metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊4Q8, Dic3.11D20, C4⋊Dic5⋊6S3, C6.16(Q8×D5), C5⋊2(D6⋊Q8), C30.50(C2×D4), C2.20(S3×D20), C6.19(C2×D20), (C2×C20).22D6, C10.19(S3×D4), C10.16(S3×Q8), C30.46(C2×Q8), Dic3⋊C4⋊19D5, C3⋊1(D10⋊2Q8), C15⋊17(C22⋊Q8), (C5×Dic3).9D4, D30⋊3C4.8C2, (C2×C12).227D10, C2.18(D15⋊Q8), C6.Dic10⋊22C2, (C2×Dic5).37D6, D30⋊4C4.14C2, C10.71(C4○D12), C30.121(C4○D4), C6.46(D4⋊2D5), (C2×C60).258C22, (C2×C30).119C23, (C2×Dic3).37D10, (C6×Dic5).72C22, C2.17(Dic3.D10), (C2×Dic15).96C22, (C10×Dic3).73C22, (C22×D15).41C22, (C2×C15⋊Q8)⋊9C2, (C2×C4).52(S3×D5), (C3×C4⋊Dic5)⋊17C2, C22.182(C2×S3×D5), (C5×Dic3⋊C4)⋊19C2, (C2×D30.C2).7C2, (C2×C6).131(C22×D5), (C2×C10).131(C22×S3), SmallGroup(480,505)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊4Q8
G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=cac-1=a-1, dad-1=a11, cbc-1=a13b, dbd-1=a10b, dcd-1=c-1 >
Subgroups: 844 in 148 conjugacy classes, 50 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, Q8, C23, D5, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, D10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C22⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, Dic3⋊C4, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, D30, D30, C2×C30, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, D6⋊Q8, D30.C2, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C22×D15, D10⋊2Q8, D30⋊4C4, C6.Dic10, C3×C4⋊Dic5, C5×Dic3⋊C4, D30⋊3C4, C2×D30.C2, C2×C15⋊Q8, D30⋊4Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, C22×S3, C22⋊Q8, D20, C22×D5, C4○D12, S3×D4, S3×Q8, S3×D5, C2×D20, D4⋊2D5, Q8×D5, D6⋊Q8, C2×S3×D5, D10⋊2Q8, D15⋊Q8, S3×D20, Dic3.D10, D30⋊4Q8
►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)
(1 239)(2 238)(3 237)(4 236)(5 235)(6 234)(7 233)(8 232)(9 231)(10 230)(11 229)(12 228)(13 227)(14 226)(15 225)(16 224)(17 223)(18 222)(19 221)(20 220)(21 219)(22 218)(23 217)(24 216)(25 215)(26 214)(27 213)(28 212)(29 211)(30 240)(31 142)(32 141)(33 140)(34 139)(35 138)(36 137)(37 136)(38 135)(39 134)(40 133)(41 132)(42 131)(43 130)(44 129)(45 128)(46 127)(47 126)(48 125)(49 124)(50 123)(51 122)(52 121)(53 150)(54 149)(55 148)(56 147)(57 146)(58 145)(59 144)(60 143)(61 190)(62 189)(63 188)(64 187)(65 186)(66 185)(67 184)(68 183)(69 182)(70 181)(71 210)(72 209)(73 208)(74 207)(75 206)(76 205)(77 204)(78 203)(79 202)(80 201)(81 200)(82 199)(83 198)(84 197)(85 196)(86 195)(87 194)(88 193)(89 192)(90 191)(91 176)(92 175)(93 174)(94 173)(95 172)(96 171)(97 170)(98 169)(99 168)(100 167)(101 166)(102 165)(103 164)(104 163)(105 162)(106 161)(107 160)(108 159)(109 158)(110 157)(111 156)(112 155)(113 154)(114 153)(115 152)(116 151)(117 180)(118 179)(119 178)(120 177)
