direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C10×D12, C60⋊12C23, C30.86C24, C6⋊1(D4×C10), (C2×C30)⋊28D4, (C2×C20)⋊36D6, C30⋊12(C2×D4), C15⋊13(C22×D4), (S3×C23)⋊3C10, (C2×C60)⋊49C22, (C22×C20)⋊16S3, (C22×C60)⋊19C2, C12⋊2(C22×C10), C20⋊10(C22×S3), (C22×C12)⋊7C10, D6⋊1(C22×C10), C6.3(C23×C10), (S3×C10)⋊10C23, C10.71(S3×C23), C23.40(S3×C10), (C2×C30).443C23, (C22×C10).154D6, (C22×C30).183C22, C3⋊1(D4×C2×C10), C4⋊2(S3×C2×C10), (C2×C6)⋊6(C5×D4), (C2×C4)⋊9(S3×C10), (S3×C22×C10)⋊9C2, (C22×C4)⋊7(C5×S3), (C2×C12)⋊12(C2×C10), C2.4(S3×C22×C10), (S3×C2×C10)⋊21C22, C22.30(S3×C2×C10), (C22×S3)⋊5(C2×C10), (C2×C6).64(C22×C10), (C22×C6).45(C2×C10), (C2×C10).377(C22×S3), SmallGroup(480,1152)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1156 in 472 conjugacy classes, 210 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×7], C22 [×32], C5, S3 [×8], C6, C6 [×6], C2×C4 [×6], D4 [×16], C23, C23 [×20], C10, C10 [×6], C10 [×8], C12 [×4], D6 [×8], D6 [×24], C2×C6 [×7], C15, C22×C4, C2×D4 [×12], C24 [×2], C20 [×4], C2×C10 [×7], C2×C10 [×32], D12 [×16], C2×C12 [×6], C22×S3 [×12], C22×S3 [×8], C22×C6, C5×S3 [×8], C30, C30 [×6], C22×D4, C2×C20 [×6], C5×D4 [×16], C22×C10, C22×C10 [×20], C2×D12 [×12], C22×C12, S3×C23 [×2], C60 [×4], S3×C10 [×8], S3×C10 [×24], C2×C30 [×7], C22×C20, D4×C10 [×12], C23×C10 [×2], C22×D12, C5×D12 [×16], C2×C60 [×6], S3×C2×C10 [×12], S3×C2×C10 [×8], C22×C30, D4×C2×C10, C10×D12 [×12], C22×C60, S3×C22×C10 [×2], C2×C10×D12
Quotients:
C1, C2 [×15], C22 [×35], C5, S3, D4 [×4], C23 [×15], C10 [×15], D6 [×7], C2×D4 [×6], C24, C2×C10 [×35], D12 [×4], C22×S3 [×7], C5×S3, C22×D4, C5×D4 [×4], C22×C10 [×15], C2×D12 [×6], S3×C23, S3×C10 [×7], D4×C10 [×6], C23×C10, C22×D12, C5×D12 [×4], S3×C2×C10 [×7], D4×C2×C10, C10×D12 [×6], S3×C22×C10, C2×C10×D12
Generators and relations
G = < a,b,c,d | a2=b10=c12=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 240 points
Generators in S240
(1 123)(2 124)(3 125)(4 126)(5 127)(6 128)(7 129)(8 130)(9 131)(10 132)(11 121)(12 122)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 25)(22 26)(23 27)(24 28)(37 163)(38 164)(39 165)(40 166)(41 167)(42 168)(43 157)(44 158)(45 159)(46 160)(47 161)(48 162)(49 141)(50 142)(51 143)(52 144)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 221)(62 222)(63 223)(64 224)(65 225)(66 226)(67 227)(68 228)(69 217)(70 218)(71 219)(72 220)(73 197)(74 198)(75 199)(76 200)(77 201)(78 202)(79 203)(80 204)(81 193)(82 194)(83 195)(84 196)(85 150)(86 151)(87 152)(88 153)(89 154)(90 155)(91 156)(92 145)(93 146)(94 147)(95 148)(96 149)(97 109)(98 110)(99 111)(100 112)(101 113)(102 114)(103 115)(104 116)(105 117)(106 118)(107 119)(108 120)(169 211)(170 212)(171 213)(172 214)(173 215)(174 216)(175 205)(176 206)(177 207)(178 208)(179 209)(180 210)(181 240)(182 229)(183 230)(184 231)(185 232)(186 233)(187 234)(188 235)(189 236)(190 237)(191 238)(192 239)
