direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C23.12D6, C60.150D4, (C6×D4).5C10, C12.17(C5×D4), C6.48(D4×C10), (C4×Dic3)⋊5C10, (D4×C10).12S3, (D4×C30).15C2, C30.431(C2×D4), (C2×C20).361D6, (C2×Dic6)⋊10C10, (C10×Dic6)⋊26C2, (Dic3×C20)⋊17C2, C6.D4⋊9C10, C15⋊24(C4.4D4), C20.72(C3⋊D4), C23.12(S3×C10), (C22×C10).24D6, C30.254(C4○D4), (C2×C30).430C23, (C2×C60).361C22, C10.121(D4⋊2S3), (C22×C30).124C22, (C10×Dic3).229C22, C4.7(C5×C3⋊D4), C3⋊3(C5×C4.4D4), (C2×D4).6(C5×S3), C6.29(C5×C4○D4), (C2×C4).50(S3×C10), C2.12(C10×C3⋊D4), C22.58(S3×C2×C10), (C2×C12).34(C2×C10), C2.16(C5×D4⋊2S3), C10.133(C2×C3⋊D4), (C5×C6.D4)⋊25C2, (C22×C6).19(C2×C10), (C2×C6).51(C22×C10), (C2×C10).364(C22×S3), (C2×Dic3).37(C2×C10), SmallGroup(480,815)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.12D6
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e6=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
Subgroups: 324 in 152 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, Dic6, C2×Dic3, C2×C12, C3×D4, C22×C6, C30, C30, C30, C4.4D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C4×Dic3, C6.D4, C2×Dic6, C6×D4, C5×Dic3, C60, C2×C30, C2×C30, C4×C20, C5×C22⋊C4, D4×C10, Q8×C10, C23.12D6, C5×Dic6, C10×Dic3, C2×C60, D4×C15, C22×C30, C5×C4.4D4, Dic3×C20, C5×C6.D4, C10×Dic6, D4×C30, C5×C23.12D6
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C4○D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C4.4D4, C5×D4, C22×C10, D4⋊2S3, C2×C3⋊D4, S3×C10, D4×C10, C5×C4○D4, C23.12D6, C5×C3⋊D4, S3×C2×C10, C5×C4.4D4, C5×D4⋊2S3, C10×C3⋊D4, C5×C23.12D6
►On 240 points
Generators in S240
(1 229 26 62 87)(2 230 27 63 88)(3 231 28 64 89)(4 232 29 65 90)(5 233 30 66 91)(6 234 31 67 92)(7 235 32 68 93)(8 236 33 69 94)(9 237 34 70 95)(10 238 35 71 96)(11 239 36 72 85)(12 240 25 61 86)(13 168 153 43 169)(14 157 154 44 170)(15 158 155 45 171)(16 159 156 46 172)(17 160 145 47 173)(18 161 146 48 174)(19 162 147 37 175)(20 163 148 38 176)(21 164 149 39 177)(22 165 150 40 178)(23 166 151 41 179)(24 167 152 42 180)(49 112 139 82 227)(50 113 140 83 228)(51 114 141 84 217)(52 115 142 73 218)(53 116 143 74 219)(54 117 144 75 220)(55 118 133 76 221)(56 119 134 77 222)(57 120 135 78 223)(58 109 136 79 224)(59 110 137 80 225)(60 111 138 81 226)(97 202 215 189 124)(98 203 216 190 125)(99 204 205 191 126)(100 193 206 192 127)(101 194 207 181 128)(102 195 208 182 129)(103 196 209 183 130)(104 197 210 184 131)(105 198 211 185 132)(106 199 212 186 121)(107 200 213 187 122)(108 201 214 188 123)
