metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.26D30, (C2×C60)⋊20C4, C60⋊5C4⋊35C2, (C2×C4)⋊4Dic15, (C2×C12)⋊6Dic5, C60.239(C2×C4), (C2×C20)⋊11Dic3, (C2×C20).417D6, (C2×C4).103D30, (C22×C4).9D15, (C4×Dic15)⋊21C2, (C2×C12).400D10, (C22×C20).14S3, (C22×C60).13C2, C20.65(C2×Dic3), C12.44(C2×Dic5), C4.15(C2×Dic15), (C22×C12).10D5, C15⋊30(C42⋊C2), C6.104(C4○D20), C30.176(C4○D4), C30.214(C22×C4), (C2×C60).481C22, (C2×C30).300C23, (C22×C10).135D6, (C22×C6).117D10, C5⋊5(C23.26D6), C10.104(C4○D12), C2.4(D60⋊11C2), C22.5(C2×Dic15), C6.25(C22×Dic5), C2.5(C22×Dic15), C30.38D4.11C2, C3⋊4(C23.21D10), C10.38(C22×Dic3), C22.22(C22×D15), (C22×C30).140C22, (C2×Dic15).169C22, (C2×C30).180(C2×C4), (C2×C6).37(C2×Dic5), (C2×C10).57(C2×Dic3), (C2×C6).296(C22×D5), (C2×C10).295(C22×S3), SmallGroup(480,891)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.26D30
G = < a,b,c,d,e | a2=b2=c2=1, d30=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=d29 >
Subgroups: 564 in 152 conjugacy classes, 87 normal (29 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, Dic3, C12, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×C12, C2×C12, C22×C6, C30, C30, C30, C42⋊C2, C2×Dic5, C2×C20, C2×C20, C22×C10, C4×Dic3, C4⋊Dic3, C6.D4, C22×C12, Dic15, C60, C2×C30, C2×C30, C2×C30, C4×Dic5, C4⋊Dic5, C23.D5, C22×C20, C23.26D6, C2×Dic15, C2×C60, C2×C60, C22×C30, C23.21D10, C4×Dic15, C60⋊5C4, C30.38D4, C22×C60, C23.26D30
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, Dic3, D6, C22×C4, C4○D4, Dic5, D10, C2×Dic3, C22×S3, D15, C42⋊C2, C2×Dic5, C22×D5, C4○D12, C22×Dic3, Dic15, D30, C4○D20, C22×Dic5, C23.26D6, C2×Dic15, C22×D15, C23.21D10, D60⋊11C2, C22×Dic15, C23.26D30
►On 240 points
Generators in S240
(1 175)(2 176)(3 177)(4 178)(5 179)(6 180)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 133)(20 134)(21 135)(22 136)(23 137)(24 138)(25 139)(26 140)(27 141)(28 142)(29 143)(30 144)(31 145)(32 146)(33 147)(34 148)(35 149)(36 150)(37 151)(38 152)(39 153)(40 154)(41 155)(42 156)(43 157)(44 158)(45 159)(46 160)(47 161)(48 162)(49 163)(50 164)(51 165)(52 166)(53 167)(54 168)(55 169)(56 170)(57 171)(58 172)(59 173)(60 174)(61 231)(62 232)(63 233)(64 234)(65 235)(66 236)(67 237)(68 238)(69 239)(70 240)(71 181)(72 182)(73 183)(74 184)(75 185)(76 186)(77 187)(78 188)(79 189)(80 190)(81 191)(82 192)(83 193)(84 194)(85 195)(86 196)(87 197)(88 198)(89 199)(90 200)(91 201)(92 202)(93 203)(94 204)(95 205)(96 206)(97 207)(98 208)(99 209)(100 210)(101 211)(102 212)(103 213)(104 214)(105 215)(106 216)(107 217)(108 218)(109 219)(110 220)(111 221)(112 222)(113 223)(114 224)(115 225)(116 226)(117 227)(118 228)(119 229)(120 230)
