metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.5D30, C40.33D6, C8.11D30, Q8.7D30, SD16⋊3D15, D30.18D4, C24.33D10, C60.70C23, C120.28C22, Dic15.50D4, D60.22C22, Dic30.24C22, (C8×D15)⋊5C2, C24⋊D5⋊8C2, D4⋊D15⋊12C2, C15⋊31(C4○D8), (C5×SD16)⋊4S3, (C3×SD16)⋊4D5, (C5×D4).17D6, C2.21(D4×D15), C6.114(D4×D5), (C5×Q8).31D6, Q8⋊3D15⋊9C2, (C15×SD16)⋊4C2, (C3×D4).17D10, C15⋊7Q16⋊10C2, C30.321(C2×D4), C10.116(S3×D4), C5⋊5(Q8.7D6), (C3×Q8).14D10, D4⋊2D15⋊10C2, C4.7(C22×D15), C3⋊5(SD16⋊3D5), C20.108(C22×S3), C15⋊3C8.32C22, (C4×D15).45C22, (D4×C15).24C22, C12.108(C22×D5), (Q8×C15).23C22, SmallGroup(480,881)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.5D30
G = < a,b,c,d | a4=b2=d2=1, c30=a2, bab=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=a2c29 >
Subgroups: 804 in 124 conjugacy classes, 41 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, D8, SD16, SD16, Q16, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, D15, C30, C30, C4○D8, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D4⋊2S3, Q8⋊3S3, Dic15, Dic15, C60, C60, D30, D30, C2×C30, C8×D5, C40⋊C2, D4⋊D5, C5⋊Q16, C5×SD16, D4⋊2D5, Q8⋊2D5, Q8.7D6, C15⋊3C8, C120, Dic30, C4×D15, C4×D15, D60, D60, C2×Dic15, C15⋊7D4, D4×C15, Q8×C15, SD16⋊3D5, C8×D15, C24⋊D5, D4⋊D15, C15⋊7Q16, C15×SD16, D4⋊2D15, Q8⋊3D15, D4.5D30
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, D15, C4○D8, C22×D5, S3×D4, D30, D4×D5, Q8.7D6, C22×D15, SD16⋊3D5, D4×D15, D4.5D30
►On 240 points
Generators in S240
(1 92 31 62)(2 63 32 93)(3 94 33 64)(4 65 34 95)(5 96 35 66)(6 67 36 97)(7 98 37 68)(8 69 38 99)(9 100 39 70)(10 71 40 101)(11 102 41 72)(12 73 42 103)(13 104 43 74)(14 75 44 105)(15 106 45 76)(16 77 46 107)(17 108 47 78)(18 79 48 109)(19 110 49 80)(20 81 50 111)(21 112 51 82)(22 83 52 113)(23 114 53 84)(24 85 54 115)(25 116 55 86)(26 87 56 117)(27 118 57 88)(28 89 58 119)(29 120 59 90)(30 91 60 61)(121 221 151 191)(122 192 152 222)(123 223 153 193)(124 194 154 224)(125 225 155 195)(126 196 156 226)(127 227 157 197)(128 198 158 228)(129 229 159 199)(130 200 160 230)(131 231 161 201)(132 202 162 232)(133 233 163 203)(134 204 164 234)(135 235 165 205)(136 206 166 236)(137 237 167 207)(138 208 168 238)(139 239 169 209)(140 210 170 240)(141 181 171 211)(142 212 172 182)(143 183 173 213)(144 214 174 184)(145 185 175 215)(146 216 176 186)(147 187 177 217)(148 218 178 188)(149 189 179 219)(150 220 180 190)
(1 186)(2 147)(3 188)(4 149)(5 190)(6 151)(7 192)(8 153)(9 194)(10 155)(11 196)(12 157)(13 198)(14 159)(15 200)(16 161)(17 202)(18 163)(19 204)(20 165)(21 206)(22 167)(23 208)(24 169)(25 210)(26 171)(27 212)(28 173)(29 214)(30 175)(31 216)(32 177)(33 218)(34 179)(35 220)(36 121)(37 222)(38 123)(39 224)(40 125)(41 226)(42 127)(43 228)(44 129)(45 230)(46 131)(47 232)(48 133)(49 234)(50 135)(51 236)(52 137)(53 238)(54 139)(55 240)(56 141)(57 182)(58 143)(59 184)(60 145)(61 215)(62 146)(63 217)(64 148)(65 219)(66 150)(67 221)(68 152)(69 223)(70 154)(71 225)(72 156)(73 227)(74 158)(75 229)(76 160)(77 231)(78 162)(79 233)(80 164)(81 235)(82 166)(83 237)(84 168)(85 239)(86 170)(87 181)(88 172)(89 183)(90 174)(91 185)(92 176)(93 187)(94 178)(95 189)(96 180)(97 191)(98 122)(99 193)(100 124)(101 195)(102 126)(103 197)(104 128)(105 199)(106 130)(107 201)(108 132)(109 203)(110 134)(111 205)(112 136)(113 207)(114 138)(115 209)(116 140)(117 211)(118 142)(119 213)(120 144)