(2 8)(4 10)(6 12)(14 20)(16 22)(18 24)(25 31)(27 33)(29 35)(38 44)(40 46)(42 48)(49 184)(50 191)(51 186)(52 181)(53 188)(54 183)(55 190)(56 185)(57 192)(58 187)(59 182)(60 189)(61 67)(63 69)(65 71)(73 194)(74 201)(75 196)(76 203)(77 198)(78 193)(79 200)(80 195)(81 202)(82 197)(83 204)(84 199)(86 92)(88 94)(90 96)(97 138)(98 133)(99 140)(100 135)(101 142)(102 137)(103 144)(104 139)(105 134)(106 141)(107 136)(108 143)(109 122)(110 129)(111 124)(112 131)(113 126)(114 121)(115 128)(116 123)(117 130)(118 125)(119 132)(120 127)(146 152)(148 154)(150 156)(157 163)(159 165)(161 167)(170 176)(172 178)(174 180)(205 228)(206 223)(207 218)(208 225)(209 220)(210 227)(211 222)(212 217)(213 224)(214 219)(215 226)(216 221)(230 236)(232 238)(234 240)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)(121 127)(122 128)(123 129)(124 130)(125 131)(126 132)(133 139)(134 140)(135 141)(136 142)(137 143)(138 144)(145 151)(146 152)(147 153)(148 154)(149 155)(150 156)(157 163)(158 164)(159 165)(160 166)(161 167)(162 168)(169 175)(170 176)(171 177)(172 178)(173 179)(174 180)(181 187)(182 188)(183 189)(184 190)(185 191)(186 192)(193 199)(194 200)(195 201)(196 202)(197 203)(198 204)(205 211)(206 212)(207 213)(208 214)(209 215)(210 216)(217 223)(218 224)(219 225)(220 226)(221 227)(222 228)(229 235)(230 236)(231 237)(232 238)(233 239)(234 240)
(1 164)(2 165)(3 166)(4 167)(5 168)(6 157)(7 158)(8 159)(9 160)(10 161)(11 162)(12 163)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 37)(49 190)(50 191)(51 192)(52 181)(53 182)(54 183)(55 184)(56 185)(57 186)(58 187)(59 188)(60 189)(61 176)(62 177)(63 178)(64 179)(65 180)(66 169)(67 170)(68 171)(69 172)(70 173)(71 174)(72 175)(73 194)(74 195)(75 196)(76 197)(77 198)(78 199)(79 200)(80 201)(81 202)(82 203)(83 204)(84 193)(97 138)(98 139)(99 140)(100 141)(101 142)(102 143)(103 144)(104 133)(105 134)(106 135)(107 136)(108 137)(109 122)(110 123)(111 124)(112 125)(113 126)(114 127)(115 128)(116 129)(117 130)(118 131)(119 132)(120 121)(145 237)(146 238)(147 239)(148 240)(149 229)(150 230)(151 231)(152 232)(153 233)(154 234)(155 235)(156 236)(205 228)(206 217)(207 218)(208 219)(209 220)(210 221)(211 222)(212 223)(213 224)(214 225)(215 226)(216 227)
(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 58 158 181)(2 51 159 186)(3 56 160 191)(4 49 161 184)(5 54 162 189)(6 59 163 182)(7 52 164 187)(8 57 165 192)(9 50 166 185)(10 55 167 190)(11 60 168 183)(12 53 157 188)(13 209 85 226)(14 214 86 219)(15 207 87 224)(16 212 88 217)(17 205 89 222)(18 210 90 227)(19 215 91 220)(20 208 92 225)(21 213 93 218)(22 206 94 223)(23 211 95 228)(24 216 96 221)(25 143 44 108)(26 136 45 101)(27 141 46 106)(28 134 47 99)(29 139 48 104)(30 144 37 97)(31 137 38 102)(32 142 39 107)(33 135 40 100)(34 140 41 105)(35 133 42 98)(36 138 43 103)(61 74 170 201)(62 79 171 194)(63 84 172 199)(64 77 173 204)(65 82 174 197)(66 75 175 202)(67 80 176 195)(68 73 177 200)(69 78 178 193)(70 83 179 198)(71 76 180 203)(72 81 169 196)(109 155 128 229)(110 148 129 234)(111 153 130 239)(112 146 131 232)(113 151 132 237)(114 156 121 230)(115 149 122 235)(116 154 123 240)(117 147 124 233)(118 152 125 238)(119 145 126 231)(120 150 127 236)
