,198),(7,37),(8,200),(9,39),(10,202),(11,41),(12,204),(13,43),(14,206),(15,45),(16,208),(17,47),(18,210),(19,49),(20,212),(21,51),(22,214),(23,53),(24,216),(25,55),(26,218),(27,57),(28,220),(29,59),(30,222),(32,224),(34,226),(36,228),(38,230),(40,232),(42,234),(44,236),(46,238),(48,240),(50,182),(52,184),(54,186),(56,188),(58,190),(60,192),(61,141),(62,92),(63,143),(64,94),(65,145),(66,96),(67,147),(68,98),(69,149),(70,100),(71,151),(72,102),(73,153),(74,104),(75,155),(76,106),(77,157),(78,108),(79,159),(80,110),(81,161),(82,112),(83,163),(84,114),(85,165),(86,116),(87,167),(88,118),(89,169),(90,120),(91,171),(93,173),(95,175),(97,177),(99,179),(101,121),(103,123),(105,125),(107,127),(109,129),(111,131),(113,133),(115,135),(117,137),(119,139),(122,152),(124,154),(126,156),(128,158),(130,160),(132,162),(134,164),(136,166),(138,168),(140,170),(142,172),(144,174),(146,176),(148,178),(150,180),(181,211),(183,213),(185,215),(187,217),(189,219),(191,221),(193,223),(195,225),(197,227),(199,229),(201,231),(203,233),(205,235),(207,237),(209,239)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,101),(72,102),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,109),(80,110),(81,111),(82,112),(83,113),(84,114),(85,115),(86,116),(87,117),(88,118),(89,119),(90,120),(121,151),(122,152),(123,153),(124,154),(125,155),(126,156),(127,157),(128,158),(129,159),(130,160),(131,161),(132,162),(133,163),(134,164),(135,165),(136,166),(137,167),(138,168),(139,169),(140,170),(141,171),(142,172),(143,173),(144,174),(145,175),(146,176),(147,177),(148,178),(149,179),(150,180),(181,211),(182,212),(183,213),(184,214),(185,215),(186,216),(187,217),(188,218),(189,219),(190,220),(191,221),(192,222),(193,223),(194,224),(195,225),(196,226),(197,227),(198,228),(199,229),(200,230),(201,231),(202,232),(203,233),(204,234),(205,235),(206,236),(207,237),(208,238),(209,239),(210,240)], [(1,223),(2,224),(3,225),(4,226),(5,227),(6,228),(7,229),(8,230),(9,231),(10,232),(11,233),(12,234),(13,235),(14,236),(15,237),(16,238),(17,239),(18,240),(19,181),(20,182),(21,183),(22,184),(23,185),(24,186),(25,187),(26,188),(27,189),(28,190),(29,191),(30,192),(31,193),(32,194),(33,195),(34,196),(35,197),(36,198),(37,199),(38,200),(39,201),(40,202),(41,203),(42,204),(43,205),(44,206),(45,207),(46,208),(47,209),(48,210),(49,211),(50,212),(51,213),(52,214),(53,215),(54,216),(55,217),(56,218),(57,219),(58,220),(59,221),(60,222),(61,171),(62,172),(63,173),(64,174),(65,175),(66,176),(67,177),(68,178),(69,179),(70,180),(71,121),(72,122),(73,123),(74,124),(75,125),(76,126),(77,127),(78,128),(79,129),(80,130),(81,131),(82,132),(83,133),(84,134),(85,135),(86,136),(87,137),(88,138),(89,139),(90,140),(91,141),(92,142),(93,143),(94,144),(95,145),(96,146),(97,147),(98,148),(99,149),(100,150),(101,151),(102,152),(103,153),(104,154),(105,155),(106,156),(107,157),(108,158),(109,159),(110,160),(111,161),(112,162),(113,163),(114,164),(115,165),(116,166),(117,167),(118,168),(119,169),(120,170)], [(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)], [(1,125,193,105),(2,154,194,74),(3,123,195,103),(4,152,196,72),(5,121,197,101),(6,150,198,70),(7,179,199,99),(8,148,200,68),(9,177,201,97),(10,146,202,66),(11,175,203,95),(12,144,204,64),(13,173,205,93),(14,142,206,62),(15,171,207,91),(16,140,208,120),(17,169,209,89),(18,138,210,118),(19,167,211,87),(20,136,212,116),(21,165,213,85),(22,134,214,114),(23,163,215,83),(24,132,216,112),(25,161,217,81),(26,130,218,110),(27,159,219,79),(28,128,220,108),(29,157,221,77),(30,126,222,106),(31,155,223,75),(32,124,224,104),(33,153,225,73),(34,122,226,102),(35,151,227,71),(36,180,228,100),(37,149,229,69),(38,178,230,98),(39,147,231,67),(40,176,232,96),(41,145,233,65),(42,174,234,94),(43,143,235,63),(44,172,236,92),(45,141,237,61),(46,170,238,90),(47,139,239,119),(48,168,240,88),(49,137,181,117),(50,166,182,86),(51,135,183,115),(52,164,184,84),(53,133,185,113),(54,162,186,82),(55,131,187,111),(56,160,188,80),(57,129,189,109),(58,158,190,78),(59,127,191,107),(60,156,192,76)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 15 | 15 | 15 | 15 | 30 | ··· | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C4×S3 | D15 | C4×D5 | D30 | D30 | C4×D15 | D4⋊2S3 | D4⋊2D5 | D4⋊2D15 |
| kernel | C23.15D30 | C4×Dic15 | C30.4Q8 | C30.38D4 | C15×C22⋊C4 | C22×Dic15 | C2×Dic15 | C5×C22⋊C4 | C3×C22⋊C4 | C2×C20 | C22×C10 | C30 | C2×C12 | C22×C6 | C2×C10 | C22⋊C4 | C2×C6 | C2×C4 | C23 | C22 | C10 | C6 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 4 | 8 | 8 | 4 | 16 | 2 | 4 | 8 |
Matrix representation of C23.15D30 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 55 | 31 | 0 | 0 |
| 21 | 34 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 7 | 25 | 0 | 0 |
| 59 | 54 | 0 | 0 |
| 0 | 0 | 0 | 50 |
| 0 | 0 | 50 | 0 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[55,21,0,0,31,34,0,0,0,0,0,1,0,0,1,0],[7,59,0,0,25,54,0,0,0,0,0,50,0,0,50,0] >;
C23.15D30 in GAP, Magma, Sage, TeX
C_2^3._{15}D_{30} % in TeX
G:=Group("C2^3.15D30"); // GroupNames label
G:=SmallGroup(480,842);
// by ID
G=gap.SmallGroup(480,842);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,219,58,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^30=b,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^29>;
// generators/relations