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 60)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(93 120)(94 119)(95 118)(96 117)(97 116)(98 115)(99 114)(100 113)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(121 170)(122 169)(123 168)(124 167)(125 166)(126 165)(127 164)(128 163)(129 162)(130 161)(131 160)(132 159)(133 158)(134 157)(135 156)(136 155)(137 154)(138 153)(139 152)(140 151)(141 150)(142 149)(143 148)(144 147)(145 146)(171 180)(172 179)(173 178)(174 177)(175 176)(181 190)(182 189)(183 188)(184 187)(185 186)(191 240)(192 239)(193 238)(194 237)(195 236)(196 235)(197 234)(198 233)(199 232)(200 231)(201 230)(202 229)(203 228)(204 227)(205 226)(206 225)(207 224)(208 223)(209 222)(210 221)(211 220)(212 219)(213 218)(214 217)(215 216)
G:=sub<Sym(240)| (1,92,31,62)(2,63,32,93)(3,94,33,64)(4,65,34,95)(5,96,35,66)(6,67,36,97)(7,98,37,68)(8,69,38,99)(9,100,39,70)(10,71,40,101)(11,102,41,72)(12,73,42,103)(13,104,43,74)(14,75,44,105)(15,106,45,76)(16,77,46,107)(17,108,47,78)(18,79,48,109)(19,110,49,80)(20,81,50,111)(21,112,51,82)(22,83,52,113)(23,114,53,84)(24,85,54,115)(25,116,55,86)(26,87,56,117)(27,118,57,88)(28,89,58,119)(29,120,59,90)(30,91,60,61)(121,221,151,191)(122,192,152,222)(123,223,153,193)(124,194,154,224)(125,225,155,195)(126,196,156,226)(127,227,157,197)(128,198,158,228)(129,229,159,199)(130,200,160,230)(131,231,161,201)(132,202,162,232)(133,233,163,203)(134,204,164,234)(135,235,165,205)(136,206,166,236)(137,237,167,207)(138,208,168,238)(139,239,169,209)(140,210,170,240)(141,181,171,211)(142,212,172,182)(143,183,173,213)(144,214,174,184)(145,185,175,215)(146,216,176,186)(147,187,177,217)(148,218,178,188)(149,189,179,219)(150,220,180,190), (1,186)(2,147)(3,188)(4,149)(5,190)(6,151)(7,192)(8,153)(9,194)(10,155)(11,196)(12,157)(13,198)(14,159)(15,200)(16,161)(17,202)(18,163)(19,204)(20,165)(21,206)(22,167)(23,208)(24,169)(25,210)(26,171)(27,212)(28,173)(29,214)(30,175)(31,216)(32,177)(33,218)(34,179)(35,220)(36,121)(37,222)(38,123)(39,224)(40,125)(41,226)(42,127)(43,228)(44,129)(45,230)(46,131)(47,232)(48,133)(49,234)(50,135)(51,236)(52,137)(53,238)(54,139)(55,240)(56,141)(57,182)(58,143)(59,184)(60,145)(61,215)(62,146)(63,217)(64,148)(65,219)(66,150)(67,221)(68,152)(69,223)(70,154)(71,225)(72,156)(73,227)(74,158)(75,229)(76,160)(77,231)(78,162)(79,233)(80,164)(81,235)(82,166)(83,237)(84,168)(85,239)(86,170)(87,181)(88,172)(89,183)(90,174)(91,185)(92,176)(93,187)(94,178)(95,189)(96,180)(97,191)(98,122)(99,193)(100,124)(101,195)(102,126)(103,197)(104,128)(105,199)(106,130)(107,201)(108,132)(109,203)(110,134)(111,205)(112,136)(113,207)(114,138)(115,209)(116,140)(117,211)(118,142)(119,213)(120,144), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(93,120)(94,119)(95,118)(96,117)(97,116)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(121,170)(122,169)(123,168)(124,167)(125,166)(126,165)(127,164)(128,163)(129,162)(130,161)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151)(141,150)(142,149)(143,148)(144,147)(145,146)(171,180)(172,179)(173,178)(174,177)(175,176)(181,190)(182,189)(183,188)(184,187)(185,186)(191,240)(192,239)(193,238)(194,237)(195,236)(196,235)(197,234)(198,233)(199,232)(200,231)(201,230)(202,229)(203,228)(204,227)(205,226)(206,225)(207,224)(208,223)(209,222)(210,221)(211,220)(212,219)(213,218)(214,217)(215,216)>;