G:=sub<Sym(240)| (1,229,26,62,87)(2,230,27,63,88)(3,231,28,64,89)(4,232,29,65,90)(5,233,30,66,91)(6,234,31,67,92)(7,235,32,68,93)(8,236,33,69,94)(9,237,34,70,95)(10,238,35,71,96)(11,239,36,72,85)(12,240,25,61,86)(13,168,153,43,169)(14,157,154,44,170)(15,158,155,45,171)(16,159,156,46,172)(17,160,145,47,173)(18,161,146,48,174)(19,162,147,37,175)(20,163,148,38,176)(21,164,149,39,177)(22,165,150,40,178)(23,166,151,41,179)(24,167,152,42,180)(49,112,139,82,227)(50,113,140,83,228)(51,114,141,84,217)(52,115,142,73,218)(53,116,143,74,219)(54,117,144,75,220)(55,118,133,76,221)(56,119,134,77,222)(57,120,135,78,223)(58,109,136,79,224)(59,110,137,80,225)(60,111,138,81,226)(97,202,215,189,124)(98,203,216,190,125)(99,204,205,191,126)(100,193,206,192,127)(101,194,207,181,128)(102,195,208,182,129)(103,196,209,183,130)(104,197,210,184,131)(105,198,211,185,132)(106,199,212,186,121)(107,200,213,187,122)(108,201,214,188,123), (2,8)(4,10)(6,12)(14,20)(16,22)(18,24)(25,31)(27,33)(29,35)(38,44)(40,46)(42,48)(49,184)(50,191)(51,186)(52,181)(53,188)(54,183)(55,190)(56,185)(57,192)(58,187)(59,182)(60,189)(61,67)(63,69)(65,71)(73,194)(74,201)(75,196)(76,203)(77,198)(78,193)(79,200)(80,195)(81,202)(82,197)(83,204)(84,199)(86,92)(88,94)(90,96)(97,138)(98,133)(99,140)(100,135)(101,142)(102,137)(103,144)(104,139)(105,134)(106,141)(107,136)(108,143)(109,122)(110,129)(111,124)(112,131)(113,126)(114,121)(115,128)(116,123)(117,130)(118,125)(119,132)(120,127)(146,152)(148,154)(150,156)(157,163)(159,165)(161,167)(170,176)(172,178)(174,180)(205,228)(206,223)(207,218)(208,225)(209,220)(210,227)(211,222)(212,217)(213,224)(214,219)(215,226)(216,221)(230,236)(232,238)(234,240), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120)(121,127)(122,128)(123,129)(124,130)(125,131)(126,132)(133,139)(134,140)(135,141)(136,142)(137,143)(138,144)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156)(157,163)(158,164)(159,165)(160,166)(161,167)(162,168)(169,175)(170,176)(171,177)(172,178)(173,179)(174,180)(181,187)(182,188)(183,189)(184,190)(185,191)(186,192)(193,199)(194,200)(195,201)(196,202)(197,203)(198,204)(205,211)(206,212)(207,213)(208,214)(209,215)(210,216)(217,223)(218,224)(219,225)(220,226)(221,227)(222,228)(229,235)(230,236)(231,237)(232,238)(233,239)(234,240), (1,164)(2,165)(3,166)(4,167)(5,168)(6,157)(7,158)(8,159)(9,160)(10,161)(11,162)(12,163)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,37)(49,190)(50,191)(51,192)(52,181)(53,182)(54,183)(55,184)(56,185)(57,186)(58,187)(59,188)(60,189)(61,176)(62,177)(63,178)(64,179)(65,180)(66,169)(67,170)(68,171)(69,172)(70,173)(71,174)(72,175)(73,194)(74,195)(75,196)(76,197)(77,198)(78,199)(79,200)(80,201)(81,202)(82,203)(83,204)(84,193)(97,138)(98,139)(99,140)(100,141)(101,142)(102,143)(103,144)(104,133)(105,134)(106,135)(107,136)(108,137)(109,122)(110,123)(111,124)(112,125)(113,126)(114,127)(115,128)(116,129)(117,130)(118,131)(119,132)(120,121)(145,237)(146,238)(147,239)(148,240)(149,229)(150,230)(151