(1 199 34 218 145 47 135 233 114 170)(2 200 35 219 146 48 136 234 115 171)(3 201 36 220 147 37 137 235 116 172)(4 202 25 221 148 38 138 236 117 173)(5 203 26 222 149 39 139 237 118 174)(6 204 27 223 150 40 140 238 119 175)(7 193 28 224 151 41 141 239 120 176)(8 194 29 225 152 42 142 240 109 177)(9 195 30 226 153 43 143 229 110 178)(10 196 31 227 154 44 144 230 111 179)(11 197 32 228 155 45 133 231 112 180)(12 198 33 217 156 46 134 232 113 169)(13 65 87 168 50 181 97 207 130 82)(14 66 88 157 51 182 98 208 131 83)(15 67 89 158 52 183 99 209 132 84)(16 68 90 159 53 184 100 210 121 73)(17 69 91 160 54 185 101 211 122 74)(18 70 92 161 55 186 102 212 123 75)(19 71 93 162 56 187 103 213 124 76)(20 72 94 163 57 188 104 214 125 77)(21 61 95 164 58 189 105 215 126 78)(22 62 96 165 59 190 106 216 127 79)(23 63 85 166 60 191 107 205 128 80)(24 64 86 167 49 192 108 206 129 81)
(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 37)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 181)(14 192)(15 191)(16 190)(17 189)(18 188)(19 187)(20 186)(21 185)(22 184)(23 183)(24 182)(25 232)(26 231)(27 230)(28 229)(29 240)(30 239)(31 238)(32 237)(33 236)(34 235)(35 234)(36 233)(49 83)(50 82)(51 81)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 84)(61 101)(62 100)(63 99)(64 98)(65 97)(66 108)(67 107)(68 106)(69 105)(70 104)(71 103)(72 102)(85 209)(86 208)(87 207)(88 206)(89 205)(90 216)(91 215)(92 214)(93 213)(94 212)(95 211)(96 210)(109 225)(110 224)(111 223)(112 222)(113 221)(114 220)(115 219)(116 218)(117 217)(118 228)(119 227)(120 226)(121 165)(122 164)(123 163)(124 162)(125 161)(126 160)(127 159)(128 158)(129 157)(130 168)(131 167)(132 166)(133 203)(134 202)(135 201)(136 200)(137 199)(138 198)(139 197)(140 196)(141 195)(142 194)(143 193)(144 204)(145 172)(146 171)(147 170)(148 169)(149 180)(150 179)(151 178)(152 177)(153 176)(154 175)(155 174)(156 173)
G:=sub<Sym(240)| (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,129)(8,130)(9,131)(10,132)(11,121)(12,122)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,25)(22,26)(23,27)(24,28)(37,163)(38,164)(39,165)(40,166)(41,167)(42,168)(43,157)(44,158)(45,159)(46,160)(47,161)(48,162)(49,141)(50,142)(51,143)(52,144)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,221)(62,222)(63,223)(64,224)(65,225)(66,226)(67,227)(68,228)(69,217)(70,218)(71,219)(72,220)(73,197)(74,198)(75,199)(76,200)(77,201)(78,202)(79,203)(80,204)(81,193)(82,194)(83,195)(84,196)(85,150)(86,151)(87,152)(88,153)(89,154)(90,155)(91,156)(92,145)(93,146)(94,147)(95,148)(96,149)(97,109)(98,110)(99,111)(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)(108,120)(169,211)(170,212)(171,213)(172,214)(173,215)(174,216)(175,205)(176,206)(177,207)(178,208)(179,209)(180,210)(181,240)(182,229)(183,230)(184,231)(185,232)(186,233)(187,234)(188,235)(189,236)(190,237)(191,238)(192,239), (1,199,34,218,145,47,135,233,114,170)(2,200,35,219,146,48,136,234,115,171)(3,201,36,220,147,37,137,235,116