G:=Group( (1,92,31,62)(2,63,32,93)(3,94,33,64)(4,65,34,95)(5,96,35,66)(6,67,36,97)(7,98,37,68)(8,69,38,99)(9,100,39,70)(10,71,40,101)(11,102,41,72)(12,73,42,103)(13,104,43,74)(14,75,44,105)(15,106,45,76)(16,77,46,107)(17,108,47,78)(18,79,48,109)(19,110,49,80)(20,81,50,111)(21,112,51,82)(22,83,52,113)(23,114,53,84)(24,85,54,115)(25,116,55,86)(26,87,56,117)(27,118,57,88)(28,89,58,119)(29,120,59,90)(30,91,60,61)(121,221,151,191)(122,192,152,222)(123,223,153,193)(124,194,154,224)(125,225,155,195)(126,196,156,226)(127,227,157,197)(128,198,158,228)(129,229,159,199)(130,200,160,230)(131,231,161,201)(132,202,162,232)(133,233,163,203)(134,204,164,234)(135,235,165,205)(136,206,166,236)(137,237,167,207)(138,208,168,238)(139,239,169,209)(140,210,170,240)(141,181,171,211)(142,212,172,182)(143,183,173,213)(144,214,174,184)(145,185,175,215)(146,216,176,186)(147,187,177,217)(148,218,178,188)(149,189,179,219)(150,220,180,190), (1,186)(2,147)(3,188)(4,149)(5,190)(6,151)(7,192)(8,153)(9,194)(10,155)(11,196)(12,157)(13,198)(14,159)(15,200)(16,161)(17,202)(18,163)(19,204)(20,165)(21,206)(22,167)(23,208)(24,169)(25,210)(26,171)(27,212)(28,173)(29,214)(30,175)(31,216)(32,177)(33,218)(34,179)(35,220)(36,121)(37,222)(38,123)(39,224)(40,125)(41,226)(42,127)(43,228)(44,129)(45,230)(46,131)(47,232)(48,133)(49,234)(50,135)(51,236)(52,137)(53,238)(54,139)(55,240)(56,141)(57,182)(58,143)(59,184)(60,145)(61,215)(62,146)(63,217)(64,148)(65,219)(66,150)(67,221)(68,152)(69,223)(70,154)(71,225)(72,156)(73,227)(74,158)(75,229)(76,160)(77,231)(78,162)(79,233)(80,164)(81,235)(82,166)(83,237)(84,168)(85,239)(86,170)(87,181)(88,172)(89,183)(90,174)(91,185)(92,176)(93,187)(94,178)(95,189)(96,180)(97,191)(98,122)(99,193)(100,124)(101,195)(102,126)(103,197)(104,128)(105,199)(106,130)(107,201)(108,132)(109,203)(110,134)(111,205)(112,136)(113,207)(114,138)(115,209)(116,140)(117,211)(118,142)(119,213)(120,144), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(93,120)(94,119)(95,118)(96,117)(97,116)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(121,170)(122,169)(123,168)(124,167)(125,166)(126,165)(127,164)(128,163)(129,162)(130,161)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151)(141,150)(142,149)(143,148)(144,147)(145,146)(171,180)(172,179)(173,178)(174,177)(175,176)(181,190)(182,189)(183,188)(184,187)(185,186)(191,240)(192,239)(193,238)(194,237)(195,236)(196,235)(197,234)(198,233)(199,232)(200,231)(201,230)(202,229)(203,228)(204,227)(205,226)(206,225)(207,224)(208,223)(209,222)(210,221)(211,220)(212,219)(213,218)(214,217)(215,216) );
G=PermutationGroup([[(1,92,31,62),(2,63,32,93),(3,94,33,64),(4,65,34,95),(5,96,35,66),(6,67,36,97),(7,98,37,68),(8,69,38,99),(9,100,39,70),(10,71,40,101),(11,102,41,72),(12,73,42,103),(13,104,43,74),(14,75,44,105),(15,106,45,76),(16,77,46,107),(17,108,47,78),(18,79,48,109),(19,110,49,80),(20,81,50,111),(21,112,51,82),(22,83,52,113),(23,114,53,84),(24,85,54,115),(25,116,55,86),(26,87,56,117),(27,118,57,88),(28,89,58,119),(29,120,59,90),(30,91,60,61),(121,221,151,191),(122,192,152,222),(123,223,153,193),(124,194,154,224),(125,225,155,195),(126,196,156,226),(127,227,157,197),(128,198,158,228),(129,229,159,199),(130,200,160,230),(131,231,161,201),(132,202,162,232),(133,233,163,203),(134,204,164,234),(135,235,165,205),(136,206,166,236),(137,237,167,207),(138,208,168,238),(139,239,169,209),(140,210,170,240),(141,181,171,211),(142,212,172,182),(143,183,173,213),(144,214,174,184),(145,185,175,215),(146,216,176,186),(147,187,177,217),(148,218,178,188),(149,189,179,219),(150,220,180,190)], [(1,186),(2,147),(3,188),(4,149),(5,190),(6,151),(7,192),(8,153),(9,194),(10,155),(11,196),(12,157),(13,198),(14,159),(15,200),(16,161),(17,202),(18,163),(19,204),(20,165),(21,206),(22,167),(23,208),(24,169),(25,210),(26,171),(27,212),(28,173