,231)(152,232)(153,233)(154,234)(155,235)(156,236)(205,228)(206,217)(207,218)(208,219)(209,220)(210,221)(211,222)(212,223)(213,224)(214,225)(215,226)(216,227), (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), (1,58,158,181)(2,51,159,186)(3,56,160,191)(4,49,161,184)(5,54,162,189)(6,59,163,182)(7,52,164,187)(8,57,165,192)(9,50,166,185)(10,55,167,190)(11,60,168,183)(12,53,157,188)(13,209,85,226)(14,214,86,219)(15,207,87,224)(16,212,88,217)(17,205,89,222)(18,210,90,227)(19,215,91,220)(20,208,92,225)(21,213,93,218)(22,206,94,223)(23,211,95,228)(24,216,96,221)(25,143,44,108)(26,136,45,101)(27,141,46,106)(28,134,47,99)(29,139,48,104)(30,144,37,97)(31,137,38,102)(32,142,39,107)(33,135,40,100)(34,140,41,105)(35,133,42,98)(36,138,43,103)(61,74,170,201)(62,79,171,194)(63,84,172,199)(64,77,173,204)(65,82,174,197)(66,75,175,202)(67,80,176,195)(68,73,177,200)(69,78,178,193)(70,83,179,198)(71,76,180,203)(72,81,169,196)(109,155,128,229)(110,148,129,234)(111,153,130,239)(112,146,131,232)(113,151,132,237)(114,156,121,230)(115,149,122,235)(116,154,123,240)(117,147,124,233)(118,152,125,238)(119,145,126,231)(120,150,127,236)>;
G:=Group( (1,229,26,62,87)(2,230,27,63,88)(3,231,28,64,89)(4,232,29,65,90)(5,233,30,66,91)(6,234,31,67,92)(7,235,32,68,93)(8,236,33,69,94)(9,237,34,70,95)(10,238,35,71,96)(11,239,36,72,85)(12,240,25,61,86)(13,168,153,43,169)(14,157,154,44,170)(15,158,155,45,171)(16,159,156,46,172)(17,160,145,47,173)(18,161,146,48,174)(19,162,147,37,175)(20,163,148,38,176)(21,164,149,39,177)(22,165,150,40,178)(23,166,151,41,179)(24,167,152,42,180)(49,112,139,82,227)(50,113,140,83,228)(51,114,141,84,217)(52,115,142,73,218)(53,116,143,74,219)(54,117,144,75,220)(55,118,133,76,221)(56,119,134,77,222)(57,120,135,78,223)(58,109,136,79,224)(59,110,137,80,225)(60,111,138,81,226)(97,202,215,189,124)(98,203,216,190,125)(99,204,205,191,126)(100,193,206,192,127)(101,194,207,181,128)(102,195,208,182,129)(103,196,209,183,130)(104,197,210,184,131)(105,198,211,185,132)(106,199,212,186,121)(107,200,213,187,122)(108,201,214,188,123), (2,8)(4,10)(6,12)(14,20)(16,22)(18,24)(25,31)(27,33)(29,35)(38,44)(40,46)(42,48)(49,184)(50,191)(51,186)(52,181)(53,188)(54,183)(55,190)(56,185)(57,192)(58,187)(59,182)(60,189)(61,67)(63,69)(65,71)(73,194)(74,201)(75,196)(76,203)(77,198)(78,193)(79,200)(80,195)(81,202)(82,197)(83,204)(84,199)(86,92)(88,94)(90,96)(97,138)(98,133)(99,140)(100,135)(101,142)(102,137)(103,144)(104,139)(105,134)(106,141)(107,136)(108,143)(109,122)(110,129)(111,124)(112,131)(113,126)(114,121)(115,128)(116,123)(117,130)(118,125)(119,132)(120,127)(146,152)(148,154)(150,156)(157,163)(159,165)(161,167)(170,176)(172,178)(174,180)(205,228)(206,223)(207,218)(208,225)(209,220)(210,227)(211,222)(212,217)(213,224)(214,219)(215,226)(216,221)(230,236)(232,238)(234,240), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120)(121,127)(122,128)(123,129)(124