,172)(4,202,25,221,148,38,138,236,117,173)(5,203,26,222,149,39,139,237,118,174)(6,204,27,223,150,40,140,238,119,175)(7,193,28,224,151,41,141,239,120,176)(8,194,29,225,152,42,142,240,109,177)(9,195,30,226,153,43,143,229,110,178)(10,196,31,227,154,44,144,230,111,179)(11,197,32,228,155,45,133,231,112,180)(12,198,33,217,156,46,134,232,113,169)(13,65,87,168,50,181,97,207,130,82)(14,66,88,157,51,182,98,208,131,83)(15,67,89,158,52,183,99,209,132,84)(16,68,90,159,53,184,100,210,121,73)(17,69,91,160,54,185,101,211,122,74)(18,70,92,161,55,186,102,212,123,75)(19,71,93,162,56,187,103,213,124,76)(20,72,94,163,57,188,104,214,125,77)(21,61,95,164,58,189,105,215,126,78)(22,62,96,165,59,190,106,216,127,79)(23,63,85,166,60,191,107,205,128,80)(24,64,86,167,49,192,108,206,129,81), (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), (1,37)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,181)(14,192)(15,191)(16,190)(17,189)(18,188)(19,187)(20,186)(21,185)(22,184)(23,183)(24,182)(25,232)(26,231)(27,230)(28,229)(29,240)(30,239)(31,238)(32,237)(33,236)(34,235)(35,234)(36,233)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,84)(61,101)(62,100)(63,99)(64,98)(65,97)(66,108)(67,107)(68,106)(69,105)(70,104)(71,103)(72,102)(85,209)(86,208)(87,207)(88,206)(89,205)(90,216)(91,215)(92,214)(93,213)(94,212)(95,211)(96,210)(109,225)(110,224)(111,223)(112,222)(113,221)(114,220)(115,219)(116,218)(117,217)(118,228)(119,227)(120,226)(121,165)(122,164)(123,163)(124,162)(125,161)(126,160)(127,159)(128,158)(129,157)(130,168)(131,167)(132,166)(133,203)(134,202)(135,201)(136,200)(137,199)(138,198)(139,197)(140,196)(141,195)(142,194)(143,193)(144,204)(145,172)(146,171)(147,170)(148,169)(149,180)(150,179)(151,178)(152,177)(153,176)(154,175)(155,174)(156,173)>;
G:=Group( (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,129)(8,130)(9,131)(10,132)(11,121)(12,122)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,25)(22,26)(23,27)(24,28)(37,163)(38,164)(39,165)(40,166)(41,167)(42,168)(43,157)(44,158)(45,159)(46,160)(47,161)(48,162)(49,141)(50,142)(51,143)(52,144)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,221)(62,222)(63,223)(64,224)(65,225)(66,226)(67,227)(68,228)(69,217)(70,218)(71,219)(72,220)(73,197)(74,198)(75,199)(76,200)(77,201)(78,202)(79,203)(80,204)(81,193)(82,194)(83,195)(84,196)(85,150)(86,151)(87,152)(88,153)(89,154)(90,155)(91,156)(92,145)(93,146)(94,147)(95,148)(96,149)(97,109)(98,110)(99,111)(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)(108,120)(169,211)(170,212)(171,213)(172,214)(173,215)(174,216)(175,205)(176,206)(177,207)(178,208)(179,209)(180,210)(181,240)(182,229)(183,230)(184,231)(185,232)(186,233)(187,234)(188,235)(189,236)(190,237)(191,238)(192,239), (1,199,34,218,145,47,135,233,114,170)(2,200,35,219,146,48,136,234,115,171)(3,201,36,220,147,37,137,235,116,172)(4,202,25,221,148,38,138,236,117,173)(5,203,26,222,149,39,139,237,118,174)(6,204,27,