direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C2×C30, C24⋊8C30, C60⋊15C23, C30.99C24, C4⋊(C22×C30), (C2×C30)⋊9C23, C20⋊4(C22×C6), (C23×C30)⋊2C2, (C22×C4)⋊7C30, C23⋊4(C2×C30), (C23×C6)⋊4C10, (C23×C10)⋊10C6, C12⋊4(C22×C10), (C22×C20)⋊16C6, (C22×C60)⋊26C2, (C2×C60)⋊53C22, C2.1(C23×C30), (C22×C12)⋊12C10, C10.16(C23×C6), C6.16(C23×C10), C22⋊2(C22×C30), (C22×C30)⋊18C22, (C2×C4)⋊4(C2×C30), (C2×C20)⋊15(C2×C6), (C2×C12)⋊15(C2×C10), (C2×C6)⋊2(C22×C10), (C2×C10)⋊4(C22×C6), (C22×C10)⋊8(C2×C6), (C22×C6)⋊6(C2×C10), SmallGroup(480,1181)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 632 in 472 conjugacy classes, 312 normal (20 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×15], C22 [×24], C5, C6, C6 [×6], C6 [×8], C2×C4 [×6], D4 [×16], C23, C23 [×12], C23 [×8], C10, C10 [×6], C10 [×8], C12 [×4], C2×C6 [×15], C2×C6 [×24], C15, C22×C4, C2×D4 [×12], C24 [×2], C20 [×4], C2×C10 [×15], C2×C10 [×24], C2×C12 [×6], C3×D4 [×16], C22×C6, C22×C6 [×12], C22×C6 [×8], C30, C30 [×6], C30 [×8], C22×D4, C2×C20 [×6], C5×D4 [×16], C22×C10, C22×C10 [×12], C22×C10 [×8], C22×C12, C6×D4 [×12], C23×C6 [×2], C60 [×4], C2×C30 [×15], C2×C30 [×24], C22×C20, D4×C10 [×12], C23×C10 [×2], D4×C2×C6, C2×C60 [×6], D4×C15 [×16], C22×C30, C22×C30 [×12], C22×C30 [×8], D4×C2×C10, C22×C60, D4×C30 [×12], C23×C30 [×2], D4×C2×C30
Quotients:
C1, C2 [×15], C3, C22 [×35], C5, C6 [×15], D4 [×4], C23 [×15], C10 [×15], C2×C6 [×35], C15, C2×D4 [×6], C24, C2×C10 [×35], C3×D4 [×4], C22×C6 [×15], C30 [×15], C22×D4, C5×D4 [×4], C22×C10 [×15], C6×D4 [×6], C23×C6, C2×C30 [×35], D4×C10 [×6], C23×C10, D4×C2×C6, D4×C15 [×4], C22×C30 [×15], D4×C2×C10, D4×C30 [×6], C23×C30, D4×C2×C30
Generators and relations
G = < a,b,c,d | a2=b30=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 240 points
Generators in S240
(1 214)(2 215)(3 216)(4 217)(5 218)(6 219)(7 220)(8 221)(9 222)(10 223)(11 224)(12 225)(13 226)(14 227)(15 228)(16 229)(17 230)(18 231)(19 232)(20 233)(21 234)(22 235)(23 236)(24 237)(25 238)(26 239)(27 240)(28 211)(29 212)(30 213)(31 146)(32 147)(33 148)(34 149)(35 150)(36 121)(37 122)(38 123)(39 124)(40 125)(41 126)(42 127)(43 128)(44 129)(45 130)(46 131)(47 132)(48 133)(49 134)(50 135)(51 136)(52 137)(53 138)(54 139)(55 140)(56 141)(57 142)(58 143)(59 144)(60 145)(61 112)(62 113)(63 114)(64 115)(65 116)(66 117)(67 118)(68 119)(69 120)(70 91)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(151 198)(152 199)(153 200)(154 201)(155 202)(156 203)(157 204)(158 205)(159 206)(160 207)(161 208)(162 209)(163 210)(164 181)(165 182)(166 183)(167 184)(168 185)(169 186)(170 187)(171 188)(172 189)(173 190)(174 191)(175 192)(176 193)(177 194)(178 195)(179 196)(180 197)
(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)