),(29,214),(30,175),(31,216),(32,177),(33,218),(34,179),(35,220),(36,121),(37,222),(38,123),(39,224),(40,125),(41,226),(42,127),(43,228),(44,129),(45,230),(46,131),(47,232),(48,133),(49,234),(50,135),(51,236),(52,137),(53,238),(54,139),(55,240),(56,141),(57,182),(58,143),(59,184),(60,145),(61,215),(62,146),(63,217),(64,148),(65,219),(66,150),(67,221),(68,152),(69,223),(70,154),(71,225),(72,156),(73,227),(74,158),(75,229),(76,160),(77,231),(78,162),(79,233),(80,164),(81,235),(82,166),(83,237),(84,168),(85,239),(86,170),(87,181),(88,172),(89,183),(90,174),(91,185),(92,176),(93,187),(94,178),(95,189),(96,180),(97,191),(98,122),(99,193),(100,124),(101,195),(102,126),(103,197),(104,128),(105,199),(106,130),(107,201),(108,132),(109,203),(110,134),(111,205),(112,136),(113,207),(114,138),(115,209),(116,140),(117,211),(118,142),(119,213),(120,144)], [(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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,60),(32,59),(33,58),(34,57),(35,56),(36,55),(37,54),(38,53),(39,52),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(61,92),(62,91),(63,90),(64,89),(65,88),(66,87),(67,86),(68,85),(69,84),(70,83),(71,82),(72,81),(73,80),(74,79),(75,78),(76,77),(93,120),(94,119),(95,118),(96,117),(97,116),(98,115),(99,114),(100,113),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107),(121,170),(122,169),(123,168),(124,167),(125,166),(126,165),(127,164),(128,163),(129,162),(130,161),(131,160),(132,159),(133,158),(134,157),(135,156),(136,155),(137,154),(138,153),(139,152),(140,151),(141,150),(142,149),(143,148),(144,147),(145,146),(171,180),(172,179),(173,178),(174,177),(175,176),(181,190),(182,189),(183,188),(184,187),(185,186),(191,240),(192,239),(193,238),(194,237),(195,236),(196,235),(197,234),(198,233),(199,232),(200,231),(201,230),(202,229),(203,228),(204,227),(205,226),(206,225),(207,224),(208,223),(209,222),(210,221),(211,220),(212,219),(213,218),(214,217),(215,216)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 60E | 60F | 60G | 60H | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 4 | 30 | 60 | 2 | 2 | 4 | 15 | 15 | 60 | 2 | 2 | 2 | 8 | 2 | 2 | 30 | 30 | 2 | 2 | 8 | 8 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | D15 | C4○D8 | D30 | D30 | D30 | S3×D4 | D4×D5 | Q8.7D6 | SD16⋊3D5 | D4×D15 | D4.5D30 |
| kernel | D4.5D30 | C8×D15 | C24⋊D5 | D4⋊D15 | C15⋊7Q16 | C15×SD16 | D4⋊2D15 | Q8⋊3D15 | C5×SD16 | Dic15 | D30 | C3×SD16 | C40 | C5×D4 | C5×Q8 | C24 | C3×D4 | C3×Q8 | SD16 | C15 | C8 | D4 | Q8 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 1 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of D4.5D30 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 58 | 177 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 233 | 134 |
| 0 | 0 | 156 | 8 |
| 177 | 93 | 0 | 0 |
| 132 | 68 | 0 | 0 |
| 0 | 0 | 240 | 77 |
| 0 | 0 | 97 | 1 |
| 211 | 131 | 0 | 0 |
| 98 | 30 | 0 | 0 |
| 0 | 0 | 1 | 164 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,64,58,0,0,0,177],[240,0,0,0,0,240,0,0,0,0,233,156,0,0,134,8],[177,132,0,0,93,68,0,0,0,0,240,97,0,0,77,1],[211,98,0,0,131,30,0,0,0,0,1,0,0,0,164,240] >;
D4.5D30 in GAP, Magma, Sage, TeX
D_4._5D_{30} % in TeX
G:=Group("D4.5D30"); // GroupNames label
G:=SmallGroup(480,881);
// by ID
G=gap.SmallGroup(480,881);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,422,135,100,346,185,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^30=a^2,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=a^2*c^29>;
// generators/relations