metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊9Q8, C60.90D4, C20.22D12, C6.9(Q8×D5), C10.9(S3×Q8), C4⋊Dic5⋊12S3, C5⋊4(C4.D12), C15⋊8(C22⋊Q8), C30.30(C2×Q8), (C2×Dic6)⋊11D5, C3⋊2(D10⋊3Q8), C10.60(C2×D12), (C2×C20).118D6, C30.120(C2×D4), (C10×Dic6)⋊11C2, D30⋊4C4.6C2, C30.46(C4○D4), (C2×C12).119D10, C4.25(C5⋊D12), C12.39(C5⋊D4), C2.12(D15⋊Q8), (C2×C30).73C23, C30.Q8⋊10C2, (C2×Dic5).24D6, C10.8(D4⋊2S3), (C2×C60).196C22, C6.28(Q8⋊2D5), (C2×Dic3).23D10, C2.12(D20⋊S3), (C6×Dic5).43C22, (C10×Dic3).42C22, (C22×D15).98C22, (C2×Dic15).193C22, (C3×C4⋊Dic5)⋊9C2, (C2×C4×D15).14C2, C6.14(C2×C5⋊D4), (C2×C4).206(S3×D5), C2.18(C2×C5⋊D12), C22.159(C2×S3×D5), (C2×C6).85(C22×D5), (C2×C10).85(C22×S3), SmallGroup(480,459)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊9Q8
G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a25b, dcd-1=c-1 >
Subgroups: 780 in 148 conjugacy classes, 54 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, Q8, C23, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, D10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C22⋊Q8, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, D30, D30, C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, C2×C4×D5, Q8×C10, C4.D12, C6×Dic5, C5×Dic6, C10×Dic3, C4×D15, C2×Dic15, C2×C60, C22×D15, D10⋊3Q8, D30⋊4C4, C30.Q8, C3×C4⋊Dic5, C10×Dic6, C2×C4×D15, D30⋊9Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, D12, C22×S3, C22⋊Q8, C5⋊D4, C22×D5, C2×D12, D4⋊2S3, S3×Q8, S3×D5, Q8×D5, Q8⋊2D5, C2×C5⋊D4, C4.D12, C5⋊D12, C2×S3×D5, D10⋊3Q8, D20⋊S3, D15⋊Q8, C2×C5⋊D12, D30⋊9Q8
►On 240 points
Generators in S240
(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)
(1 102)(2 101)(3 100)(4 99)(5 98)(6 97)(7 96)(8 95)(9 94)(10 93)(11 92)(12 91)(13 120)(14 119)(15 118)(16 117)(17 116)(18 115)(19 114)(20 113)(21 112)(22 111)(23 110)(24 109)(25 108)(26 107)(27 106)(28 105)(29 104)(30 103)(31 157)(32 156)(33 155)(34 154)(35 153)(36 152)(37 151)(38 180)(39 179)(40 178)(41 177)(42 176)(43 175)(44 174)(45 173)(46 172)(47 171)(48 170)(49 169)(50 168)(51 167)(52 166)(53 165)(54 164)(55 163)(56 162)(57 161)(58 160)(59 159)(60 158)(61 206)(62 205)(63 204)(64 203)(65 202)(66 201)(67 200)(68 199)(69 198)(70 197)(71 196)(72 195)(73 194)(74 193)(75 192)(76 191)(77 190)(78 189)(79 188)(80 187)(81 186)(82 185)(83 184)(84 183)(85 182)(86 181)(87 210)(88 209)(89 208)(90 207)(121 234)(122 233)(123 232)(124 231)(125 230)(126 229)(127 228)(128 227)(129 226)(130 225)(131 224)(132 223)(133 222)(134 221)(135 220)(136 219)(137 218)(138 217)(139 216)(140 215)(141 214)(142 213)(143 212)(144 211)(145 240)(146 239)(147 238)(148 237)(149 236)(150 235)