(1 132 202 86)(2 133 203 87)(3 134 204 88)(4 135 205 89)(5 136 206 90)(6 137 207 61)(7 138 208 62)(8 139 209 63)(9 140 210 64)(10 141 181 65)(11 142 182 66)(12 143 183 67)(13 144 184 68)(14 145 185 69)(15 146 186 70)(16 147 187 71)(17 148 188 72)(18 149 189 73)(19 150 190 74)(20 121 191 75)(21 122 192 76)(22 123 193 77)(23 124 194 78)(24 125 195 79)(25 126 196 80)(26 127 197 81)(27 128 198 82)(28 129 199 83)(29 130 200 84)(30 131 201 85)(31 169 91 228)(32 170 92 229)(33 171 93 230)(34 172 94 231)(35 173 95 232)(36 174 96 233)(37 175 97 234)(38 176 98 235)(39 177 99 236)(40 178 100 237)(41 179 101 238)(42 180 102 239)(43 151 103 240)(44 152 104 211)(45 153 105 212)(46 154 106 213)(47 155 107 214)(48 156 108 215)(49 157 109 216)(50 158 110 217)(51 159 111 218)(52 160 112 219)(53 161 113 220)(54 162 114 221)(55 163 115 222)(56 164 116 223)(57 165 117 224)(58 166 118 225)(59 167 119 226)(60 168 120 227)
(1 170)(2 171)(3 172)(4 173)(5 174)(6 175)(7 176)(8 177)(9 178)(10 179)(11 180)(12 151)(13 152)(14 153)(15 154)(16 155)(17 156)(18 157)(19 158)(20 159)(21 160)(22 161)(23 162)(24 163)(25 164)(26 165)(27 166)(28 167)(29 168)(30 169)(31 131)(32 132)(33 133)(34 134)(35 135)(36 136)(37 137)(38 138)(39 139)(40 140)(41 141)(42 142)(43 143)(44 144)(45 145)(46 146)(47 147)(48 148)(49 149)(50 150)(51 121)(52 122)(53 123)(54 124)(55 125)(56 126)(57 127)(58 128)(59 129)(60 130)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 113)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(181 238)(182 239)(183 240)(184 211)(185 212)(186 213)(187 214)(188 215)(189 216)(190 217)(191 218)(192 219)(193 220)(194 221)(195 222)(196 223)(197 224)(198 225)(199 226)(200 227)(201 228)(202 229)(203 230)(204 231)(205 232)(206 233)(207 234)(208 235)(209 236)(210 237)
G:=sub<Sym(240)| (1,214)(2,215)(3,216)(4,217)(5,218)(6,219)(7,220)(8,221)(9,222)(10,223)(11,224)(12,225)(13,226)(14,227)(15,228)(16,229)(17,230)(18,231)(19,232)(20,233)(21,234)(22,235)(23,236)(24,237)(25,238)(26,239)(27,240)(28,211)(29,212)(30,213)(31,146)(32,147)(33,148)(34,149)(35,150)(36,121)(37,122)(38,123)(39,124)(40,125)(41,126)(42,127)(43,128)(44,129)(45,130)(46,131)(47,132)(48,133)(49,134)(50,135)(51,136)(52,137)(53,138)(54,139)(55,140)(56,141)(57,142)(58,143)(59,144)(60,145)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(151,198)(152,199)(153,200)(154,201)(155,202)(156,203)(157,204)(158,205)(159,206)(160,207)(161,208)(162,209)(163,210)(164,181)(165,182)(166,183)(167,184)(168,185)(169,186)(170,187)(171,188)(172,189)(173,190)(174,191)(175,192)(176,193)(177,194)(178,195)(179,196)(180,197), (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), (1,132,202,86)(2,133,203,87)(3,134,204,88)(4,135,205,89)(5,136,206,90)(6,137,207,61)(7,138,208,62)(8,139,209,63)(9,140,210,64)(10,141,181,65)(11,142,182,66)(12,143,183,67)(13,144,184,68)(14,145,185,69)(15,146,186,70)(16,147,187,71)(17,148,188,72)(18,149,189,73)(19,150,190,74)(20,121,191,75)(21,122,192,76)(22,123,193,77)(23,124,194,78)(24,125,195,79)(25,126,196,80)(26,127,197,81)(27