direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C12.23D4, C60.156D4, (C6×Q8)⋊4C10, D6⋊C4⋊16C10, (Q8×C10)⋊15S3, (Q8×C30)⋊18C2, C12.23(C5×D4), C6.58(D4×C10), (C4×Dic3)⋊7C10, (C2×D12).9C10, C30.441(C2×D4), (C2×C20).247D6, (Dic3×C20)⋊19C2, (C10×D12).19C2, C20.76(C3⋊D4), C15⋊27(C4.4D4), C30.273(C4○D4), (C2×C30).438C23, (C2×C60).368C22, C10.55(Q8⋊3S3), (C10×Dic3).233C22, (C2×Q8)⋊6(C5×S3), C3⋊4(C5×C4.4D4), (C5×D6⋊C4)⋊38C2, C6.37(C5×C4○D4), C4.11(C5×C3⋊D4), (C2×C4).57(S3×C10), C2.22(C10×C3⋊D4), C22.65(S3×C2×C10), C2.9(C5×Q8⋊3S3), (C2×C12).42(C2×C10), C10.143(C2×C3⋊D4), (S3×C2×C10).74C22, (C2×C6).59(C22×C10), (C22×S3).13(C2×C10), (C2×C10).372(C22×S3), (C2×Dic3).42(C2×C10), SmallGroup(480,826)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C12.23D4
G = < a,b,c,d | a5=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=b6c-1 >
Subgroups: 388 in 152 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C5×S3, C30, C30, C4.4D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C4×Dic3, D6⋊C4, C2×D12, C6×Q8, C5×Dic3, C60, C60, S3×C10, C2×C30, C4×C20, C5×C22⋊C4, D4×C10, Q8×C10, C12.23D4, C5×D12, C10×Dic3, C2×C60, C2×C60, Q8×C15, S3×C2×C10, C5×C4.4D4, Dic3×C20, C5×D6⋊C4, C10×D12, Q8×C30, C5×C12.23D4
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, Q8⋊3S3, C2×C3⋊D4, S3×C10, D4×C10, C5×C4○D4, C12.23D4, C5×C3⋊D4, S3×C2×C10, C5×C4.4D4, C5×Q8⋊3S3, C10×C3⋊D4, C5×C12.23D4
►On 240 points
Generators in S240
(1 114 197 75 152)(2 115 198 76 153)(3 116 199 77 154)(4 117 200 78 155)(5 118 201 79 156)(6 119 202 80 145)(7 120 203 81 146)(8 109 204 82 147)(9 110 193 83 148)(10 111 194 84 149)(11 112 195 73 150)(12 113 196 74 151)(13 58 215 136 219)(14 59 216 137 220)(15 60 205 138 221)(16 49 206 139 222)(17 50 207 140 223)(18 51 208 141 224)(19 52 209 142 225)(20 53 210 143 226)(21 54 211 144 227)(22 55 212 133 228)(23 56 213 134 217)(24 57 214 135 218)(25 40 163 96 235)(26 41 164 85 236)(27 42 165 86 237)(28 43 166 87 238)(29 44 167 88 239)(30 45 168 89 240)(31 46 157 90 229)(32 47 158 91 230)(33 48 159 92 231)(34 37 160 93 232)(35 38 161 94 233)(36 39 162 95 234)(61 173 99 183 129)(62 174 100 184 130)(63 175 101 185 131)(64 176 102 186 132)(65 177 103 187 121)(66 178 104 188 122)(67 179 105 189 123)(68 180 106 190 124)(69 169 107 191 125)(70 170 108 192 126)(71 171 97 181 127)(72 172 98 182 128)
(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 52 106 232)(2 57 107 237)(3 50 108 230)(4 55 97 235)(5 60 98 240)(6 53 99 233)(7 58 100 238)(8 51 101 231)(9 56 102 236)(10 49 103 229)(11 54 104 234)(12 59 105 239)(13 174 87 146)(14 179 88 151)(15 172 89 156)(16 177 90 149)(17 170 91 154)(18 175 92 147)(19 180 93 152)(20 173 94 145)(21 178 95 150)(22 171 96 155)(23 176 85 148)(24 169 86 153)(25 117 212 181)(26 110 213 186)(27 115 214 191)(28 120 215 184)(29 113 216 189)(30 118 205 182)(31 111 206 187)(32 116 207 192)(33 109 208 185)(34 114 209 190)(35 119 210 183)(36 112 211 188)(37 197 142 124)(38 202 143 129)(39 195 144 122)(40 200 133 127)(41 193 134 132)(42 198 135 125)(43 203 136 130)(44 196 137 123)(45 201 138 128)(46 194 139 121)(47 199 140 126)(48 204 141 131)(61 161 80 226)(62 166 81 219)(63 159 82 224)(64 164 83 217)(65 157 84 222)(66 162 73 227)(67 167 74 220)(68 160 75 225)(69 165 76 218)(70 158 77 223)(71 163 78 228)(72 168 79 221)