(1 228 103 128)(2 229 104 129)(3 230 105 130)(4 231 106 131)(5 232 107 132)(6 233 108 133)(7 234 109 134)(8 235 110 135)(9 236 111 136)(10 237 112 137)(11 238 113 138)(12 239 114 139)(13 240 115 140)(14 211 116 141)(15 212 117 142)(16 213 118 143)(17 214 119 144)(18 215 120 145)(19 216 91 146)(20 217 92 147)(21 218 93 148)(22 219 94 149)(23 220 95 150)(24 221 96 121)(25 222 97 122)(26 223 98 123)(27 224 99 124)(28 225 100 125)(29 226 101 126)(30 227 102 127)(31 69 165 206)(32 70 166 207)(33 71 167 208)(34 72 168 209)(35 73 169 210)(36 74 170 181)(37 75 171 182)(38 76 172 183)(39 77 173 184)(40 78 174 185)(41 79 175 186)(42 80 176 187)(43 81 177 188)(44 82 178 189)(45 83 179 190)(46 84 180 191)(47 85 151 192)(48 86 152 193)(49 87 153 194)(50 88 154 195)(51 89 155 196)(52 90 156 197)(53 61 157 198)(54 62 158 199)(55 63 159 200)(56 64 160 201)(57 65 161 202)(58 66 162 203)(59 67 163 204)(60 68 164 205)
(1 210 103 73)(2 191 104 84)(3 202 105 65)(4 183 106 76)(5 194 107 87)(6 205 108 68)(7 186 109 79)(8 197 110 90)(9 208 111 71)(10 189 112 82)(11 200 113 63)(12 181 114 74)(13 192 115 85)(14 203 116 66)(15 184 117 77)(16 195 118 88)(17 206 119 69)(18 187 120 80)(19 198 91 61)(20 209 92 72)(21 190 93 83)(22 201 94 64)(23 182 95 75)(24 193 96 86)(25 204 97 67)(26 185 98 78)(27 196 99 89)(28 207 100 70)(29 188 101 81)(30 199 102 62)(31 214 165 144)(32 225 166 125)(33 236 167 136)(34 217 168 147)(35 228 169 128)(36 239 170 139)(37 220 171 150)(38 231 172 131)(39 212 173 142)(40 223 174 123)(41 234 175 134)(42 215 176 145)(43 226 177 126)(44 237 178 137)(45 218 179 148)(46 229 180 129)(47 240 151 140)(48 221 152 121)(49 232 153 132)(50 213 154 143)(51 224 155 124)(52 235 156 135)(53 216 157 146)(54 227 158 127)(55 238 159 138)(56 219 160 149)(57 230 161 130)(58 211 162 141)(59 222 163 122)(60 233 164 133)
G:=sub<Sym(240)| (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), (1,102)(2,101)(3,100)(4,99)(5,98)(6,97)(7,96)(8,95)(9,94)(10,93)(11,92)(12,91)(13,120)(14,119)(15,118)(16,117)(17,116)(18,115)(19,114)(20,113)(21,112)(22,111)(23,110)(24,109)(25,108)(26,107)(27,106)(28,105)(29,104)(30,103)(31,157)(32,156)(33,155)(34,154)(35,153)(36,152)(37,151)(38,180)(39,179)(40,178)(41,177)(42,176)(43,175)(44,174)(45,173)(46,172)(47,171)(48,170)(49,169)(50,168)(51,167)(52,166)(53,165)(54,164)(55,163)(56,162)(57,161)(58,160)(59,159)(60,158)(61,206)(62,205)(63,204)(64,203)(65,202)(66,201)(67,200)(68,199)(69,198)(70,197)(71,196)(72,195)(73,194)(74,193)(75,192)(76,191)(77,190)(78,189)(79,188)(80,187)(81,186)(82,185)(83,184)(84,183)(85,182)(86,181)(87,210)(88,209)(89,208)(90,207)(121,234)(122,233)(123,232)(124,231)(125,230)(126,229)(127,228)(128,227)(129,226)(130,225)(131,224)(132,223)(133,222)(134,221)(135,220)(136,219)(137,218)(138,217)(139,216)(140,215)(141,214)(142,213)(143,212)(144,211)(145,240)(146,239)(147,238)(148,237)(149,236)(150,235), (1,228,103,128)(2,229,104,129)(3,230,105,130)(4,231,106,131)(5,232,107,132)(6,233,108,133)(7,234,109,134)(8,235,110,135)(9,236,111,136)(10,237,112,137)(11,238,113,138)(12,239,114,139)(13,240,115,140)(14,211,116,141)(15,212,117,142)(16,213,118,143)(17,214,119,144)(18,215,120,145)(19,216,91,146)(20,217,92,147)(21,218,93,148)(22,219,94,149)(23,220,95,150)(24,221,96,121)(25,222,97,122)(26,223,98,123)(27,224,99,124)(28,225,100,125)(29,226,101,126)(30,227,102,127)(31,69,165,206)(32,70,166,207)(33,71,167,208)(34,72,168,209)(35,73,169,210)(36,74,170,181