(1 182 240 85)(2 181 211 84)(3 210 212 83)(4 209 213 82)(5 208 214 81)(6 207 215 80)(7 206 216 79)(8 205 217 78)(9 204 218 77)(10 203 219 76)(11 202 220 75)(12 201 221 74)(13 200 222 73)(14 199 223 72)(15 198 224 71)(16 197 225 70)(17 196 226 69)(18 195 227 68)(19 194 228 67)(20 193 229 66)(21 192 230 65)(22 191 231 64)(23 190 232 63)(24 189 233 62)(25 188 234 61)(26 187 235 90)(27 186 236 89)(28 185 237 88)(29 184 238 87)(30 183 239 86)(31 104 138 154)(32 103 139 153)(33 102 140 152)(34 101 141 151)(35 100 142 180)(36 99 143 179)(37 98 144 178)(38 97 145 177)(39 96 146 176)(40 95 147 175)(41 94 148 174)(42 93 149 173)(43 92 150 172)(44 91 121 171)(45 120 122 170)(46 119 123 169)(47 118 124 168)(48 117 125 167)(49 116 126 166)(50 115 127 165)(51 114 128 164)(52 113 129 163)(53 112 130 162)(54 111 131 161)(55 110 132 160)(56 109 133 159)(57 108 134 158)(58 107 135 157)(59 106 136 156)(60 105 137 155)
(1 159 240 109)(2 170 211 120)(3 151 212 101)(4 162 213 112)(5 173 214 93)(6 154 215 104)(7 165 216 115)(8 176 217 96)(9 157 218 107)(10 168 219 118)(11 179 220 99)(12 160 221 110)(13 171 222 91)(14 152 223 102)(15 163 224 113)(16 174 225 94)(17 155 226 105)(18 166 227 116)(19 177 228 97)(20 158 229 108)(21 169 230 119)(22 180 231 100)(23 161 232 111)(24 172 233 92)(25 153 234 103)(26 164 235 114)(27 175 236 95)(28 156 237 106)(29 167 238 117)(30 178 239 98)(31 207 138 80)(32 188 139 61)(33 199 140 72)(34 210 141 83)(35 191 142 64)(36 202 143 75)(37 183 144 86)(38 194 145 67)(39 205 146 78)(40 186 147 89)(41 197 148 70)(42 208 149 81)(43 189 150 62)(44 200 121 73)(45 181 122 84)(46 192 123 65)(47 203 124 76)(48 184 125 87)(49 195 126 68)(50 206 127 79)(51 187 128 90)(52 198 129 71)(53 209 130 82)(54 190 131 63)(55 201 132 74)(56 182 133 85)(57 193 134 66)(58 204 135 77)(59 185 136 88)(60 196 137 69)
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), (1,239)(2,238)(3,237)(4,236)(5,235)(6,234)(7,233)(8,232)(9,231)(10,230)(11,229)(12,228)(13,227)(14,226)(15,225)(16,224)(17,223)(18,222)(19,221)(20,220)(21,219)(22,218)(23,217)(24,216)(25,215)(26,214)(27,213)(28,212)(29,211)(30,240)(31,142)(32,141)(33,140)(34,139)(35,138)(36,137)(37,136)(38,135)(39,134)(40,133)(41,132)(42,131)(43,130)(44,129)(45,128)(46,127)(47,126)(48,125)(49,124)(50,123)(51,122)(52,121)(53,150)(54,149)(55,148)(56,147)(57,146)(58,145)(59,144)(60,143)(61,190)(62,189)(63,188)(64,187)(65,186)(66,185)(67,184)(68,183)(69,182)(70,181)(71,210)(72,209)(73,208)(74,207)(75,206)(76,205)(77,204)(78,203)(79,202)(80,201)(81,200)(82,199)(83,198)(84,197)(85,196)(86,195)(87,194)(88,193)(89,192)(90,191)(91,176)(92,175)(93,174)(94,173)(95,172)(96,171)(97,170)(98,169)(99,168)(100,167)(101,166)(102,165)(103,164)(104,163)(105,162)(106,161)(107,160)(108,159)(109,158)(110,157)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,180)(118,179)(119,178)(120,177), (1,182,240,85)(2,181,211,84)(3,210,212,83)(4,209,213,82)(5,208,214,81)(6,207,215,80)(7,206,216,79)(8,205,217,78)(9,204,218,77)(10,203,219,76)(11,202,220,75)(12,201,221,74)(13,200,222,73)(14,199,223,72)(15,198,224,71)(16,197,225,70)(17,196,226,69)(18,195,227,68)(19,194,228,67)(20,193,229,66)(21,192,230,65)(22,191,231,64)(23,190,232,63)(24,189,233,62)(25,188,234,61)(26,187