223,150,40,140,238,119,175)(7,193,28,224,151,41,141,239,120,176)(8,194,29,225,152,42,142,240,109,177)(9,195,30,226,153,43,143,229,110,178)(10,196,31,227,154,44,144,230,111,179)(11,197,32,228,155,45,133,231,112,180)(12,198,33,217,156,46,134,232,113,169)(13,65,87,168,50,181,97,207,130,82)(14,66,88,157,51,182,98,208,131,83)(15,67,89,158,52,183,99,209,132,84)(16,68,90,159,53,184,100,210,121,73)(17,69,91,160,54,185,101,211,122,74)(18,70,92,161,55,186,102,212,123,75)(19,71,93,162,56,187,103,213,124,76)(20,72,94,163,57,188,104,214,125,77)(21,61,95,164,58,189,105,215,126,78)(22,62,96,165,59,190,106,216,127,79)(23,63,85,166,60,191,107,205,128,80)(24,64,86,167,49,192,108,206,129,81), (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), (1,37)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,181)(14,192)(15,191)(16,190)(17,189)(18,188)(19,187)(20,186)(21,185)(22,184)(23,183)(24,182)(25,232)(26,231)(27,230)(28,229)(29,240)(30,239)(31,238)(32,237)(33,236)(34,235)(35,234)(36,233)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,84)(61,101)(62,100)(63,99)(64,98)(65,97)(66,108)(67,107)(68,106)(69,105)(70,104)(71,103)(72,102)(85,209)(86,208)(87,207)(88,206)(89,205)(90,216)(91,215)(92,214)(93,213)(94,212)(95,211)(96,210)(109,225)(110,224)(111,223)(112,222)(113,221)(114,220)(115,219)(116,218)(117,217)(118,228)(119,227)(120,226)(121,165)(122,164)(123,163)(124,162)(125,161)(126,160)(127,159)(128,158)(129,157)(130,168)(131,167)(132,166)(133,203)(134,202)(135,201)(136,200)(137,199)(138,198)(139,197)(140,196)(141,195)(142,194)(143,193)(144,204)(145,172)(146,171)(147,170)(148,169)(149,180)(150,179)(151,178)(152,177)(153,176)(154,175)(155,174)(156,173) );
G=PermutationGroup([(1,123),(2,124),(3,125),(4,126),(5,127),(6,128),(7,129),(8,130),(9,131),(10,132),(11,121),(12,122),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(21,25),(22,26),(23,27),(24,28),(37,163),(38,164),(39,165),(40,166),(41,167),(42,168),(43,157),(44,158),(45,159),(46,160),(47,161),(48,162),(49,141),(50,142),(51,143),(52,144),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,221),(62,222),(63,223),(64,224),(65,225),(66,226),(67,227),(68,228),(69,217),(70,218),(71,219),(72,220),(73,197),(74,198),(75,199),(76,200),(77,201),(78,202),(79,203),(80,204),(81,193),(82,194),(83,195),(84,196),(85,150),(86,151),(87,152),(88,153),(89,154),(90,155),(91,156),(92,145),(93,146),(94,147),(95,148),(96,149),(97,109),(98,110),(99,111),(100,112),(101,113),(102,114),(103,115),(104,116),(105,117),(106,118),(107,119),(108,120),(169,211),(170,212),(171,213),(172,214),(173,215),(174,216),(175,205),(176,206),(177,207),(178,208),(179,209),(180,210),(181,240),(182,229),(183,230),(184,231),(185,232),(186,233),(187,234),(188,235),(189,236),(190,237),(191,238),(192,239)], [(1,199,34,218,145,47,135,233,114,170),(2,200,35,219,146,48,136,234,115,171),(3,201,36,220,147,37,137,235,116,172),(4,202,25,221,148,38,138,236,117,173),(5,203