)(37,75,171,182)(38,76,172,183)(39,77,173,184)(40,78,174,185)(41,79,175,186)(42,80,176,187)(43,81,177,188)(44,82,178,189)(45,83,179,190)(46,84,180,191)(47,85,151,192)(48,86,152,193)(49,87,153,194)(50,88,154,195)(51,89,155,196)(52,90,156,197)(53,61,157,198)(54,62,158,199)(55,63,159,200)(56,64,160,201)(57,65,161,202)(58,66,162,203)(59,67,163,204)(60,68,164,205), (1,210,103,73)(2,191,104,84)(3,202,105,65)(4,183,106,76)(5,194,107,87)(6,205,108,68)(7,186,109,79)(8,197,110,90)(9,208,111,71)(10,189,112,82)(11,200,113,63)(12,181,114,74)(13,192,115,85)(14,203,116,66)(15,184,117,77)(16,195,118,88)(17,206,119,69)(18,187,120,80)(19,198,91,61)(20,209,92,72)(21,190,93,83)(22,201,94,64)(23,182,95,75)(24,193,96,86)(25,204,97,67)(26,185,98,78)(27,196,99,89)(28,207,100,70)(29,188,101,81)(30,199,102,62)(31,214,165,144)(32,225,166,125)(33,236,167,136)(34,217,168,147)(35,228,169,128)(36,239,170,139)(37,220,171,150)(38,231,172,131)(39,212,173,142)(40,223,174,123)(41,234,175,134)(42,215,176,145)(43,226,177,126)(44,237,178,137)(45,218,179,148)(46,229,180,129)(47,240,151,140)(48,221,152,121)(49,232,153,132)(50,213,154,143)(51,224,155,124)(52,235,156,135)(53,216,157,146)(54,227,158,127)(55,238,159,138)(56,219,160,149)(57,230,161,130)(58,211,162,141)(59,222,163,122)(60,233,164,133)>;
G:=Group( (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), (1,102)(2,101)(3,100)(4,99)(5,98)(6,97)(7,96)(8,95)(9,94)(10,93)(11,92)(12,91)(13,120)(14,119)(15,118)(16,117)(17,116)(18,115)(19,114)(20,113)(21,112)(22,111)(23,110)(24,109)(25,108)(26,107)(27,106)(28,105)(29,104)(30,103)(31,157)(32,156)(33,155)(34,154)(35,153)(36,152)(37,151)(38,180)(39,179)(40,178)(41,177)(42,176)(43,175)(44,174)(45,173)(46,172)(47,171)(48,170)(49,169)(50,168)(51,167)(52,166)(53,165)(54,164)(55,163)(56,162)(57,161)(58,160)(59,159)(60,158)(61,206)(62,205)(63,204)(64,203)(65,202)(66,201)(67,200)(68,199)(69,198)(70,197)(71,196)(72,195)(73,194)(74,193)(75,192)(76,191)(77,190)(78,189)(79,188)(80,187)(81,186)(82,185)(83,184)(84,183)(85,182)(86,181)(87,210)(88,209)(89,208)(90,207)(121,234)(122,233)(123,232)(124,231)(125,230)(126,229)(127,228)(128,227)(129,226)(130,225)(131,224)(132,223)(133,222)(134,221)(135,220)(136,219)(137,218)(138,217)(139,216)(140,215)(141,214)(142,213)(143,212)(144,211)(145,240)(146,239)(147,238)(148,237)(149,236)(150,235), (1,228,103,128)(2,229,104,129)(3,230,105,130)(4,231,106,131)(5,232,107,132)(6,233,108,133)(7,234,109,134)(8,235,110,135)(9,236,111,136)(10,237,112,137)(11,238,113,138)(12,239,114,139)(13,240,115,140)(14,211,116,141)(15,212,117,142)(16,213,118,143)(17,214,119,144)(18,215,120,145)(19,216,91,146)(20,217,92,147)(21,218,93,148)(22,219,94,149)(23,220,95,150)(24,221,96,121)(25,222,97,122)(26,223,98,123)(27,224,99,124)(28,225,100,125)(29,226,101,126)(30,227,102,127)(31,69,165,206)(32,70,166,207)(33,71,167,208)(34,72,168,209)(35,73,169,210)(36,74,170,181)(37,75,171,182)(38,76,172,183)(39,77,173,184)(40,78,174,185)(41,79,175,186)(42,80,176,187)(43,81,177,188)(44,82,178,189)(45,83,179,190)(46,84,180,191)(47,85