,235,90)(27,186,236,89)(28,185,237,88)(29,184,238,87)(30,183,239,86)(31,104,138,154)(32,103,139,153)(33,102,140,152)(34,101,141,151)(35,100,142,180)(36,99,143,179)(37,98,144,178)(38,97,145,177)(39,96,146,176)(40,95,147,175)(41,94,148,174)(42,93,149,173)(43,92,150,172)(44,91,121,171)(45,120,122,170)(46,119,123,169)(47,118,124,168)(48,117,125,167)(49,116,126,166)(50,115,127,165)(51,114,128,164)(52,113,129,163)(53,112,130,162)(54,111,131,161)(55,110,132,160)(56,109,133,159)(57,108,134,158)(58,107,135,157)(59,106,136,156)(60,105,137,155), (1,159,240,109)(2,170,211,120)(3,151,212,101)(4,162,213,112)(5,173,214,93)(6,154,215,104)(7,165,216,115)(8,176,217,96)(9,157,218,107)(10,168,219,118)(11,179,220,99)(12,160,221,110)(13,171,222,91)(14,152,223,102)(15,163,224,113)(16,174,225,94)(17,155,226,105)(18,166,227,116)(19,177,228,97)(20,158,229,108)(21,169,230,119)(22,180,231,100)(23,161,232,111)(24,172,233,92)(25,153,234,103)(26,164,235,114)(27,175,236,95)(28,156,237,106)(29,167,238,117)(30,178,239,98)(31,207,138,80)(32,188,139,61)(33,199,140,72)(34,210,141,83)(35,191,142,64)(36,202,143,75)(37,183,144,86)(38,194,145,67)(39,205,146,78)(40,186,147,89)(41,197,148,70)(42,208,149,81)(43,189,150,62)(44,200,121,73)(45,181,122,84)(46,192,123,65)(47,203,124,76)(48,184,125,87)(49,195,126,68)(50,206,127,79)(51,187,128,90)(52,198,129,71)(53,209,130,82)(54,190,131,63)(55,201,132,74)(56,182,133,85)(57,193,134,66)(58,204,135,77)(59,185,136,88)(60,196,137,69)>;
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), (1,239)(2,238)(3,237)(4,236)(5,235)(6,234)(7,233)(8,232)(9,231)(10,230)(11,229)(12,228)(13,227)(14,226)(15,225)(16,224)(17,223)(18,222)(19,221)(20,220)(21,219)(22,218)(23,217)(24,216)(25,215)(26,214)(27,213)(28,212)(29,211)(30,240)(31,142)(32,141)(33,140)(34,139)(35,138)(36,137)(37,136)(38,135)(39,134)(40,133)(41,132)(42,131)(43,130)(44,129)(45,128)(46,127)(47,126)(48,125)(49,124)(50,123)(51,122)(52,121)(53,150)(54,149)(55,148)(56,147)(57,146)(58,145)(59,144)(60,143)(61,190)(62,189)(63,188)(64,187)(65,186)(66,185)(67,184)(68,183)(69,182)(70,181)(71,210)(72,209)(73,208)(74,207)(75,206)(76,205)(77,204)(78,203)(79,202)(80,201)(81,200)(82,199)(83,198)(84,197)(85,196)(86,195)(87,194)(88,193)(89,192)(90,191)(91,176)(92,175)(93,174)(94,173)(95,172)(96,171)(97,170)(98,169)(99,168)(100,167)(101,166)(102,165)(103,164)(104,163)(105,162)(106,161)(107,160)(108,159)(109,158)(110,157)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,180)(118,179)(119,178)(120,177), (1,182,240,85)(2,181,211,84)(3,210,212,83)(4,209,213,82)(5,208,214,81)(6,207,215,80)(7,206,216,79)(8,205,217,78)(9,204,218,77)(10,203,219,76)(11,202,220,75)(12,201,221,74)(13,200,222,73)(14,199,223,72)(15,198,224,71)(16,197,225,70)(17,196,226,69)(18,195,227,68)(19,194,228,67)(20,193,229,66)(21,192,230,65)(22,191,231,64)(23,190,232,63)(24,189,233,62)(25,188,234,61)(26,187,235,90)(27,186,236,89)(28,185,237,88)(29,184,238,87)(30,183,239,86)(31,104,138,154)(32,103,139,153)(33,102,140