metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.8D30, C60.214D4, Q8.13D30, C60.81C23, D60.41C22, Dic30.42C22, C4○D4⋊4D15, (C2×C30).9D4, D4⋊D15⋊15C2, C15⋊37(C4○D8), (C5×D4).32D6, (C2×C4).59D30, (C5×Q8).51D6, D4.D15⋊15C2, (C3×D4).32D10, C15⋊7Q16⋊15C2, C30.392(C2×D4), (C2×C20).159D6, C5⋊7(Q8.13D6), (C3×Q8).34D10, Q8⋊2D15⋊15C2, (C2×C12).157D10, C3⋊7(D4.8D10), C4.32(C15⋊7D4), (C2×C60).84C22, D60⋊11C2⋊14C2, C4.18(C22×D15), C12.111(C5⋊D4), C20.111(C3⋊D4), C20.119(C22×S3), C15⋊3C8.36C22, (D4×C15).37C22, C12.119(C22×D5), C22.1(C15⋊7D4), (Q8×C15).39C22, (C3×C4○D4)⋊2D5, (C5×C4○D4)⋊6S3, (C2×C15⋊3C8)⋊8C2, (C15×C4○D4)⋊2C2, C6.119(C2×C5⋊D4), C2.24(C2×C15⋊7D4), (C2×C6).21(C5⋊D4), C10.119(C2×C3⋊D4), (C2×C10).20(C3⋊D4), SmallGroup(480,915)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.8D30
G = < a,b,c,d | a4=b2=1, c30=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c29 >
Subgroups: 644 in 124 conjugacy classes, 47 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, D15, C30, C30, C4○D8, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C4○D12, C3×C4○D4, Dic15, C60, C60, D30, C2×C30, C2×C30, C2×C5⋊2C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, Q8.13D6, C15⋊3C8, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, D4.8D10, C2×C15⋊3C8, D4⋊D15, D4.D15, Q8⋊2D15, C15⋊7Q16, D60⋊11C2, C15×C4○D4, D4.8D30
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, D15, C4○D8, C5⋊D4, C22×D5, C2×C3⋊D4, D30, C2×C5⋊D4, Q8.13D6, C15⋊7D4, C22×D15, D4.8D10, C2×C15⋊7D4, D4.8D30
►On 240 points
Generators in S240
(1 16 31 46)(2 17 32 47)(3 18 33 48)(4 19 34 49)(5 20 35 50)(6 21 36 51)(7 22 37 52)(8 23 38 53)(9 24 39 54)(10 25 40 55)(11 26 41 56)(12 27 42 57)(13 28 43 58)(14 29 44 59)(15 30 45 60)(61 106 91 76)(62 107 92 77)(63 108 93 78)(64 109 94 79)(65 110 95 80)(66 111 96 81)(67 112 97 82)(68 113 98 83)(69 114 99 84)(70 115 100 85)(71 116 101 86)(72 117 102 87)(73 118 103 88)(74 119 104 89)(75 120 105 90)(121 166 151 136)(122 167 152 137)(123 168 153 138)(124 169 154 139)(125 170 155 140)(126 171 156 141)(127 172 157 142)(128 173 158 143)(129 174 159 144)(130 175 160 145)(131 176 161 146)(132 177 162 147)(133 178 163 148)(134 179 164 149)(135 180 165 150)(181 196 211 226)(182 197 212 227)(183 198 213 228)(184 199 214 229)(185 200 215 230)(186 201 216 231)(187 202 217 232)(188 203 218 233)(189 204 219 234)(190 205 220 235)(191 206 221 236)(192 207 222 237)(193 208 223 238)(194 209 224 239)(195 210 225 240)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 120)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 84)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(57 92)(58 93)(59 94)(60 95)(121 235)(122 236)(123 237)(124 238)(125 239)(126 240)(127 181)(128 182)(129 183)(130 184)(131 185)(132 186)(133 187)(134 188)(135 189)(136 190)(137 191)(138 192)(139 193)(140 194)(141 195)(142 196)(143 197)(144 198)(145 199)(146 200)(147 201)(148 202)(149 203)(150 204)(151 205)(152 206)(153 207)(154 208)(155 209)(156 210)(157 211)(158 212)(159 213)(160 214)(161 215)(162 216)(163 217)(164 218)(165 219)(166 220)(167 221)(168 222)(169 223)(170 224)(171 225)(172 226)(173 227)(174 228)(175 229)(176 230)(177 231)(178 232)(179 233)(180 234)
(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)