(1 175)(2 176)(3 177)(4 178)(5 179)(6 180)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 133)(20 134)(21 135)(22 136)(23 137)(24 138)(25 139)(26 140)(27 141)(28 142)(29 143)(30 144)(31 145)(32 146)(33 147)(34 148)(35 149)(36 150)(37 151)(38 152)(39 153)(40 154)(41 155)(42 156)(43 157)(44 158)(45 159)(46 160)(47 161)(48 162)(49 163)(50 164)(51 165)(52 166)(53 167)(54 168)(55 169)(56 170)(57 171)(58 172)(59 173)(60 174)(61 201)(62 202)(63 203)(64 204)(65 205)(66 206)(67 207)(68 208)(69 209)(70 210)(71 211)(72 212)(73 213)(74 214)(75 215)(76 216)(77 217)(78 218)(79 219)(80 220)(81 221)(82 222)(83 223)(84 224)(85 225)(86 226)(87 227)(88 228)(89 229)(90 230)(91 231)(92 232)(93 233)(94 234)(95 235)(96 236)(97 237)(98 238)(99 239)(100 240)(101 181)(102 182)(103 183)(104 184)(105 185)(106 186)(107 187)(108 188)(109 189)(110 190)(111 191)(112 192)(113 193)(114 194)(115 195)(116 196)(117 197)(118 198)(119 199)(120 200)
(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 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)
(1 206 145 96)(2 235 146 65)(3 204 147 94)(4 233 148 63)(5 202 149 92)(6 231 150 61)(7 200 151 90)(8 229 152 119)(9 198 153 88)(10 227 154 117)(11 196 155 86)(12 225 156 115)(13 194 157 84)(14 223 158 113)(15 192 159 82)(16 221 160 111)(17 190 161 80)(18 219 162 109)(19 188 163 78)(20 217 164 107)(21 186 165 76)(22 215 166 105)(23 184 167 74)(24 213 168 103)(25 182 169 72)(26 211 170 101)(27 240 171 70)(28 209 172 99)(29 238 173 68)(30 207 174 97)(31 236 175 66)(32 205 176 95)(33 234 177 64)(34 203 178 93)(35 232 179 62)(36 201 180 91)(37 230 121 120)(38 199 122 89)(39 228 123 118)(40 197 124 87)(41 226 125 116)(42 195 126 85)(43 224 127 114)(44 193 128 83)(45 222 129 112)(46 191 130 81)(47 220 131 110)(48 189 132 79)(49 218 133 108)(50 187 134 77)(51 216 135 106)(52 185 136 75)(53 214 137 104)(54 183 138 73)(55 212 139 102)(56 181 140 71)(57 210 141 100)(58 239 142 69)(59 208 143 98)(60 237 144 67)
G:=sub<Sym(240)| (1,175)(2,176)(3,177)(4,178)(5,179)(6,180)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,134)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,141)(28,142)(29,143)(30,144)(31,145)(32,146)(33,147)(34,148)(35,149)(36,150)(37,151)(38,152)(39,153)(40,154)(41,155)(42,156)(43,157)(44,158)(45,159)(46,160)(47,161)(48,162)(49,163)(50,164)(51,165)(52,166)(53,167)(54,168)(55,169)(56,170)(57,171)(58,172)(59,173)(60,174)(61,231)(62,232)(63,233)(64,234)(65,235)(66,236)(67,237)(68,238)(69,239)(70,240)(71,181)(72,182)(73,183)(74,184)(75,185)(76,186)(77,187)(78,188)(79,189)(80,190)(81,191)(82,192)(83,193)(84,194)(85,195)(86,196)(87,197)(88,198)(89,199)(90,200)(91,201)(92,202)(93,203)(94,204)(95,205)(96,206)(97,207)(98,208)(99,209)(100,210)(101,211)(102,212)(103,213)(104,214)(105,215)(106,216)(107,217)(108,218)(109,219)(110,220)(111,221)(112,222)(113,223)(114,224)(115,225)(116,226)(117,227)(118,228)(119,229)(120,230), (1,175)(2,176)(3,177)(4,178)(5,179)(6,180)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,134)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,141)(28,142)(29,143)(30,144)(31,145)(32,146)(33,147)(34,148)(35,149)(36,150)(37,151)(38,152)(39,153)(40,154)(41,155)(42,156)(43,157)(44,158)(45,159)(46,160)(47,161)(48,162)(49,163)(50,164)(51,165)(52,166)(53,167)(54,168)(55,169)(56,170)(57,171)(58,172)(59,173)(60,174)(61,201)(62,202)(63,203)(64,204)(65,205)(66,206)(67,207)(68,208)(69,209)(70,210)(71,211)(72,212)(73,213)(74,214)(75,215)(76,216)(77,217