,26,222,149,39,139,237,118,174),(6,204,27,223,150,40,140,238,119,175),(7,193,28,224,151,41,141,239,120,176),(8,194,29,225,152,42,142,240,109,177),(9,195,30,226,153,43,143,229,110,178),(10,196,31,227,154,44,144,230,111,179),(11,197,32,228,155,45,133,231,112,180),(12,198,33,217,156,46,134,232,113,169),(13,65,87,168,50,181,97,207,130,82),(14,66,88,157,51,182,98,208,131,83),(15,67,89,158,52,183,99,209,132,84),(16,68,90,159,53,184,100,210,121,73),(17,69,91,160,54,185,101,211,122,74),(18,70,92,161,55,186,102,212,123,75),(19,71,93,162,56,187,103,213,124,76),(20,72,94,163,57,188,104,214,125,77),(21,61,95,164,58,189,105,215,126,78),(22,62,96,165,59,190,106,216,127,79),(23,63,85,166,60,191,107,205,128,80),(24,64,86,167,49,192,108,206,129,81)], [(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)], [(1,37),(2,48),(3,47),(4,46),(5,45),(6,44),(7,43),(8,42),(9,41),(10,40),(11,39),(12,38),(13,181),(14,192),(15,191),(16,190),(17,189),(18,188),(19,187),(20,186),(21,185),(22,184),(23,183),(24,182),(25,232),(26,231),(27,230),(28,229),(29,240),(30,239),(31,238),(32,237),(33,236),(34,235),(35,234),(36,233),(49,83),(50,82),(51,81),(52,80),(53,79),(54,78),(55,77),(56,76),(57,75),(58,74),(59,73),(60,84),(61,101),(62,100),(63,99),(64,98),(65,97),(66,108),(67,107),(68,106),(69,105),(70,104),(71,103),(72,102),(85,209),(86,208),(87,207),(88,206),(89,205),(90,216),(91,215),(92,214),(93,213),(94,212),(95,211),(96,210),(109,225),(110,224),(111,223),(112,222),(113,221),(114,220),(115,219),(116,218),(117,217),(118,228),(119,227),(120,226),(121,165),(122,164),(123,163),(124,162),(125,161),(126,160),(127,159),(128,158),(129,157),(130,168),(131,167),(132,166),(133,203),(134,202),(135,201),(136,200),(137,199),(138,198),(139,197),(140,196),(141,195),(142,194),(143,193),(144,204),(145,172),(146,171),(147,170),(148,169),(149,180),(150,179),(151,178),(152,177),(153,176),(154,175),(155,174),(156,173)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 27 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 38 | 23 |
| 0 | 0 | 38 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,38,38,0,0,23,15],[1,0,0,0,0,60,0,0,0,0,1,0,0,0,1,60] >;
180 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | ··· | 6G | 10A | ··· | 10AB | 10AC | ··· | 10BH | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | D4 | D6 | D6 | D12 | C5×S3 | C5×D4 | S3×C10 | S3×C10 | C5×D12 |
| kernel | C2×C10×D12 | C10×D12 | C22×C60 | S3×C22×C10 | C22×D12 | C2×D12 | C22×C12 | S3×C23 | C22×C20 | C2×C30 | C2×C20 | C22×C10 | C2×C10 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 2 | 4 | 48 | 4 | 8 | 1 | 4 | 6 | 1 | 8 | 4 | 16 | 24 | 4 | 32 |
In GAP, Magma, Sage, TeX
C_2\times C_{10}\times D_{12} % in TeX
G:=Group("C2xC10xD12"); // GroupNames label
G:=SmallGroup(480,1152);
// by ID
G=gap.SmallGroup(480,1152);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,2467,304,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