,152)(34,101,141,151)(35,100,142,180)(36,99,143,179)(37,98,144,178)(38,97,145,177)(39,96,146,176)(40,95,147,175)(41,94,148,174)(42,93,149,173)(43,92,150,172)(44,91,121,171)(45,120,122,170)(46,119,123,169)(47,118,124,168)(48,117,125,167)(49,116,126,166)(50,115,127,165)(51,114,128,164)(52,113,129,163)(53,112,130,162)(54,111,131,161)(55,110,132,160)(56,109,133,159)(57,108,134,158)(58,107,135,157)(59,106,136,156)(60,105,137,155), (1,159,240,109)(2,170,211,120)(3,151,212,101)(4,162,213,112)(5,173,214,93)(6,154,215,104)(7,165,216,115)(8,176,217,96)(9,157,218,107)(10,168,219,118)(11,179,220,99)(12,160,221,110)(13,171,222,91)(14,152,223,102)(15,163,224,113)(16,174,225,94)(17,155,226,105)(18,166,227,116)(19,177,228,97)(20,158,229,108)(21,169,230,119)(22,180,231,100)(23,161,232,111)(24,172,233,92)(25,153,234,103)(26,164,235,114)(27,175,236,95)(28,156,237,106)(29,167,238,117)(30,178,239,98)(31,207,138,80)(32,188,139,61)(33,199,140,72)(34,210,141,83)(35,191,142,64)(36,202,143,75)(37,183,144,86)(38,194,145,67)(39,205,146,78)(40,186,147,89)(41,197,148,70)(42,208,149,81)(43,189,150,62)(44,200,121,73)(45,181,122,84)(46,192,123,65)(47,203,124,76)(48,184,125,87)(49,195,126,68)(50,206,127,79)(51,187,128,90)(52,198,129,71)(53,209,130,82)(54,190,131,63)(55,201,132,74)(56,182,133,85)(57,193,134,66)(58,204,135,77)(59,185,136,88)(60,196,137,69) );
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)], [(1,239),(2,238),(3,237),(4,236),(5,235),(6,234),(7,233),(8,232),(9,231),(10,230),(11,229),(12,228),(13,227),(14,226),(15,225),(16,224),(17,223),(18,222),(19,221),(20,220),(21,219),(22,218),(23,217),(24,216),(25,215),(26,214),(27,213),(28,212),(29,211),(30,240),(31,142),(32,141),(33,140),(34,139),(35,138),(36,137),(37,136),(38,135),(39,134),(40,133),(41,132),(42,131),(43,130),(44,129),(45,128),(46,127),(47,126),(48,125),(49,124),(50,123),(51,122),(52,121),(53,150),(54,149),(55,148),(56,147),(57,146),(58,145),(59,144),(60,143),(61,190),(62,189),(63,188),(64,187),(65,186),(66,185),(67,184),(68,183),(69,182),(70,181),(71,210),(72,209),(73,208),(74,207),(75,206),(76,205),(77,204),(78,203),(79,202),(80,201),(81,200),(82,199),(83,198),(84,197),(85,196),(86,195),(87,194),(88,193),(89,192),(90,191),(91,176),(92,175),(93,174),(94,173),(95,172),(96,171),(97,170),(98,169),(99,168),(100,167),(101,166),(102,165),(103,164),(104,163),(105,162),(106,161),(107,160),(108,159),(109,158),(110,157),(111,156),(112,155),(113,154),(114,153),(115,152),(116,151),(117,180),(118,179),(119,178),(120,177)], [(1,182,240,85),(2,181,211,84),(3,210,212,83),(4,209,213,82),(5,208,214,81),(6,207,215,80),(7,206,216,79),(8,205,217,78),(9,204,218,77),(10,203,219,76),(11,202,220,75),(12,201,221,74),(13,200,222,73),(14,199,223,72),(15,198,224,71),(16,197,225,70),(17,196,226,69),(18,195,227,68),(19,194,228,67),(20,193,229,66),(21,192,230,65),(22,191,231,64),(23,190,232,63),(24,189,233,62),(25,188,234,61),(26,187,235,90),(27,186,236,89),(28,185,237,88),(29,184,238,87),(30,183,