)(78,218)(79,219)(80,220)(81,221)(82,222)(83,223)(84,224)(85,225)(86,226)(87,227)(88,228)(89,229)(90,230)(91,231)(92,232)(93,233)(94,234)(95,235)(96,236)(97,237)(98,238)(99,239)(100,240)(101,181)(102,182)(103,183)(104,184)(105,185)(106,186)(107,187)(108,188)(109,189)(110,190)(111,191)(112,192)(113,193)(114,194)(115,195)(116,196)(117,197)(118,198)(119,199)(120,200), (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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(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), (1,206,145,96)(2,235,146,65)(3,204,147,94)(4,233,148,63)(5,202,149,92)(6,231,150,61)(7,200,151,90)(8,229,152,119)(9,198,153,88)(10,227,154,117)(11,196,155,86)(12,225,156,115)(13,194,157,84)(14,223,158,113)(15,192,159,82)(16,221,160,111)(17,190,161,80)(18,219,162,109)(19,188,163,78)(20,217,164,107)(21,186,165,76)(22,215,166,105)(23,184,167,74)(24,213,168,103)(25,182,169,72)(26,211,170,101)(27,240,171,70)(28,209,172,99)(29,238,173,68)(30,207,174,97)(31,236,175,66)(32,205,176,95)(33,234,177,64)(34,203,178,93)(35,232,179,62)(36,201,180,91)(37,230,121,120)(38,199,122,89)(39,228,123,118)(40,197,124,87)(41,226,125,116)(42,195,126,85)(43,224,127,114)(44,193,128,83)(45,222,129,112)(46,191,130,81)(47,220,131,110)(48,189,132,79)(49,218,133,108)(50,187,134,77)(51,216,135,106)(52,185,136,75)(53,214,137,104)(54,183,138,73)(55,212,139,102)(56,181,140,71)(57,210,141,100)(58,239,142,69)(59,208,143,98)(60,237,144,67)>;
G:=Group( (1,175)(2,176)(3,177)(4,178)(5,179)(6,180)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,134)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,141)(28,142)(29,143)(30,144)(31,145)(32,146)(33,147)(34,148)(35,149)(36,150)(37,151)(38,152)(39,153)(40,154)(41,155)(42,156)(43,157)(44,158)(45,159)(46,160)(47,161)(48,162)(49,163)(50,164)(51,165)(52,166)(53,167)(54,168)(55,169)(56,170)(57,171)(58,172)(59,173)(60,174)(61,231)(62,232)(63,233)(64,234)(65,235)(66,236)(67,237)(68,238)(69,239)(70,240)(71,181)(72,182)(73,183)(74,184)(75,185)(76,186)(77,187)(78,188)(79,189)(80,190)(81,191)(82,192)(83,193)(84,194)(85,195)(86,196)(87,197)(88,198)(89,199)(90,200)(91,201)(92,202)(93,203)(94,204)(95,205)(96,206)(97,207)(98,208)(99,209)(100,210)(101,211)(102,212)(103,213)(104,214)(105,215)(106,216)(107,217)(108,218)(109,219)(110,220)(111,221)(112,222)(113,223)(114,224)(115,225)(116,226)(117,227)(118,228)(119,229)(120,230), (1,175)(2,176)(3,177)(4,178)(5,179)(6,180)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,134)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,141)(28,142)(29,143)(30,144)(31,145)(32,146)(33,147)(34,148)(35,149)(36,150)(37,151)(38,152)(39,153)(40,154)(41,155)(42,156)(43,157)(44,158)(45,159)(46,160)(47,161)(48,162)(49,163)(50,164)(51,165)(52,166)(53,167)(54,168)(55,169)(56,170)(57,171)(58,172)(59,173)(60,174)(61,201)(62,202)(63,203)(64,204)(65,205)(66,206)(67,207)(68,208)(69,209)(70,210)(71,211)(72,212)(73,213)(74,214)(75,215)(76,216)(77,217)(78,218)(79,219)(80,220)(81,221)(82,222)(83,223)(84,224)(85,225)(86,226)(87,227)(88,228)(89,229)(90,230