,128,198,82)(28,129,199,83)(29,130,200,84)(30,131,201,85)(31,169,91,228)(32,170,92,229)(33,171,93,230)(34,172,94,231)(35,173,95,232)(36,174,96,233)(37,175,97,234)(38,176,98,235)(39,177,99,236)(40,178,100,237)(41,179,101,238)(42,180,102,239)(43,151,103,240)(44,152,104,211)(45,153,105,212)(46,154,106,213)(47,155,107,214)(48,156,108,215)(49,157,109,216)(50,158,110,217)(51,159,111,218)(52,160,112,219)(53,161,113,220)(54,162,114,221)(55,163,115,222)(56,164,116,223)(57,165,117,224)(58,166,118,225)(59,167,119,226)(60,168,120,227), (1,170)(2,171)(3,172)(4,173)(5,174)(6,175)(7,176)(8,177)(9,178)(10,179)(11,180)(12,151)(13,152)(14,153)(15,154)(16,155)(17,156)(18,157)(19,158)(20,159)(21,160)(22,161)(23,162)(24,163)(25,164)(26,165)(27,166)(28,167)(29,168)(30,169)(31,131)(32,132)(33,133)(34,134)(35,135)(36,136)(37,137)(38,138)(39,139)(40,140)(41,141)(42,142)(43,143)(44,144)(45,145)(46,146)(47,147)(48,148)(49,149)(50,150)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,127)(58,128)(59,129)(60,130)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(181,238)(182,239)(183,240)(184,211)(185,212)(186,213)(187,214)(188,215)(189,216)(190,217)(191,218)(192,219)(193,220)(194,221)(195,222)(196,223)(197,224)(198,225)(199,226)(200,227)(201,228)(202,229)(203,230)(204,231)(205,232)(206,233)(207,234)(208,235)(209,236)(210,237)>;
G:=Group( (1,214)(2,215)(3,216)(4,217)(5,218)(6,219)(7,220)(8,221)(9,222)(10,223)(11,224)(12,225)(13,226)(14,227)(15,228)(16,229)(17,230)(18,231)(19,232)(20,233)(21,234)(22,235)(23,236)(24,237)(25,238)(26,239)(27,240)(28,211)(29,212)(30,213)(31,146)(32,147)(33,148)(34,149)(35,150)(36,121)(37,122)(38,123)(39,124)(40,125)(41,126)(42,127)(43,128)(44,129)(45,130)(46,131)(47,132)(48,133)(49,134)(50,135)(51,136)(52,137)(53,138)(54,139)(55,140)(56,141)(57,142)(58,143)(59,144)(60,145)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(151,198)(152,199)(153,200)(154,201)(155,202)(156,203)(157,204)(158,205)(159,206)(160,207)(161,208)(162,209)(163,210)(164,181)(165,182)(166,183)(167,184)(168,185)(169,186)(170,187)(171,188)(172,189)(173,190)(174,191)(175,192)(176,193)(177,194)(178,195)(179,196)(180,197), (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), (1,132,202,86)(2,133,203,87)(3,134,204,88)(4,135,205,89)(5,136,206,90)(6,137,207,61)(7,138,208,62)(8,139,209,63)(9,140,210,64)(10,141,181,65)(11,142,182,66)(12,143,183,67)(13,144,184,68)(14,145,185,69)(15,146,186,70)(16,147,187,71)(17,148,188,72)(18,149,189,73)(19,150,190,74)(20,121,191,75)(21,122,192,76)(22,123,193,77)(23,124,194,78)(24,125,195,79)(25,126,196,80)(26,127,197,81)(27,128,198,82)(28,129,199,83)(29,130,200,84)(30,131,201,85)(31,169,91,228)(32,170,92,229)(33,171,93,230)(34,172,94,231)(35,173,95,232)(36,174,96,233)(37,175,97,234)(38,176,98,235)(39,177,99,236)(40,178,100,237)(41,179,101,238)(42,180,102,239)(43,151,103,240)(44,152,104,211)(45,153,105,212)(46,154