,130)(125,131)(126,132)(133,139)(134,140)(135,141)(136,142)(137,143)(138,144)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156)(157,163)(158,164)(159,165)(160,166)(161,167)(162,168)(169,175)(170,176)(171,177)(172,178)(173,179)(174,180)(181,187)(182,188)(183,189)(184,190)(185,191)(186,192)(193,199)(194,200)(195,201)(196,202)(197,203)(198,204)(205,211)(206,212)(207,213)(208,214)(209,215)(210,216)(217,223)(218,224)(219,225)(220,226)(221,227)(222,228)(229,235)(230,236)(231,237)(232,238)(233,239)(234,240), (1,164)(2,165)(3,166)(4,167)(5,168)(6,157)(7,158)(8,159)(9,160)(10,161)(11,162)(12,163)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,37)(49,190)(50,191)(51,192)(52,181)(53,182)(54,183)(55,184)(56,185)(57,186)(58,187)(59,188)(60,189)(61,176)(62,177)(63,178)(64,179)(65,180)(66,169)(67,170)(68,171)(69,172)(70,173)(71,174)(72,175)(73,194)(74,195)(75,196)(76,197)(77,198)(78,199)(79,200)(80,201)(81,202)(82,203)(83,204)(84,193)(97,138)(98,139)(99,140)(100,141)(101,142)(102,143)(103,144)(104,133)(105,134)(106,135)(107,136)(108,137)(109,122)(110,123)(111,124)(112,125)(113,126)(114,127)(115,128)(116,129)(117,130)(118,131)(119,132)(120,121)(145,237)(146,238)(147,239)(148,240)(149,229)(150,230)(151,231)(152,232)(153,233)(154,234)(155,235)(156,236)(205,228)(206,217)(207,218)(208,219)(209,220)(210,221)(211,222)(212,223)(213,224)(214,225)(215,226)(216,227), (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), (1,58,158,181)(2,51,159,186)(3,56,160,191)(4,49,161,184)(5,54,162,189)(6,59,163,182)(7,52,164,187)(8,57,165,192)(9,50,166,185)(10,55,167,190)(11,60,168,183)(12,53,157,188)(13,209,85,226)(14,214,86,219)(15,207,87,224)(16,212,88,217)(17,205,89,222)(18,210,90,227)(19,215,91,220)(20,208,92,225)(21,213,93,218)(22,206,94,223)(23,211,95,228)(24,216,96,221)(25,143,44,108)(26,136,45,101)(27,141,46,106)(28,134,47,99)(29,139,48,104)(30,144,37,97)(31,137,38,102)(32,142,39,107)(33,135,40,100)(34,140,41,105)(35,133,42,98)(36,138,43,103)(61,74,170,201)(62,79,171,194)(63,84,172,199)(64,77,173,204)(65,82,174,197)(66,75,175,202)(67,80,176,195)(68,73,177,200)(69,78,178,193)(70,83,179,198)(71,76,180,203)(72,81,169,196)(109,155,128,229)(110,148,129,234)(111,153,130,239)(112,146,131,232)(113,151,132,237)(114,156,121,230)(115,149,122,235)(116,154,123,240)(117,147,124,233)(118,152,125,238)(119,145,126,231)(120,150,127,236) );
G=PermutationGroup([[(1,229,26,62,87),(2,230,27,63,88),(3,231,28,64,89),(4,232,29,65,90),(5,233,30,66,91),(6,234,31,67,92),(7,235,32,68,93),(8,236,33,69,94),(9,237,34,70,95),(10,238,35,71,96),(11,239,36,72,85),(12,240,25,61,86),(13,168,153,43,169),(14,157,154,44,170),(15,158,155,45,171),(16,159,156,46,172),(17,160,145,47,173),(18,161,146,48,174),(19,162,147,37,175),(20,163,148,38,176),(21,164,149,39,177),(22,165,150,40,178),(23,166,151,41,179),(24,167,152,42,180),(49,112,139,82,227),(50,113,140,83,228),(51,114,141,84,217