(1 152 31 122)(2 121 32 151)(3 150 33 180)(4 179 34 149)(5 148 35 178)(6 177 36 147)(7 146 37 176)(8 175 38 145)(9 144 39 174)(10 173 40 143)(11 142 41 172)(12 171 42 141)(13 140 43 170)(14 169 44 139)(15 138 45 168)(16 167 46 137)(17 136 47 166)(18 165 48 135)(19 134 49 164)(20 163 50 133)(21 132 51 162)(22 161 52 131)(23 130 53 160)(24 159 54 129)(25 128 55 158)(26 157 56 127)(27 126 57 156)(28 155 58 125)(29 124 59 154)(30 153 60 123)(61 196 91 226)(62 225 92 195)(63 194 93 224)(64 223 94 193)(65 192 95 222)(66 221 96 191)(67 190 97 220)(68 219 98 189)(69 188 99 218)(70 217 100 187)(71 186 101 216)(72 215 102 185)(73 184 103 214)(74 213 104 183)(75 182 105 212)(76 211 106 181)(77 240 107 210)(78 209 108 239)(79 238 109 208)(80 207 110 237)(81 236 111 206)(82 205 112 235)(83 234 113 204)(84 203 114 233)(85 232 115 202)(86 201 116 231)(87 230 117 200)(88 199 118 229)(89 228 119 198)(90 197 120 227)
G:=sub<Sym(240)| (1,16,31,46)(2,17,32,47)(3,18,33,48)(4,19,34,49)(5,20,35,50)(6,21,36,51)(7,22,37,52)(8,23,38,53)(9,24,39,54)(10,25,40,55)(11,26,41,56)(12,27,42,57)(13,28,43,58)(14,29,44,59)(15,30,45,60)(61,106,91,76)(62,107,92,77)(63,108,93,78)(64,109,94,79)(65,110,95,80)(66,111,96,81)(67,112,97,82)(68,113,98,83)(69,114,99,84)(70,115,100,85)(71,116,101,86)(72,117,102,87)(73,118,103,88)(74,119,104,89)(75,120,105,90)(121,166,151,136)(122,167,152,137)(123,168,153,138)(124,169,154,139)(125,170,155,140)(126,171,156,141)(127,172,157,142)(128,173,158,143)(129,174,159,144)(130,175,160,145)(131,176,161,146)(132,177,162,147)(133,178,163,148)(134,179,164,149)(135,180,165,150)(181,196,211,226)(182,197,212,227)(183,198,213,228)(184,199,214,229)(185,200,215,230)(186,201,216,231)(187,202,217,232)(188,203,218,233)(189,204,219,234)(190,205,220,235)(191,206,221,236)(192,207,222,237)(193,208,223,238)(194,209,224,239)(195,210,225,240), (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,92)(58,93)(59,94)(60,95)(121,235)(122,236)(123,237)(124,238)(125,239)(126,240)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)(162,216)(163,217)(164,218)(165,219)(166,220)(167,221)(168,222)(169,223)(170,224)(171,225)(172,226)(173,227)(174,228)(175,229)(176,230)(177,231)(178,232)(179,233)(180,234), (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), (1,152,31,122)(2,121,32,151)(3,150,33,180)(4,179,34,149)(5,148,35,178)(6,177,36,147)(7,146,37,176)(8,175,38,145)(9,144,39,174)(10,173,40,143)(11,142,41,172)(12,171,42,141)(13,140,43,170)(14,169,44,139)(15,138,45,168)(16,167,46,137)(17,136,47,166)(18,165,48,135)(19,134,49,164)(20,163,50,133)(21,132,51,162)(22,161,52,131)(23,130,53,160)(24,159,54,129)(25,128,55,158)(26,157,56,127)(27,126,57,156)(28,155,58,125)(29,124,59,154)(30,153,60,123)(61,196,91,226)(62,225,92,195)(63,194,93,224)(64,223,94,193)(65,192,95,222)(66,221,96,191)(67,190,97,220)(68,219,98,189)(69,188,99,218)(70,217,100,187)(71,186,101,216)(72,215,102,185)(73,184,103,214)(74,213,104,183)(75,182,105,212)(76,211,106,181)(77,240,107,210)(78,209,108,239)(79,238,109,208)(80,207,110,237)(81,236,111,206)(82,205,112,235)(83,234,113,204)(84,203,114,233)(85,232,115,202)(86,201,116,231)(87,230,117,200)(88,199,118,229)(89,228,119,198)(90,197,120,227)>;