(1 103)(2 102)(3 101)(4 100)(5 99)(6 98)(7 97)(8 108)(9 107)(10 106)(11 105)(12 104)(13 16)(14 15)(17 24)(18 23)(19 22)(20 21)(25 34)(26 33)(27 32)(28 31)(29 30)(35 36)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)(49 58)(50 57)(51 56)(52 55)(53 54)(59 60)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 84)(69 83)(70 82)(71 81)(72 80)(85 92)(86 91)(87 90)(88 89)(93 96)(94 95)(109 192)(110 191)(111 190)(112 189)(113 188)(114 187)(115 186)(116 185)(117 184)(118 183)(119 182)(120 181)(121 197)(122 196)(123 195)(124 194)(125 193)(126 204)(127 203)(128 202)(129 201)(130 200)(131 199)(132 198)(133 142)(134 141)(135 140)(136 139)(137 138)(143 144)(145 172)(146 171)(147 170)(148 169)(149 180)(150 179)(151 178)(152 177)(153 176)(154 175)(155 174)(156 173)(157 166)(158 165)(159 164)(160 163)(161 162)(167 168)(205 216)(206 215)(207 214)(208 213)(209 212)(210 211)(217 224)(218 223)(219 222)(220 221)(225 228)(226 227)(229 238)(230 237)(231 236)(232 235)(233 234)(239 240)
G:=sub<Sym(240)| (1,114,197,75,152)(2,115,198,76,153)(3,116,199,77,154)(4,117,200,78,155)(5,118,201,79,156)(6,119,202,80,145)(7,120,203,81,146)(8,109,204,82,147)(9,110,193,83,148)(10,111,194,84,149)(11,112,195,73,150)(12,113,196,74,151)(13,58,215,136,219)(14,59,216,137,220)(15,60,205,138,221)(16,49,206,139,222)(17,50,207,140,223)(18,51,208,141,224)(19,52,209,142,225)(20,53,210,143,226)(21,54,211,144,227)(22,55,212,133,228)(23,56,213,134,217)(24,57,214,135,218)(25,40,163,96,235)(26,41,164,85,236)(27,42,165,86,237)(28,43,166,87,238)(29,44,167,88,239)(30,45,168,89,240)(31,46,157,90,229)(32,47,158,91,230)(33,48,159,92,231)(34,37,160,93,232)(35,38,161,94,233)(36,39,162,95,234)(61,173,99,183,129)(62,174,100,184,130)(63,175,101,185,131)(64,176,102,186,132)(65,177,103,187,121)(66,178,104,188,122)(67,179,105,189,123)(68,180,106,190,124)(69,169,107,191,125)(70,170,108,192,126)(71,171,97,181,127)(72,172,98,182,128), (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), (1,52,106,232)(2,57,107,237)(3,50,108,230)(4,55,97,235)(5,60,98,240)(6,53,99,233)(7,58,100,238)(8,51,101,231)(9,56,102,236)(10,49,103,229)(11,54,104,234)(12,59,105,239)(13,174,87,146)(14,179,88,151)(15,172,89,156)(16,177,90,149)(17,170,91,154)(18,175,92,147)(19,180,93,152)(20,173,94,145)(21,178,95,150)(22,171,96,155)(23,176,85,148)(24,169,86,153)(25,117,212,181)(26,110,213,186)(27,115,214,191)(28,120,215,184)(29,113,216,189)(30,118,205,182)(31,111,206,187)(32,116,207,192)(33,109,208,185)(34,114,209,190)(35,119,210,183)(36,112,211,188)(37,197,142,124)(38,202,143,129)(