,151,192)(48,86,152,193)(49,87,153,194)(50,88,154,195)(51,89,155,196)(52,90,156,197)(53,61,157,198)(54,62,158,199)(55,63,159,200)(56,64,160,201)(57,65,161,202)(58,66,162,203)(59,67,163,204)(60,68,164,205), (1,210,103,73)(2,191,104,84)(3,202,105,65)(4,183,106,76)(5,194,107,87)(6,205,108,68)(7,186,109,79)(8,197,110,90)(9,208,111,71)(10,189,112,82)(11,200,113,63)(12,181,114,74)(13,192,115,85)(14,203,116,66)(15,184,117,77)(16,195,118,88)(17,206,119,69)(18,187,120,80)(19,198,91,61)(20,209,92,72)(21,190,93,83)(22,201,94,64)(23,182,95,75)(24,193,96,86)(25,204,97,67)(26,185,98,78)(27,196,99,89)(28,207,100,70)(29,188,101,81)(30,199,102,62)(31,214,165,144)(32,225,166,125)(33,236,167,136)(34,217,168,147)(35,228,169,128)(36,239,170,139)(37,220,171,150)(38,231,172,131)(39,212,173,142)(40,223,174,123)(41,234,175,134)(42,215,176,145)(43,226,177,126)(44,237,178,137)(45,218,179,148)(46,229,180,129)(47,240,151,140)(48,221,152,121)(49,232,153,132)(50,213,154,143)(51,224,155,124)(52,235,156,135)(53,216,157,146)(54,227,158,127)(55,238,159,138)(56,219,160,149)(57,230,161,130)(58,211,162,141)(59,222,163,122)(60,233,164,133) );
G=PermutationGroup([[(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)], [(1,102),(2,101),(3,100),(4,99),(5,98),(6,97),(7,96),(8,95),(9,94),(10,93),(11,92),(12,91),(13,120),(14,119),(15,118),(16,117),(17,116),(18,115),(19,114),(20,113),(21,112),(22,111),(23,110),(24,109),(25,108),(26,107),(27,106),(28,105),(29,104),(30,103),(31,157),(32,156),(33,155),(34,154),(35,153),(36,152),(37,151),(38,180),(39,179),(40,178),(41,177),(42,176),(43,175),(44,174),(45,173),(46,172),(47,171),(48,170),(49,169),(50,168),(51,167),(52,166),(53,165),(54,164),(55,163),(56,162),(57,161),(58,160),(59,159),(60,158),(61,206),(62,205),(63,204),(64,203),(65,202),(66,201),(67,200),(68,199),(69,198),(70,197),(71,196),(72,195),(73,194),(74,193),(75,192),(76,191),(77,190),(78,189),(79,188),(80,187),(81,186),(82,185),(83,184),(84,183),(85,182),(86,181),(87,210),(88,209),(89,208),(90,207),(121,234),(122,233),(123,232),(124,231),(125,230),(126,229),(127,228),(128,227),(129,226),(130,225),(131,224),(132,223),(133,222),(134,221),(135,220),(136,219),(137,218),(138,217),(139,216),(140,215),(141,214),(142,213),(143,212),(144,211),(145,240),(146,239),(147,238),(148,237),(149,236),(150,235)], [(1,228,103,128),(2,229,104,129),(3,230,105,130),(4,231,106,131),(5,232,107,132),(6,233,108,133),(7,234,109,134),(8,235,110,135),(9,236,111,136),(10,237,112,137),(11,238,113,138),(12,239,114,139),(13,240,115,140),(14,211,116,141),(15,212,117,142),(16,213,118,143),(17,214,119,144),(18,215,120,145),(19,216,91,146),(20,217,92,147),(21,218,93,148),(22,219,94,149),(23,220,95,150),(24,221,96,121),(25,222,97,122),(26,223,98,123),(27,224,99,124),(28,225,100,125),(29,226,101,126),(30,227,102,127),(31,69,165,206),(32,70,166,207),(33,71,167,208),(34,72,168,209),(35,73,169,210),(36,74,170,181),(37,75,171,182),(38,76,172,183),(39,77,173,184),(40,78,174,185),(41,79,175,186),(42,80,176,187),(43,81,177,188),(44,82,178,189),(45,83,179,190