G:=Group( (1,16,31,46)(2,17,32,47)(3,18,33,48)(4,19,34,49)(5,20,35,50)(6,21,36,51)(7,22,37,52)(8,23,38,53)(9,24,39,54)(10,25,40,55)(11,26,41,56)(12,27,42,57)(13,28,43,58)(14,29,44,59)(15,30,45,60)(61,106,91,76)(62,107,92,77)(63,108,93,78)(64,109,94,79)(65,110,95,80)(66,111,96,81)(67,112,97,82)(68,113,98,83)(69,114,99,84)(70,115,100,85)(71,116,101,86)(72,117,102,87)(73,118,103,88)(74,119,104,89)(75,120,105,90)(121,166,151,136)(122,167,152,137)(123,168,153,138)(124,169,154,139)(125,170,155,140)(126,171,156,141)(127,172,157,142)(128,173,158,143)(129,174,159,144)(130,175,160,145)(131,176,161,146)(132,177,162,147)(133,178,163,148)(134,179,164,149)(135,180,165,150)(181,196,211,226)(182,197,212,227)(183,198,213,228)(184,199,214,229)(185,200,215,230)(186,201,216,231)(187,202,217,232)(188,203,218,233)(189,204,219,234)(190,205,220,235)(191,206,221,236)(192,207,222,237)(193,208,223,238)(194,209,224,239)(195,210,225,240), (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,92)(58,93)(59,94)(60,95)(121,235)(122,236)(123,237)(124,238)(125,239)(126,240)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)(162,216)(163,217)(164,218)(165,219)(166,220)(167,221)(168,222)(169,223)(170,224)(171,225)(172,226)(173,227)(174,228)(175,229)(176,230)(177,231)(178,232)(179,233)(180,234), (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), (1,152,31,122)(2,121,32,151)(3,150,33,180)(4,179,34,149)(5,148,35,178)(6,177,36,147)(7,146,37,176)(8,175,38,145)(9,144,39,174)(10,173,40,143)(11,142,41,172)(12,171,42,141)(13,140,43,170)(14,169,44,139)(15,138,45,168)(16,167,46,137)(17,136,47,166)(18,165,48,135)(19,134,49,164)(20,163,50,133)(21,132,51,162)(22,161,52,131)(23,130,53,160)(24,159,54,129)(25,128,55,158)(26,157,56,127)(27,126,57,156)(28,155,58,125)(29,124,59,154)(30,153,60,123)(61,196,91,226)(62,225,92,195)(63,194,93,224)(64,223,94,193)(65,192,95,222)(66,221,96,191)(67,190,97,220)(68,219,98,189)(69,188,99,218)(70,217,100,187)(71,186,101,216)(72,215,102,185)(73,184,103,214)(74,213,104,183)(75,182,105,212)(76,211,106,181)(77,240,107,210)(78,209,108,239)(79,238,109,208)(80,207,110,237)(81,236,111,206)(82,205,112,235)(83,234,113,204)(84,203,114,233)(85,232,115,202)(86,201,116,231)(87,230,117,200)(88,199,118,229)(89,228,119,198)(90,197,120,227) );
G=PermutationGroup([[(1,16,31,46),(2,17,32,47),(3,18,33,48),(4,19,34,49),(5,20,35,50),(6,21,36,51),(7,22,37,52),(8,23,38,53),(9,24,39,54),(10,25,40,55),(11,26,41,56),(12,27,42,57),(13,28,43,58),(14,29,44,59),(15,30,45,60),(61,106,91,76),(62,107,92,77),(63,108,93,78),(64,109,94,79),(65,110,95,80),(66,111,96,81),(67,112,97,82),(68,113,98,83),(69,114,99,84),(70,115,100,85),(71,116,101,86),(72,117,102,87),(73,118,103,88),(74,119,104,89),(75,120,105,90),(121,166,151,136),(122,167,152,137),(123,168,153,138),(124,169,154,139),(125,170,155,140),(126,171,156,141),(127,172,157,142),(128,173,158,143),(129,174,159,144),(130,175,160,145),(131,176,161,146),(132,177,162,147),(133,178,163,148),(134,179,164,149),(135,180,165,150),(181,196,211,226),(182,197,212,227),(183,198,213,228),(184,199,214,229),(185,200,215,230),(186,201,216,231),(187,202,217,232),(188,203,218,233),(189,204,219,234),(190,205,220,235),(191,206,221,236),(192,207,222,237),(193,208,223,238),(194,209,224,239),(195,210,225,240)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,101),(7,102),(8,103),(9,104),(10,105),(11,106),(12,107),(13,108),(14,109),(15,110),(16,111),(17,112),(18,113),(19,114),(20,115),(21,116),(22,117),(23,118),(24,119),(25,120),(26,61),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,71),(37,72),(38,73),(39,