239,86),(31,104,138,154),(32,103,139,153),(33,102,140,152),(34,101,141,151),(35,100,142,180),(36,99,143,179),(37,98,144,178),(38,97,145,177),(39,96,146,176),(40,95,147,175),(41,94,148,174),(42,93,149,173),(43,92,150,172),(44,91,121,171),(45,120,122,170),(46,119,123,169),(47,118,124,168),(48,117,125,167),(49,116,126,166),(50,115,127,165),(51,114,128,164),(52,113,129,163),(53,112,130,162),(54,111,131,161),(55,110,132,160),(56,109,133,159),(57,108,134,158),(58,107,135,157),(59,106,136,156),(60,105,137,155)], [(1,159,240,109),(2,170,211,120),(3,151,212,101),(4,162,213,112),(5,173,214,93),(6,154,215,104),(7,165,216,115),(8,176,217,96),(9,157,218,107),(10,168,219,118),(11,179,220,99),(12,160,221,110),(13,171,222,91),(14,152,223,102),(15,163,224,113),(16,174,225,94),(17,155,226,105),(18,166,227,116),(19,177,228,97),(20,158,229,108),(21,169,230,119),(22,180,231,100),(23,161,232,111),(24,172,233,92),(25,153,234,103),(26,164,235,114),(27,175,236,95),(28,156,237,106),(29,167,238,117),(30,178,239,98),(31,207,138,80),(32,188,139,61),(33,199,140,72),(34,210,141,83),(35,191,142,64),(36,202,143,75),(37,183,144,86),(38,194,145,67),(39,205,146,78),(40,186,147,89),(41,197,148,70),(42,208,149,81),(43,189,150,62),(44,200,121,73),(45,181,122,84),(46,192,123,65),(47,203,124,76),(48,184,125,87),(49,195,126,68),(50,206,127,79),(51,187,128,90),(52,198,129,71),(53,209,130,82),(54,190,131,63),(55,201,132,74),(56,182,133,85),(57,193,134,66),(58,204,135,77),(59,185,136,88),(60,196,137,69)]])
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 | 4 | 6 | 6 | 10 | 10 | 12 | 20 | 60 | 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 | 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 | C2 | C2 | S3 | D4 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D20 | C4○D12 | S3×D4 | S3×Q8 | S3×D5 | D4⋊2D5 | Q8×D5 | C2×S3×D5 | D15⋊Q8 | S3×D20 | Dic3.D10 |
| kernel | D30⋊4Q8 | D30⋊4C4 | C6.Dic10 | C3×C4⋊Dic5 | C5×Dic3⋊C4 | D30⋊3C4 | C2×D30.C2 | C2×C15⋊Q8 | C4⋊Dic5 | C5×Dic3 | D30 | Dic3⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | Dic3 | C10 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 8 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D30⋊4Q8 ►in GL4(𝔽61) generated by
| 17 | 1 | 0 | 0 |
| 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 1 |
| 0 | 43 | 0 | 0 |
| 44 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 2 | 25 | 0 | 0 |
| 56 | 59 | 0 | 0 |
| 0 | 0 | 9 | 18 |
| 0 | 0 | 9 | 52 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 11 | 11 |
G:=sub<GL(4,GF(61))| [17,16,0,0,1,1,0,0,0,0,0,1,0,0,60,1],[0,44,0,0,43,0,0,0,0,0,0,1,0,0,1,0],[2,56,0,0,25,59,0,0,0,0,9,9,0,0,18,52],[60,0,0,0,0,60,0,0,0,0,50,11,0,0,0,11] >;
D30⋊4Q8 in GAP, Magma, Sage, TeX
D_{30}\rtimes_4Q_8 % in TeX
G:=Group("D30:4Q8"); // GroupNames label
G:=SmallGroup(480,505);
// by ID
G=gap.SmallGroup(480,505);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=a^11,c*b*c^-1=a^13*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations