metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.3D4, D20.5D6, C40.29D6, Dic12⋊6D5, C24.26D10, Dic15.3D4, Dic10.3D6, Dic6.4D10, C120.49C22, C60.146C23, C8.5(S3×D5), C40⋊C2⋊4S3, D15⋊Q8⋊9C2, C6.35(D4×D5), C40⋊S3⋊8C2, C30.28(C2×D4), C10.35(S3×D4), C5⋊2(D4.D6), (C5×Dic12)⋊8C2, C15⋊Q16⋊11C2, C3⋊2(Q16⋊D5), C15⋊5(C8.C22), D20⋊S3.2C2, C30.D4⋊12C2, C20.81(C22×S3), C12.81(C22×D5), C2.13(C20⋊D6), C15⋊3C8.23C22, (C4×D15).32C22, (C3×D20).29C22, (C5×Dic6).30C22, (C3×Dic10).29C22, C4.119(C2×S3×D5), (C3×C40⋊C2)⋊8C2, SmallGroup(480,354)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.3D4
G = < a,b,c,d | a20=b2=1, c6=a15, d2=a10, bab=a-1, ac=ca, dad-1=a11, cbc-1=dbd-1=a15b, dcd-1=a5c5 >
Subgroups: 684 in 120 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C3⋊C8, C24, Dic6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C3×D5, D15, C30, C8.C22, C5⋊2C8, C40, Dic10, Dic10, C4×D5, D20, D20, C5×Q8, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D4⋊2S3, S3×Q8, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, D30, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q8⋊2D5, D4.D6, C15⋊3C8, C120, D5×Dic3, D30.C2, C3⋊D20, C15⋊Q8, C3×Dic10, C3×D20, C5×Dic6, C4×D15, Q16⋊D5, C30.D4, C15⋊Q16, C3×C40⋊C2, C5×Dic12, C40⋊S3, D20⋊S3, D15⋊Q8, D30.3D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8.C22, C22×D5, S3×D4, S3×D5, D4×D5, D4.D6, C2×S3×D5, Q16⋊D5, C20⋊D6, D30.3D4
►On 240 points
Generators in S240
(1 217 38 148 122 19 235 32 166 140 13 229 26 160 134 7 223 44 154 128)(2 218 39 149 123 20 236 33 167 141 14 230 27 161 135 8 224 45 155 129)(3 219 40 150 124 21 237 34 168 142 15 231 28 162 136 9 225 46 156 130)(4 220 41 151 125 22 238 35 145 143 16 232 29 163 137 10 226 47 157 131)(5 221 42 152 126 23 239 36 146 144 17 233 30 164 138 11 227 48 158 132)(6 222 43 153 127 24 240 37 147 121 18 234 31 165 139 12 228 25 159 133)(49 215 73 174 120 67 209 91 192 114 61 203 85 186 108 55 197 79 180 102)(50 216 74 175 97 68 210 92 169 115 62 204 86 187 109 56 198 80 181 103)(51 193 75 176 98 69 211 93 170 116 63 205 87 188 110 57 199 81 182 104)(52 194 76 177 99 70 212 94 171 117 64 206 88 189 111 58 200 82 183 105)(53 195 77 178 100 71 213 95 172 118 65 207 89 190 112 59 201 83 184 106)(54 196 78 179 101 72 214 96 173 119 66 208 90 191 113 60 202 84 185 107)
(1 76)(2 95)(3 90)(4 85)(5 80)(6 75)(7 94)(8 89)(9 84)(10 79)(11 74)(12 93)(13 88)(14 83)(15 78)(16 73)(17 92)(18 87)(19 82)(20 77)(21 96)(22 91)(23 86)(24 81)(25 69)(26 64)(27 59)(28 54)(29 49)(30 68)(31 63)(32 58)(33 53)(34 72)(35 67)(36 62)(37 57)(38 52)(39 71)(40 66)(41 61)(42 56)(43 51)(44 70)(45 65)(46 60)(47 55)(48 50)(97 164)(98 159)(99 154)(100 149)(101 168)(102 163)(103 158)(104 153)(105 148)(106 167)(107 162)(108 157)(109 152)(110 147)(111 166)(112 161)(113 156)(114 151)(115 146)(116 165)(117 160)(118 155)(119 150)(120 145)(121 188)(122 183)(123 178)(124 173)(125 192)(126 187)(127 182)(128 177)(129 172)(130 191)(131 186)(132 181)(133 176)(134 171)(135 190)(136 185)(137 180)(138 175)(139 170)(140 189)(141 184)(142 179)(143 174)(144 169)(193 222)(194 217)(195 236)(196 231)(197 226)(198 221)(199 240)(200 235)(201 230)(202 225)(203 220)(204 239)(205 234)(206 229)(207 224)(208 219)(209 238)(210 233)(211 228)(212 223)(213 218)(214 237)(215 232)(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)
(1 188 13 176)(2 187 14 175)(3 186 15 174)(4 185 16 173)(5 184 17 172)(6 183 18 171)(7 182 19 170)(8 181 20 169)(9 180 21 192)(10 179 22 191)(11 178 23 190)(12 177 24 189)(25 70 37 58)(26 69 38 57)(27 68 39 56)(28 67 40 55)(29 66 41 54)(30 65 42 53)(31 64 43 52)(32 63 44 51)(33 62 45 50)(34 61 46 49)(35 60 47 72)(36 59 48 71)(73 142 85 130)(74 141 86 129)(75 140 87 128)(76 139 88 127)(77 138 89 126)(78 137 90 125)(79 136 91 124)(80 135 92 123)(81 134 93 122)(82 133 94 121)(83 132 95 144)(84 131 96 143)(97 230 109 218)(98 229 110 217)(99 228 111 240)(100 227 112 239)(101 226 113 238)(102 225 114 237)(103 224 115 236)(104 223 116 235)(105 222 117 234)(106 221 118 233)(107 220 119 232)(108 219 120 231)(145 214 157 202)(146 213 158 201)(147 212 159 200)(148 211 160 199)(149 210 161 198)(150 209 162 197)(151 208 163 196)(152 207 164 195)(153 206 165 194)(154 205 166 193)(155 204 167 216)(156 203 168 215)
G:=sub<Sym(240)| (1,217,38,148,122,19,235,32,166,140,13,229,26,160,134,7,223,44,154,128)(2,218,39,149,123,20,236,33,167,141,14,230,27,161,135,8,224,45,155,129)(3,219,40,150,124,21,237,34,168,142,15,231,28,162,136,9,225,46,156,130)(4,220,41,151,125,22,238,35,145,143,16,232,29,163,137,10,226,47,157,131)(5,221,42,152,126,23,239,36,146,144,17,233,30,164,138,11,227,48,158,132)(6,222,43,153,127,24,240,37,147,121,18,234,31,165,139,12,228,25,159,133)(49,215,73,174,120,67,209,91,192,114,61,203,85,186,108,55,197,79,180,102)(50,216,74,175,97,68,210,92,169,115,62,204,86,187,109,56,198,80,181,103)(51,193,75,176,98,69,211,93,170,116,63,205,87,188,110,57,199,81,182,104)(52,194,76,177,99,70,212,94,171,117,64,206,88,189,111,58,200,82,183,105)(53,195,77,178,100,71,213,95,172,118,65,207,89,190,112,59,201,83,184,106)(54,196,78,179,101,72,214,96,173,119,66,208,90,191,113,60,202,84,185,107), (1,76)(2,95)(3,90)(4,85)(5,80)(6,75)(7,94)(8,89)(9,84)(10,79)(11,74)(12,93)(13,88)(14,83)(15,78)(16,73)(17,92)(18,87)(19,82)(20,77)(21,96)(22,91)(23,86)(24,81)(25,69)(26,64)(27,59)(28,54)(29,49)(30,68)(31,63)(32,58)(33,53)(34,72)(35,67)(36,62)(37,57)(38,52)(39,71)(40,66)(41,61)(42,56)(43,51)(44,70)(45,65)(46,60)(47,55)(48,50)(97,164)(98,159)(99,154)(100,149)(101,168)(102,163)(103,158)(104,153)(105,148)(106,167)(107,162)(108,157)(109,152)(110,147)(111,166)(112,161)(113,156)(114,151)(115,146)(116,165)(117,160)(118,155)(119,150)(120,145)(121,188)(122,183)(123,178)(124,173)(125,192)(126,187)(127,182)(128,177)(129,172)(130,191)(131,186)(132,181)(133,176)(134,171)(135,190)(136,185)(137,180)(138,175)(139,170)(140,189)(141,184)(142,179)(143,174)(144,169)(193,222)(194,217)(195,236)(196,231)(197,226)(198,221)(199,240)(200,235)(201,230)(202,225)(203,220)(204,239)(205,234)(206,229)(207,224)(208,219)(209,238)(210,233)(211,228)(212,223)(213,218)(214,237)(215,232)(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), (1,188,13,176)(2,187,14,175)(3,186,15,174)(4,185,16,173)(5,184,17,172)(6,183,18,171)(7,182,19,170)(8,181,20,169)(9,180,21,192)(10,179,22,191)(11,178,23,190)(12,177,24,189)(25,70,37,58)(26,69,38,57)(27,68,39,56)(28,67,40,55)(29,66,41,54)(30,65,42,53)(31,64,43,52)(32,63,44,51)(33,62,45,50)(34,61,46,49)(35,60,47,72)(36,59,48,71)(73,142,85,130)(74,141,86,129)(75,140,87,128)(76,139,88,127)(77,138,89,126)(78,137,90,125)(79,136,91,124)(80,135,92,123)(81,134,93,122)(82,133,94,121)(83,132,95,144)(84,131,96,143)(97,230,109,218)(98,229,110,217)(99,228,111,240)(100,227,112,239)(101,226,113,238)(102,225,114,237)(103,224,115,236)(104,223,116,235)(105,222,117,234)(106,221,118,233)(107,220,119,232)(108,219,120,231)(145,214,157,202)(146,213,158,201)(147,212,159,200)(148,211,160,199)(149,210,161,198)(150,209,162,197)(151,208,163,196)(152,207,164,195)(153,206,165,194)(154,205,166,193)(155,204,167,216)(156,203,168,215)>;
G:=Group( (1,217,38,148,122,19,235,32,166,140,13,229,26,160,134,7,223,44,154,128)(2,218,39,149,123,20,236,33,167,141,14,230,27,161,135,8,224,45,155,129)(3,219,40,150,124,21,237,34,168,142,15,231,28,162,136,9,225,46,156,130)(4,220,41,151,125,22,238,35,145,143,16,232,29,163,137,10,226,47,157,131)(5,221,42,152,126,23,239,36,146,144,17,233,30,164,138,11,227,48,158,132)(6,222,43,153,127,24,240,37,147,121,18,234,31,165,139,12,228,25,159,133)(49,215,73,174,120,67,209,91,192,114,61,203,85,186,108,55,197,79,180,102)(50,216,74,175,97,68,210,92,169,115,62,204,86,187,109,56,198,80,181,103)(51,193,75,176,98,69,211,93,170,116,63,205,87,188,110,57,199,81,182,104)(52,194,76,177,99,70,212,94,171,117,64,206,88,189,111,58,200,82,183,105)(53,195,77,178,100,71,213,95,172,118,65,207,89,190,112,59,201,83,184,106)(54,196,78,179,101,72,214,96,173,119,66,208,90,191,113,60,202,84,185,107), (1,76)(2,95)(3,90)(4,85)(5,80)(6,75)(7,94)(8,89)(9,84)(10,79)(11,74)(12,93)(13,88)(14,83)(15,78)(16,73)(17,92)(18,87)(19,82)(20,77)(21,96)(22,91)(23,86)(24,81)(25,69)(26,64)(27,59)(28,54)(29,49)(30,68)(31,63)(32,58)(33,53)(34,72)(35,67)(36,62)(37,57)(38,52)(39,71)(40,66)(41,61)(42,56)(43,51)(44,70)(45,65)(46,60)(47,55)(48,50)(97,164)(98,159)(99,154)(100,149)(101,168)(102,163)(103,158)(104,153)(105,148)(106,167)(107,162)(108,157)(109,152)(110,147)(111,166)(112,161)(113,156)(114,151)(115,146)(116,165)(117,160)(118,155)(119,150)(120,145)(121,188)(122,183)(123,178)(124,173)(125,192)(126,187)(127,182)(128,177)(129,172)(130,191)(131,186)(132,181)(133,176)(134,171)(135,190)(136,185)(137,180)(138,175)(139,170)(140,189)(141,184)(142,179)(143,174)(144,169)(193,222)(194,217)(195,236)(196,231)(197,226)(198,221)(199,240)(200,235)(201,230)(202,225)(203,220)(204,239)(205,234)(206,229)(207,224)(208,219)(209,238)(210,233)(211,228)(212,223)(213,218)(214,237)(215,232)(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), (1,188,13,176)(2,187,14,175)(3,186,15,174)(4,185,16,173)(5,184,17,172)(6,183,18,171)(7,182,19,170)(8,181,20,169)(9,180,21,192)(10,179,22,191)(11,178,23,190)(12,177,24,189)(25,70,37,58)(26,69,38,57)(27,68,39,56)(28,67,40,55)(29,66,41,54)(30,65,42,53)(31,64,43,52)(32,63,44,51)(33,62,45,50)(34,61,46,49)(35,60,47,72)(36,59,48,71)(73,142,85,130)(74,141,86,129)(75,140,87,128)(76,139,88,127)(77,138,89,126)(78,137,90,125)(79,136,91,124)(80,135,92,123)(81,134,93,122)(82,133,94,121)(83,132,95,144)(84,131,96,143)(97,230,109,218)(98,229,110,217)(99,228,111,240)(100,227,112,239)(101,226,113,238)(102,225,114,237)(103,224,115,236)(104,223,116,235)(105,222,117,234)(106,221,118,233)(107,220,119,232)(108,219,120,231)(145,214,157,202)(146,213,158,201)(147,212,159,200)(148,211,160,199)(149,210,161,198)(150,209,162,197)(151,208,163,196)(152,207,164,195)(153,206,165,194)(154,205,166,193)(155,204,167,216)(156,203,168,215) );
G=PermutationGroup([[(1,217,38,148,122,19,235,32,166,140,13,229,26,160,134,7,223,44,154,128),(2,218,39,149,123,20,236,33,167,141,14,230,27,161,135,8,224,45,155,129),(3,219,40,150,124,21,237,34,168,142,15,231,28,162,136,9,225,46,156,130),(4,220,41,151,125,22,238,35,145,143,16,232,29,163,137,10,226,47,157,131),(5,221,42,152,126,23,239,36,146,144,17,233,30,164,138,11,227,48,158,132),(6,222,43,153,127,24,240,37,147,121,18,234,31,165,139,12,228,25,159,133),(49,215,73,174,120,67,209,91,192,114,61,203,85,186,108,55,197,79,180,102),(50,216,74,175,97,68,210,92,169,115,62,204,86,187,109,56,198,80,181,103),(51,193,75,176,98,69,211,93,170,116,63,205,87,188,110,57,199,81,182,104),(52,194,76,177,99,70,212,94,171,117,64,206,88,189,111,58,200,82,183,105),(53,195,77,178,100,71,213,95,172,118,65,207,89,190,112,59,201,83,184,106),(54,196,78,179,101,72,214,96,173,119,66,208,90,191,113,60,202,84,185,107)], [(1,76),(2,95),(3,90),(4,85),(5,80),(6,75),(7,94),(8,89),(9,84),(10,79),(11,74),(12,93),(13,88),(14,83),(15,78),(16,73),(17,92),(18,87),(19,82),(20,77),(21,96),(22,91),(23,86),(24,81),(25,69),(26,64),(27,59),(28,54),(29,49),(30,68),(31,63),(32,58),(33,53),(34,72),(35,67),(36,62),(37,57),(38,52),(39,71),(40,66),(41,61),(42,56),(43,51),(44,70),(45,65),(46,60),(47,55),(48,50),(97,164),(98,159),(99,154),(100,149),(101,168),(102,163),(103,158),(104,153),(105,148),(106,167),(107,162),(108,157),(109,152),(110,147),(111,166),(112,161),(113,156),(114,151),(115,146),(116,165),(117,160),(118,155),(119,150),(120,145),(121,188),(122,183),(123,178),(124,173),(125,192),(126,187),(127,182),(128,177),(129,172),(130,191),(131,186),(132,181),(133,176),(134,171),(135,190),(136,185),(137,180),(138,175),(139,170),(140,189),(141,184),(142,179),(143,174),(144,169),(193,222),(194,217),(195,236),(196,231),(197,226),(198,221),(199,240),(200,235),(201,230),(202,225),(203,220),(204,239),(205,234),(206,229),(207,224),(208,219),(209,238),(210,233),(211,228),(212,223),(213,218),(214,237),(215,232),(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)], [(1,188,13,176),(2,187,14,175),(3,186,15,174),(4,185,16,173),(5,184,17,172),(6,183,18,171),(7,182,19,170),(8,181,20,169),(9,180,21,192),(10,179,22,191),(11,178,23,190),(12,177,24,189),(25,70,37,58),(26,69,38,57),(27,68,39,56),(28,67,40,55),(29,66,41,54),(30,65,42,53),(31,64,43,52),(32,63,44,51),(33,62,45,50),(34,61,46,49),(35,60,47,72),(36,59,48,71),(73,142,85,130),(74,141,86,129),(75,140,87,128),(76,139,88,127),(77,138,89,126),(78,137,90,125),(79,136,91,124),(80,135,92,123),(81,134,93,122),(82,133,94,121),(83,132,95,144),(84,131,96,143),(97,230,109,218),(98,229,110,217),(99,228,111,240),(100,227,112,239),(101,226,113,238),(102,225,114,237),(103,224,115,236),(104,223,116,235),(105,222,117,234),(106,221,118,233),(107,220,119,232),(108,219,120,231),(145,214,157,202),(146,213,158,201),(147,212,159,200),(148,211,160,199),(149,210,161,198),(150,209,162,197),(151,208,163,196),(152,207,164,195),(153,206,165,194),(154,205,166,193),(155,204,167,216),(156,203,168,215)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 20 | 30 | 2 | 2 | 12 | 12 | 20 | 30 | 2 | 2 | 2 | 40 | 4 | 60 | 2 | 2 | 4 | 40 | 4 | 4 | 4 | 4 | 24 | 24 | 24 | 24 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | C8.C22 | S3×D4 | S3×D5 | D4×D5 | D4.D6 | C2×S3×D5 | Q16⋊D5 | C20⋊D6 | D30.3D4 |
| kernel | D30.3D4 | C30.D4 | C15⋊Q16 | C3×C40⋊C2 | C5×Dic12 | C40⋊S3 | D20⋊S3 | D15⋊Q8 | C40⋊C2 | Dic15 | D30 | Dic12 | C40 | Dic10 | D20 | C24 | Dic6 | C15 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of D30.3D4 ►in GL6(𝔽241)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 240 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 144 | 47 | 133 | 0 |
| 0 | 0 | 194 | 97 | 0 | 133 |
| 0 | 0 | 118 | 0 | 97 | 194 |
| 0 | 0 | 0 | 118 | 47 | 144 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 130 | 104 | 208 |
| 0 | 0 | 111 | 110 | 33 | 137 |
| 0 | 0 | 96 | 27 | 1 | 111 |
| 0 | 0 | 214 | 69 | 130 | 131 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 132 | 141 | 237 | 4 |
| 0 | 0 | 100 | 232 | 237 | 233 |
| 0 | 0 | 49 | 192 | 100 | 109 |
| 0 | 0 | 49 | 98 | 132 | 232 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 156 | 179 | 142 | 142 |
| 0 | 0 | 23 | 85 | 0 | 99 |
| 0 | 0 | 173 | 173 | 62 | 85 |
| 0 | 0 | 0 | 68 | 23 | 179 |
G:=sub<GL(6,GF(241))| [0,240,0,0,0,0,1,51,0,0,0,0,0,0,144,194,118,0,0,0,47,97,0,118,0,0,133,0,97,47,0,0,0,133,194,144],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,240,111,96,214,0,0,130,110,27,69,0,0,104,33,1,130,0,0,208,137,111,131],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,132,100,49,49,0,0,141,232,192,98,0,0,237,237,100,132,0,0,4,233,109,232],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,156,23,173,0,0,0,179,85,173,68,0,0,142,0,62,23,0,0,142,99,85,179] >;
D30.3D4 in GAP, Magma, Sage, TeX
D_{30}._3D_4 % in TeX
G:=Group("D30.3D4"); // GroupNames label
G:=SmallGroup(480,354);
// by ID
G=gap.SmallGroup(480,354);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,303,58,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^6=a^15,d^2=a^10,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,c*b*c^-1=d*b*d^-1=a^15*b,d*c*d^-1=a^5*c^5>;
// generators/relations