),(52,115,142,73,218),(53,116,143,74,219),(54,117,144,75,220),(55,118,133,76,221),(56,119,134,77,222),(57,120,135,78,223),(58,109,136,79,224),(59,110,137,80,225),(60,111,138,81,226),(97,202,215,189,124),(98,203,216,190,125),(99,204,205,191,126),(100,193,206,192,127),(101,194,207,181,128),(102,195,208,182,129),(103,196,209,183,130),(104,197,210,184,131),(105,198,211,185,132),(106,199,212,186,121),(107,200,213,187,122),(108,201,214,188,123)], [(2,8),(4,10),(6,12),(14,20),(16,22),(18,24),(25,31),(27,33),(29,35),(38,44),(40,46),(42,48),(49,184),(50,191),(51,186),(52,181),(53,188),(54,183),(55,190),(56,185),(57,192),(58,187),(59,182),(60,189),(61,67),(63,69),(65,71),(73,194),(74,201),(75,196),(76,203),(77,198),(78,193),(79,200),(80,195),(81,202),(82,197),(83,204),(84,199),(86,92),(88,94),(90,96),(97,138),(98,133),(99,140),(100,135),(101,142),(102,137),(103,144),(104,139),(105,134),(106,141),(107,136),(108,143),(109,122),(110,129),(111,124),(112,131),(113,126),(114,121),(115,128),(116,123),(117,130),(118,125),(119,132),(120,127),(146,152),(148,154),(150,156),(157,163),(159,165),(161,167),(170,176),(172,178),(174,180),(205,228),(206,223),(207,218),(208,225),(209,220),(210,227),(211,222),(212,217),(213,224),(214,219),(215,226),(216,221),(230,236),(232,238),(234,240)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96),(97,103),(98,104),(99,105),(100,106),(101,107),(102,108),(109,115),(110,116),(111,117),(112,118),(113,119),(114,120),(121,127),(122,128),(123,129),(124,130),(125,131),(126,132),(133,139),(134,140),(135,141),(136,142),(137,143),(138,144),(145,151),(146,152),(147,153),(148,154),(149,155),(150,156),(157,163),(158,164),(159,165),(160,166),(161,167),(162,168),(169,175),(170,176),(171,177),(172,178),(173,179),(174,180),(181,187),(182,188),(183,189),(184,190),(185,191),(186,192),(193,199),(194,200),(195,201),(196,202),(197,203),(198,204),(205,211),(206,212),(207,213),(208,214),(209,215),(210,216),(217,223),(218,224),(219,225),(220,226),(221,227),(222,228),(229,235),(230,236),(231,237),(232,238),(233,239),(234,240)], [(1,164),(2,165),(3,166),(4,167),(5,168),(6,157),(7,158),(8,159),(9,160),(10,161),(11,162),(12,163),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,85),(20,86),(21,87),(22,88),(23,89),(24,90),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,37),(49,190),(50,191),(51,192),(52,181),(53,182),(54,183),(55,184),(56,185),(57,186),(58,187),(59,188),(60,189),(61,176),(62,177),(63,178),(64,179),(65,180),(66,169),(67,170),(68,171),(69,172),(70,173),(71,174),(72,175),(73,194),(74,195),(75,196),(76,197),(77,198),(78,199),(79,200),(80,201),(81,202),(82,203),(83,204),(84,193),(97,138),(98,139),(99,140),(100,141),(101,142),(102,143),(103,144),(104,133),(105,134),(106,135),(107,136),(108,137),(109,122),(110,123),(111,124),(112,125),(113,126),(114,127),(115,128),(116,129),(117,130),(118,131),(119,132),(120,121),(145