),(46,84,180,191),(47,85,151,192),(48,86,152,193),(49,87,153,194),(50,88,154,195),(51,89,155,196),(52,90,156,197),(53,61,157,198),(54,62,158,199),(55,63,159,200),(56,64,160,201),(57,65,161,202),(58,66,162,203),(59,67,163,204),(60,68,164,205)], [(1,210,103,73),(2,191,104,84),(3,202,105,65),(4,183,106,76),(5,194,107,87),(6,205,108,68),(7,186,109,79),(8,197,110,90),(9,208,111,71),(10,189,112,82),(11,200,113,63),(12,181,114,74),(13,192,115,85),(14,203,116,66),(15,184,117,77),(16,195,118,88),(17,206,119,69),(18,187,120,80),(19,198,91,61),(20,209,92,72),(21,190,93,83),(22,201,94,64),(23,182,95,75),(24,193,96,86),(25,204,97,67),(26,185,98,78),(27,196,99,89),(28,207,100,70),(29,188,101,81),(30,199,102,62),(31,214,165,144),(32,225,166,125),(33,236,167,136),(34,217,168,147),(35,228,169,128),(36,239,170,139),(37,220,171,150),(38,231,172,131),(39,212,173,142),(40,223,174,123),(41,234,175,134),(42,215,176,145),(43,226,177,126),(44,237,178,137),(45,218,179,148),(46,229,180,129),(47,240,151,140),(48,221,152,121),(49,232,153,132),(50,213,154,143),(51,224,155,124),(52,235,156,135),(53,216,157,146),(54,227,158,127),(55,238,159,138),(56,219,160,149),(57,230,161,130),(58,211,162,141),(59,222,163,122),(60,233,164,133)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 2 | 2 | 12 | 12 | 20 | 20 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 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 | S3 | D4 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D12 | C5⋊D4 | D4⋊2S3 | S3×Q8 | S3×D5 | Q8×D5 | Q8⋊2D5 | C5⋊D12 | C2×S3×D5 | D20⋊S3 | D15⋊Q8 |
| kernel | D30⋊9Q8 | D30⋊4C4 | C30.Q8 | C3×C4⋊Dic5 | C10×Dic6 | C2×C4×D15 | C4⋊Dic5 | C60 | D30 | C2×Dic6 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C20 | C12 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of D30⋊9Q8 ►in GL6(𝔽61)
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 15 | 0 | 0 |
| 0 | 0 | 12 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 18 | 60 | 0 | 0 | 0 | 0 |
| 18 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 15 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 52 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 0 | 38 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 53 | 19 | 0 | 0 |
| 0 | 0 | 32 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 16 |
| 0 | 0 | 0 | 0 | 0 | 50 |
G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,43,0,0,0,0,0,0,60,12,0,0,0,0,15,2,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[18,18,0,0,0,0,60,43,0,0,0,0,0,0,60,0,0,0,0,0,15,1,0,0,0,0,0,0,1,52,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,38,0,0,0,0,0,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,53,32,0,0,0,0,19,8,0,0,0,0,0,0,11,0,0,0,0,0,16,50] >;
D30⋊9Q8 in GAP, Magma, Sage, TeX
D_{30}\rtimes_9Q_8 % in TeX
G:=Group("D30:9Q8"); // GroupNames label
G:=SmallGroup(480,459);
// by ID
G=gap.SmallGroup(480,459);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,120,422,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^25*b,d*c*d^-1=c^-1>;
// generators/relations