39,195,144,122)(40,200,133,127)(41,193,134,132)(42,198,135,125)(43,203,136,130)(44,196,137,123)(45,201,138,128)(46,194,139,121)(47,199,140,126)(48,204,141,131)(61,161,80,226)(62,166,81,219)(63,159,82,224)(64,164,83,217)(65,157,84,222)(66,162,73,227)(67,167,74,220)(68,160,75,225)(69,165,76,218)(70,158,77,223)(71,163,78,228)(72,168,79,221), (1,103)(2,102)(3,101)(4,100)(5,99)(6,98)(7,97)(8,108)(9,107)(10,106)(11,105)(12,104)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21)(25,34)(26,33)(27,32)(28,31)(29,30)(35,36)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,84)(69,83)(70,82)(71,81)(72,80)(85,92)(86,91)(87,90)(88,89)(93,96)(94,95)(109,192)(110,191)(111,190)(112,189)(113,188)(114,187)(115,186)(116,185)(117,184)(118,183)(119,182)(120,181)(121,197)(122,196)(123,195)(124,194)(125,193)(126,204)(127,203)(128,202)(129,201)(130,200)(131,199)(132,198)(133,142)(134,141)(135,140)(136,139)(137,138)(143,144)(145,172)(146,171)(147,170)(148,169)(149,180)(150,179)(151,178)(152,177)(153,176)(154,175)(155,174)(156,173)(157,166)(158,165)(159,164)(160,163)(161,162)(167,168)(205,216)(206,215)(207,214)(208,213)(209,212)(210,211)(217,224)(218,223)(219,222)(220,221)(225,228)(226,227)(229,238)(230,237)(231,236)(232,235)(233,234)(239,240)>;
G:=Group( (1,114,197,75,152)(2,115,198,76,153)(3,116,199,77,154)(4,117,200,78,155)(5,118,201,79,156)(6,119,202,80,145)(7,120,203,81,146)(8,109,204,82,147)(9,110,193,83,148)(10,111,194,84,149)(11,112,195,73,150)(12,113,196,74,151)(13,58,215,136,219)(14,59,216,137,220)(15,60,205,138,221)(16,49,206,139,222)(17,50,207,140,223)(18,51,208,141,224)(19,52,209,142,225)(20,53,210,143,226)(21,54,211,144,227)(22,55,212,133,228)(23,56,213,134,217)(24,57,214,135,218)(25,40,163,96,235)(26,41,164,85,236)(27,42,165,86,237)(28,43,166,87,238)(29,44,167,88,239)(30,45,168,89,240)(31,46,157,90,229)(32,47,158,91,230)(33,48,159,92,231)(34,37,160,93,232)(35,38,161,94,233)(36,39,162,95,234)(61,173,99,183,129)(62,174,100,184,130)(63,175,101,185,131)(64,176,102,186,132)(65,177,103,187,121)(66,178,104,188,122)(67,179,105,189,123)(68,180,106,190,124)(69,169,107,191,125)(70,170,108,192,126)(71,171,97,181,127)(72,172,98,182,128), (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), (1,52,106,232)(2,57,107,237)(3,50,108,230)(4,55,97,235)(5,60,98,240)(6,53,99,233)(7,58,100,238)(8,51,101,231)(9,56,102,236)(10,49,103,229)(11,54,104,234)(12,59,105,239)(13,174,87,146)(14,179,88,151)(15,172,89,156)(16,177,90,149)(17,170,91,154)(18,175,92,147)(19,180,93,152)(20,173,94,145)(21,178,95,150)(22,171,96,155)(23,176,85,148)(24,169,86,153)(25,117,212,181)(26,110,213,186)(27,115,214,191)(28,120,215,184)(29,113,216,189)(30,118,205,182)(31,111,206,187)(32,116,207,192)(33,109,208,185)(34,114,209,190)(35,119,210,183)(36,112,211,188)(37,197,142,124)(38,202,143,129)(39,195,144,122)(40,200,133,127)(41,193,134,132)(42,198,135,125)(43,203,136,130)(44,196,137,123)(45,201,138,128)(46,194,139,121