,106,213)(47,155,107,214)(48,156,108,215)(49,157,109,216)(50,158,110,217)(51,159,111,218)(52,160,112,219)(53,161,113,220)(54,162,114,221)(55,163,115,222)(56,164,116,223)(57,165,117,224)(58,166,118,225)(59,167,119,226)(60,168,120,227), (1,170)(2,171)(3,172)(4,173)(5,174)(6,175)(7,176)(8,177)(9,178)(10,179)(11,180)(12,151)(13,152)(14,153)(15,154)(16,155)(17,156)(18,157)(19,158)(20,159)(21,160)(22,161)(23,162)(24,163)(25,164)(26,165)(27,166)(28,167)(29,168)(30,169)(31,131)(32,132)(33,133)(34,134)(35,135)(36,136)(37,137)(38,138)(39,139)(40,140)(41,141)(42,142)(43,143)(44,144)(45,145)(46,146)(47,147)(48,148)(49,149)(50,150)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,127)(58,128)(59,129)(60,130)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(181,238)(182,239)(183,240)(184,211)(185,212)(186,213)(187,214)(188,215)(189,216)(190,217)(191,218)(192,219)(193,220)(194,221)(195,222)(196,223)(197,224)(198,225)(199,226)(200,227)(201,228)(202,229)(203,230)(204,231)(205,232)(206,233)(207,234)(208,235)(209,236)(210,237) );
G=PermutationGroup([(1,214),(2,215),(3,216),(4,217),(5,218),(6,219),(7,220),(8,221),(9,222),(10,223),(11,224),(12,225),(13,226),(14,227),(15,228),(16,229),(17,230),(18,231),(19,232),(20,233),(21,234),(22,235),(23,236),(24,237),(25,238),(26,239),(27,240),(28,211),(29,212),(30,213),(31,146),(32,147),(33,148),(34,149),(35,150),(36,121),(37,122),(38,123),(39,124),(40,125),(41,126),(42,127),(43,128),(44,129),(45,130),(46,131),(47,132),(48,133),(49,134),(50,135),(51,136),(52,137),(53,138),(54,139),(55,140),(56,141),(57,142),(58,143),(59,144),(60,145),(61,112),(62,113),(63,114),(64,115),(65,116),(66,117),(67,118),(68,119),(69,120),(70,91),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105),(85,106),(86,107),(87,108),(88,109),(89,110),(90,111),(151,198),(152,199),(153,200),(154,201),(155,202),(156,203),(157,204),(158,205),(159,206),(160,207),(161,208),(162,209),(163,210),(164,181),(165,182),(166,183),(167,184),(168,185),(169,186),(170,187),(171,188),(172,189),(173,190),(174,191),(175,192),(176,193),(177,194),(178,195),(179,196),(180,197)], [(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)], [(1,132,202,86),(2,133,203,87),(3,134,204,88),(4,135,205,89),(5,136,206,90),(6,137,207,61),(7,138,208,62),(8,139,209,63),(9,140,210,64),(10,141,181,65),(11,142,182,66),(12,143,183,67),(13,144,184,68),(14,145,185,69),(15,146,186,70),(16,147,187,71),(17,148,188,72),(18,149,189,73),(19,150,190,74),(20,121,191,75),(21,122,192,76),(22,123,193,77),(23,124,194,78),(24,125,195,79),(25,126,196,80),(26,127,197,81),(27,128,198,82),(28,129,199,83),(29,130,200,84),(30,131,201,85),(31,169,91,228),(32,170,92,229),(33,171,93,230),(34,172,94,231),(35,173,95,232),(36,174,96,233),(37,175,97,234),(38,176,98,235),(39,177,99,236),(40,178,100,237),(41,179,101,238),(42,180,102,239),(43,151,103,240),(44,152,104,211),(45,153,105,212),(46,