74),(40,75),(41,76),(42,77),(43,78),(44,79),(45,80),(46,81),(47,82),(48,83),(49,84),(50,85),(51,86),(52,87),(53,88),(54,89),(55,90),(56,91),(57,92),(58,93),(59,94),(60,95),(121,235),(122,236),(123,237),(124,238),(125,239),(126,240),(127,181),(128,182),(129,183),(130,184),(131,185),(132,186),(133,187),(134,188),(135,189),(136,190),(137,191),(138,192),(139,193),(140,194),(141,195),(142,196),(143,197),(144,198),(145,199),(146,200),(147,201),(148,202),(149,203),(150,204),(151,205),(152,206),(153,207),(154,208),(155,209),(156,210),(157,211),(158,212),(159,213),(160,214),(161,215),(162,216),(163,217),(164,218),(165,219),(166,220),(167,221),(168,222),(169,223),(170,224),(171,225),(172,226),(173,227),(174,228),(175,229),(176,230),(177,231),(178,232),(179,233),(180,234)], [(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)], [(1,152,31,122),(2,121,32,151),(3,150,33,180),(4,179,34,149),(5,148,35,178),(6,177,36,147),(7,146,37,176),(8,175,38,145),(9,144,39,174),(10,173,40,143),(11,142,41,172),(12,171,42,141),(13,140,43,170),(14,169,44,139),(15,138,45,168),(16,167,46,137),(17,136,47,166),(18,165,48,135),(19,134,49,164),(20,163,50,133),(21,132,51,162),(22,161,52,131),(23,130,53,160),(24,159,54,129),(25,128,55,158),(26,157,56,127),(27,126,57,156),(28,155,58,125),(29,124,59,154),(30,153,60,123),(61,196,91,226),(62,225,92,195),(63,194,93,224),(64,223,94,193),(65,192,95,222),(66,221,96,191),(67,190,97,220),(68,219,98,189),(69,188,99,218),(70,217,100,187),(71,186,101,216),(72,215,102,185),(73,184,103,214),(74,213,104,183),(75,182,105,212),(76,211,106,181),(77,240,107,210),(78,209,108,239),(79,238,109,208),(80,207,110,237),(81,236,111,206),(82,205,112,235),(83,234,113,204),(84,203,114,233),(85,232,115,202),(86,201,116,231),(87,230,117,200),(88,199,118,229),(89,228,119,198),(90,197,120,227)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 4 | 60 | 2 | 1 | 1 | 2 | 4 | 60 | 2 | 2 | 2 | 4 | 4 | 4 | 30 | 30 | 30 | 30 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C3⋊D4 | C3⋊D4 | D15 | C4○D8 | C5⋊D4 | C5⋊D4 | D30 | D30 | D30 | C15⋊7D4 | C15⋊7D4 | Q8.13D6 | D4.8D10 | D4.8D30 |
| kernel | D4.8D30 | C2×C15⋊3C8 | D4⋊D15 | D4.D15 | Q8⋊2D15 | C15⋊7Q16 | D60⋊11C2 | C15×C4○D4 | C5×C4○D4 | C60 | C2×C30 | C3×C4○D4 | C2×C20 | C5×D4 | C5×Q8 | C2×C12 | C3×D4 | C3×Q8 | C20 | C2×C10 | C4○D4 | C15 | C12 | C2×C6 | C2×C4 | D4 | Q8 | C4 | C22 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 4 | 8 |
Matrix representation of D4.8D30 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 64 | 64 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 177 | 113 |
| 0 | 0 | 64 | 64 |
| 225 | 64 | 0 | 0 |
| 177 | 30 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 78 | 72 | 0 | 0 |
| 113 | 163 | 0 | 0 |
| 0 | 0 | 233 | 225 |
| 0 | 0 | 230 | 8 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,177,64,0,0,0,64],[240,0,0,0,0,240,0,0,0,0,177,64,0,0,113,64],[225,177,0,0,64,30,0,0,0,0,64,0,0,0,0,64],[78,113,0,0,72,163,0,0,0,0,233,230,0,0,225,8] >;
D4.8D30 in GAP, Magma, Sage, TeX
D_4._8D_{30} % in TeX
G:=Group("D4.8D30"); // GroupNames label
G:=SmallGroup(480,915);
// by ID
G=gap.SmallGroup(480,915);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,675,185,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^30=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^29>;
// generators/relations