)(47,199,140,126)(48,204,141,131)(61,161,80,226)(62,166,81,219)(63,159,82,224)(64,164,83,217)(65,157,84,222)(66,162,73,227)(67,167,74,220)(68,160,75,225)(69,165,76,218)(70,158,77,223)(71,163,78,228)(72,168,79,221), (1,103)(2,102)(3,101)(4,100)(5,99)(6,98)(7,97)(8,108)(9,107)(10,106)(11,105)(12,104)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21)(25,34)(26,33)(27,32)(28,31)(29,30)(35,36)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,84)(69,83)(70,82)(71,81)(72,80)(85,92)(86,91)(87,90)(88,89)(93,96)(94,95)(109,192)(110,191)(111,190)(112,189)(113,188)(114,187)(115,186)(116,185)(117,184)(118,183)(119,182)(120,181)(121,197)(122,196)(123,195)(124,194)(125,193)(126,204)(127,203)(128,202)(129,201)(130,200)(131,199)(132,198)(133,142)(134,141)(135,140)(136,139)(137,138)(143,144)(145,172)(146,171)(147,170)(148,169)(149,180)(150,179)(151,178)(152,177)(153,176)(154,175)(155,174)(156,173)(157,166)(158,165)(159,164)(160,163)(161,162)(167,168)(205,216)(206,215)(207,214)(208,213)(209,212)(210,211)(217,224)(218,223)(219,222)(220,221)(225,228)(226,227)(229,238)(230,237)(231,236)(232,235)(233,234)(239,240) );
G=PermutationGroup([[(1,114,197,75,152),(2,115,198,76,153),(3,116,199,77,154),(4,117,200,78,155),(5,118,201,79,156),(6,119,202,80,145),(7,120,203,81,146),(8,109,204,82,147),(9,110,193,83,148),(10,111,194,84,149),(11,112,195,73,150),(12,113,196,74,151),(13,58,215,136,219),(14,59,216,137,220),(15,60,205,138,221),(16,49,206,139,222),(17,50,207,140,223),(18,51,208,141,224),(19,52,209,142,225),(20,53,210,143,226),(21,54,211,144,227),(22,55,212,133,228),(23,56,213,134,217),(24,57,214,135,218),(25,40,163,96,235),(26,41,164,85,236),(27,42,165,86,237),(28,43,166,87,238),(29,44,167,88,239),(30,45,168,89,240),(31,46,157,90,229),(32,47,158,91,230),(33,48,159,92,231),(34,37,160,93,232),(35,38,161,94,233),(36,39,162,95,234),(61,173,99,183,129),(62,174,100,184,130),(63,175,101,185,131),(64,176,102,186,132),(65,177,103,187,121),(66,178,104,188,122),(67,179,105,189,123),(68,180,106,190,124),(69,169,107,191,125),(70,170,108,192,126),(71,171,97,181,127),(72,172,98,182,128)], [(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)], [(1,52,106,232),(2,57,107,237),(3,50,108,230),(4,55,97,235),(5,60,98,240),(6,53,99,233),(7,58,100,238),(8,51,101,231),(9,56,102,236),(10,49,103,229),(11,54,104,234),(12,59,105,239),(13,174,87,146),(14,179,88,151),(15,172,89,156),(16,177,90,149),(17,170,91,154),(18,175,92,147),(19,180,93,152),(20,173,94,145),(21,178,95,150),(22,171,96,155),(23,176,85,148),(24,169,86,153),(25,117,212,181),(26,110,213,186),(27,115,214,191),(28,120,215,184),(29,113,216,189),(30,118,205,182),(31,111,206,187),(32,116,207,192),(33,109,208,185),(34,114,209,190),(35,119,210,183),(36,112,211,188),(37,197,142,124),(38,202,143,129),(39,195,144,122),(40,200,133,127),(41,193,134,132),(42,198,135,125),(43,203,136,130),(44,196,137,123),(45,201,138,128),(46,194,139,121),(47,199,140,126