154,106,213),(47,155,107,214),(48,156,108,215),(49,157,109,216),(50,158,110,217),(51,159,111,218),(52,160,112,219),(53,161,113,220),(54,162,114,221),(55,163,115,222),(56,164,116,223),(57,165,117,224),(58,166,118,225),(59,167,119,226),(60,168,120,227)], [(1,170),(2,171),(3,172),(4,173),(5,174),(6,175),(7,176),(8,177),(9,178),(10,179),(11,180),(12,151),(13,152),(14,153),(15,154),(16,155),(17,156),(18,157),(19,158),(20,159),(21,160),(22,161),(23,162),(24,163),(25,164),(26,165),(27,166),(28,167),(29,168),(30,169),(31,131),(32,132),(33,133),(34,134),(35,135),(36,136),(37,137),(38,138),(39,139),(40,140),(41,141),(42,142),(43,143),(44,144),(45,145),(46,146),(47,147),(48,148),(49,149),(50,150),(51,121),(52,122),(53,123),(54,124),(55,125),(56,126),(57,127),(58,128),(59,129),(60,130),(61,97),(62,98),(63,99),(64,100),(65,101),(66,102),(67,103),(68,104),(69,105),(70,106),(71,107),(72,108),(73,109),(74,110),(75,111),(76,112),(77,113),(78,114),(79,115),(80,116),(81,117),(82,118),(83,119),(84,120),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96),(181,238),(182,239),(183,240),(184,211),(185,212),(186,213),(187,214),(188,215),(189,216),(190,217),(191,218),(192,219),(193,220),(194,221),(195,222),(196,223),(197,224),(198,225),(199,226),(200,227),(201,228),(202,229),(203,230),(204,231),(205,232),(206,233),(207,234),(208,235),(209,236),(210,237)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 |
| 0 | 58 | 0 | 0 |
| 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 47 |
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 51 | 2 |
| 0 | 0 | 41 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 51 | 1 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[3,0,0,0,0,58,0,0,0,0,47,0,0,0,0,47],[60,0,0,0,0,1,0,0,0,0,51,41,0,0,2,10],[1,0,0,0,0,1,0,0,0,0,60,51,0,0,0,1] >;
300 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | ··· | 6N | 6O | ··· | 6AD | 10A | ··· | 10AB | 10AC | ··· | 10BH | 12A | ··· | 12H | 15A | ··· | 15H | 20A | ··· | 20P | 30A | ··· | 30BD | 30BE | ··· | 30DP | 60A | ··· | 60AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
300 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C5 | C6 | C6 | C6 | C10 | C10 | C10 | C15 | C30 | C30 | C30 | D4 | C3×D4 | C5×D4 | D4×C15 |
| kernel | D4×C2×C30 | C22×C60 | D4×C30 | C23×C30 | D4×C2×C10 | D4×C2×C6 | C22×C20 | D4×C10 | C23×C10 | C22×C12 | C6×D4 | C23×C6 | C22×D4 | C22×C4 | C2×D4 | C24 | C2×C30 | C2×C10 | C2×C6 | C22 |
| # reps | 1 | 1 | 12 | 2 | 2 | 4 | 2 | 24 | 4 | 4 | 48 | 8 | 8 | 8 | 96 | 16 | 4 | 8 | 16 | 32 |
In GAP, Magma, Sage, TeX
D_4\times C_2\times C_{30} % in TeX
G:=Group("D4xC2xC30"); // GroupNames label
G:=SmallGroup(480,1181);
// by ID
G=gap.SmallGroup(480,1181);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-5,-2,3389]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^30=c^4=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
○
≀