)(91,231)(92,232)(93,233)(94,234)(95,235)(96,236)(97,237)(98,238)(99,239)(100,240)(101,181)(102,182)(103,183)(104,184)(105,185)(106,186)(107,187)(108,188)(109,189)(110,190)(111,191)(112,192)(113,193)(114,194)(115,195)(116,196)(117,197)(118,198)(119,199)(120,200), (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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(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), (1,206,145,96)(2,235,146,65)(3,204,147,94)(4,233,148,63)(5,202,149,92)(6,231,150,61)(7,200,151,90)(8,229,152,119)(9,198,153,88)(10,227,154,117)(11,196,155,86)(12,225,156,115)(13,194,157,84)(14,223,158,113)(15,192,159,82)(16,221,160,111)(17,190,161,80)(18,219,162,109)(19,188,163,78)(20,217,164,107)(21,186,165,76)(22,215,166,105)(23,184,167,74)(24,213,168,103)(25,182,169,72)(26,211,170,101)(27,240,171,70)(28,209,172,99)(29,238,173,68)(30,207,174,97)(31,236,175,66)(32,205,176,95)(33,234,177,64)(34,203,178,93)(35,232,179,62)(36,201,180,91)(37,230,121,120)(38,199,122,89)(39,228,123,118)(40,197,124,87)(41,226,125,116)(42,195,126,85)(43,224,127,114)(44,193,128,83)(45,222,129,112)(46,191,130,81)(47,220,131,110)(48,189,132,79)(49,218,133,108)(50,187,134,77)(51,216,135,106)(52,185,136,75)(53,214,137,104)(54,183,138,73)(55,212,139,102)(56,181,140,71)(57,210,141,100)(58,239,142,69)(59,208,143,98)(60,237,144,67) );
G=PermutationGroup([[(1,175),(2,176),(3,177),(4,178),(5,179),(6,180),(7,121),(8,122),(9,123),(10,124),(11,125),(12,126),(13,127),(14,128),(15,129),(16,130),(17,131),(18,132),(19,133),(20,134),(21,135),(22,136),(23,137),(24,138),(25,139),(26,140),(27,141),(28,142),(29,143),(30,144),(31,145),(32,146),(33,147),(34,148),(35,149),(36,150),(37,151),(38,152),(39,153),(40,154),(41,155),(42,156),(43,157),(44,158),(45,159),(46,160),(47,161),(48,162),(49,163),(50,164),(51,165),(52,166),(53,167),(54,168),(55,169),(56,170),(57,171),(58,172),(59,173),(60,174),(61,231),(62,232),(63,233),(64,234),(65,235),(66,236),(67,237),(68,238),(69,239),(70,240),(71,181),(72,182),(73,183),(74,184),(75,185),(76,186),(77,187),(78,188),(79,189),(80,190),(81,191),(82,192),(83,193),(84,194),(85,195),(86,196),(87,197),(88,198),(89,199),(90,200),(91,201),(92,202),(93,203),(94,204),(95,205),(96,206),(97,207),(98,208),(99,209),(100,210),(101,211),(102,212),(103,213),(104,214),(105,215),(106,216),(107,217),(108,218),(109,219),(110,220),(111,221),(112,222),(113,223),(114,224),(115,225),(116,226),(117,227),(118,228),(119,229),(120,230)], [(1,175),(2,176),(3,177),(4,178),(5,179),(6,180),(7,121),(8,122),(9,123),(10,124),(11,125),(12,126),(13,127),(14,128),(15,129),(16,130),(17,131),(18,132),(19,133),(20,134),(21,135),(22,136),(23,137),(24,138),(25,139),(26,140),(27,141),(28,142),(29,143),(30,144),(31,145),(32,146),(33,147),(34,148),(35,149),(36,150),(37,151),(38,152),(39,153),(40,154),(41,155),(42,156),(43,157),(44,158),(45,159),(46,160),(47,161),(48,162),(49,163),(50,164),(51,165),(52,166),(53,167),(54,168),(55,169),(56,170),(57,171),(58,172),(59,173),(60,174),(61,201),(62,202),(63,203),(64,204),(65,205),(66,206