),(48,204,141,131),(61,161,80,226),(62,166,81,219),(63,159,82,224),(64,164,83,217),(65,157,84,222),(66,162,73,227),(67,167,74,220),(68,160,75,225),(69,165,76,218),(70,158,77,223),(71,163,78,228),(72,168,79,221)], [(1,103),(2,102),(3,101),(4,100),(5,99),(6,98),(7,97),(8,108),(9,107),(10,106),(11,105),(12,104),(13,16),(14,15),(17,24),(18,23),(19,22),(20,21),(25,34),(26,33),(27,32),(28,31),(29,30),(35,36),(37,40),(38,39),(41,48),(42,47),(43,46),(44,45),(49,58),(50,57),(51,56),(52,55),(53,54),(59,60),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,84),(69,83),(70,82),(71,81),(72,80),(85,92),(86,91),(87,90),(88,89),(93,96),(94,95),(109,192),(110,191),(111,190),(112,189),(113,188),(114,187),(115,186),(116,185),(117,184),(118,183),(119,182),(120,181),(121,197),(122,196),(123,195),(124,194),(125,193),(126,204),(127,203),(128,202),(129,201),(130,200),(131,199),(132,198),(133,142),(134,141),(135,140),(136,139),(137,138),(143,144),(145,172),(146,171),(147,170),(148,169),(149,180),(150,179),(151,178),(152,177),(153,176),(154,175),(155,174),(156,173),(157,166),(158,165),(159,164),(160,163),(161,162),(167,168),(205,216),(206,215),(207,214),(208,213),(209,212),(210,211),(217,224),(218,223),(219,222),(220,221),(225,228),(226,227),(229,238),(230,237),(231,236),(232,235),(233,234),(239,240)]])
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 | 10A | ··· | 10L | 10M | ··· | 10T | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 20Q | ··· | 20AF | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 12 | ··· | 12 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 2 | ··· | 2 | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | S3 | D4 | D6 | C4○D4 | C3⋊D4 | C5×S3 | C5×D4 | S3×C10 | C5×C4○D4 | C5×C3⋊D4 | Q8⋊3S3 | C5×Q8⋊3S3 |
| kernel | C5×C12.23D4 | Dic3×C20 | C5×D6⋊C4 | C10×D12 | Q8×C30 | C12.23D4 | C4×Dic3 | D6⋊C4 | C2×D12 | C6×Q8 | Q8×C10 | C60 | C2×C20 | C30 | C20 | C2×Q8 | C12 | C2×C4 | C6 | C4 | C10 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 16 | 4 | 4 | 1 | 2 | 3 | 4 | 4 | 4 | 8 | 12 | 16 | 16 | 2 | 8 |
Matrix representation of C5×C12.23D4 ►in GL4(𝔽61) generated by
| 20 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 60 | 1 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 11 | 0 |
| 52 | 52 | 0 | 0 |
| 43 | 9 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
| 1 | 60 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 50 | 0 |
G:=sub<GL(4,GF(61))| [20,0,0,0,0,20,0,0,0,0,1,0,0,0,0,1],[0,60,0,0,1,1,0,0,0,0,0,11,0,0,11,0],[52,43,0,0,52,9,0,0,0,0,50,0,0,0,0,50],[1,0,0,0,60,60,0,0,0,0,0,50,0,0,11,0] >;
C5×C12.23D4 in GAP, Magma, Sage, TeX
C_5\times C_{12}._{23}D_4 % in TeX
G:=Group("C5xC12.23D4"); // GroupNames label
G:=SmallGroup(480,826);
// by ID
G=gap.SmallGroup(480,826);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,568,926,891,436,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=b^6*c^-1>;
// generators/relations