,237),(146,238),(147,239),(148,240),(149,229),(150,230),(151,231),(152,232),(153,233),(154,234),(155,235),(156,236),(205,228),(206,217),(207,218),(208,219),(209,220),(210,221),(211,222),(212,223),(213,224),(214,225),(215,226),(216,227)], [(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)], [(1,58,158,181),(2,51,159,186),(3,56,160,191),(4,49,161,184),(5,54,162,189),(6,59,163,182),(7,52,164,187),(8,57,165,192),(9,50,166,185),(10,55,167,190),(11,60,168,183),(12,53,157,188),(13,209,85,226),(14,214,86,219),(15,207,87,224),(16,212,88,217),(17,205,89,222),(18,210,90,227),(19,215,91,220),(20,208,92,225),(21,213,93,218),(22,206,94,223),(23,211,95,228),(24,216,96,221),(25,143,44,108),(26,136,45,101),(27,141,46,106),(28,134,47,99),(29,139,48,104),(30,144,37,97),(31,137,38,102),(32,142,39,107),(33,135,40,100),(34,140,41,105),(35,133,42,98),(36,138,43,103),(61,74,170,201),(62,79,171,194),(63,84,172,199),(64,77,173,204),(65,82,174,197),(66,75,175,202),(67,80,176,195),(68,73,177,200),(69,78,178,193),(70,83,179,198),(71,76,180,203),(72,81,169,196),(109,155,128,229),(110,148,129,234),(111,153,130,239),(112,146,131,232),(113,151,132,237),(114,156,121,230),(115,149,122,235),(116,154,123,240),(117,147,124,233),(118,152,125,238),(119,145,126,231),(120,150,127,236)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10L | 10M | ··· | 10T | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20X | 20Y | ··· | 20AF | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | S3 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C5×S3 | C5×D4 | S3×C10 | S3×C10 | C5×C4○D4 | C5×C3⋊D4 | D4⋊2S3 | C5×D4⋊2S3 |
| kernel | C5×C23.12D6 | Dic3×C20 | C5×C6.D4 | C10×Dic6 | D4×C30 | C23.12D6 | C4×Dic3 | C6.D4 | C2×Dic6 | C6×D4 | D4×C10 | C60 | C2×C20 | C22×C10 | C30 | C20 | C2×D4 | C12 | C2×C4 | C23 | C6 | C4 | C10 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 16 | 4 | 4 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 16 | 16 | 2 | 8 |
Matrix representation of C5×C23.12D6 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 10 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 |
| 13 | 47 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 0 |
| 47 | 15 | 0 | 0 |
| 56 | 14 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[1,10,0,0,0,60,0,0,0,0,60,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[13,13,0,0,0,47,0,0,0,0,0,1,0,0,60,0],[47,56,0,0,15,14,0,0,0,0,50,0,0,0,0,50] >;
C5×C23.12D6 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{12}D_6 % in TeX
G:=Group("C5xC2^3.12D6"); // GroupNames label
G:=SmallGroup(480,815);
// by ID
G=gap.SmallGroup(480,815);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,288,2606,471,15686]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^6=c,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=b*c=c*b,f*b*f^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;
// generators/relations