),(67,207),(68,208),(69,209),(70,210),(71,211),(72,212),(73,213),(74,214),(75,215),(76,216),(77,217),(78,218),(79,219),(80,220),(81,221),(82,222),(83,223),(84,224),(85,225),(86,226),(87,227),(88,228),(89,229),(90,230),(91,231),(92,232),(93,233),(94,234),(95,235),(96,236),(97,237),(98,238),(99,239),(100,240),(101,181),(102,182),(103,183),(104,184),(105,185),(106,186),(107,187),(108,188),(109,189),(110,190),(111,191),(112,192),(113,193),(114,194),(115,195),(116,196),(117,197),(118,198),(119,199),(120,200)], [(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)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(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)], [(1,206,145,96),(2,235,146,65),(3,204,147,94),(4,233,148,63),(5,202,149,92),(6,231,150,61),(7,200,151,90),(8,229,152,119),(9,198,153,88),(10,227,154,117),(11,196,155,86),(12,225,156,115),(13,194,157,84),(14,223,158,113),(15,192,159,82),(16,221,160,111),(17,190,161,80),(18,219,162,109),(19,188,163,78),(20,217,164,107),(21,186,165,76),(22,215,166,105),(23,184,167,74),(24,213,168,103),(25,182,169,72),(26,211,170,101),(27,240,171,70),(28,209,172,99),(29,238,173,68),(30,207,174,97),(31,236,175,66),(32,205,176,95),(33,234,177,64),(34,203,178,93),(35,232,179,62),(36,201,180,91),(37,230,121,120),(38,199,122,89),(39,228,123,118),(40,197,124,87),(41,226,125,116),(42,195,126,85),(43,224,127,114),(44,193,128,83),(45,222,129,112),(46,191,130,81),(47,220,131,110),(48,189,132,79),(49,218,133,108),(50,187,134,77),(51,216,135,106),(52,185,136,75),(53,214,137,104),(54,183,138,73),(55,212,139,102),(56,181,140,71),(57,210,141,100),(58,239,142,69),(59,208,143,98),(60,237,144,67)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10N | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 30 | ··· | 30 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | - | + | + | + | - | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D5 | Dic3 | D6 | D6 | C4○D4 | Dic5 | D10 | D10 | D15 | C4○D12 | Dic15 | D30 | D30 | C4○D20 | D60⋊11C2 |
| kernel | C23.26D30 | C4×Dic15 | C60⋊5C4 | C30.38D4 | C22×C60 | C2×C60 | C22×C20 | C22×C12 | C2×C20 | C2×C20 | C22×C10 | C30 | C2×C12 | C2×C12 | C22×C6 | C22×C4 | C10 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 1 | 2 | 4 | 2 | 1 | 4 | 8 | 4 | 2 | 4 | 8 | 16 | 8 | 4 | 16 | 32 |
Matrix representation of C23.26D30 ►in GL3(𝔽61) generated by
| 1 | 0 | 0 |
| 0 | 60 | 0 |
| 0 | 0 | 1 |
| 60 | 0 | 0 |
| 0 | 60 | 0 |
| 0 | 0 | 60 |
| 1 | 0 | 0 |
| 0 | 60 | 0 |
| 0 | 0 | 60 |
| 60 | 0 | 0 |
| 0 | 31 | 0 |
| 0 | 0 | 59 |
| 11 | 0 | 0 |
| 0 | 0 | 59 |
| 0 | 30 | 0 |
G:=sub<GL(3,GF(61))| [1,0,0,0,60,0,0,0,1],[60,0,0,0,60,0,0,0,60],[1,0,0,0,60,0,0,0,60],[60,0,0,0,31,0,0,0,59],[11,0,0,0,0,30,0,59,0] >;
C23.26D30 in GAP, Magma, Sage, TeX
C_2^3._{26}D_{30} % in TeX
G:=Group("C2^3.26D30"); // GroupNames label
G:=SmallGroup(480,891);
// by ID
G=gap.SmallGroup(480,891);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^30=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*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