metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊3Q8, C6.15(Q8×D5), (C2×Dic6)⋊5D5, C5⋊6(D6⋊Q8), C30.44(C2×Q8), C10.15(S3×Q8), C3⋊1(D10⋊3Q8), (C2×C20).228D6, (C2×C12).21D10, C10.132(S3×D4), C30.136(C2×D4), C15⋊16(C22⋊Q8), (C5×Dic3).8D4, (C10×Dic6)⋊15C2, C30.66(C4○D4), C10.D4⋊19S3, C2.17(D15⋊Q8), Dic15⋊5C4⋊18C2, C6.Dic10⋊19C2, (C2×Dic5).36D6, D30⋊4C4.13C2, D30⋊3C4.11C2, C10.12(C4○D12), (C2×C30).114C23, (C2×C60).321C22, C6.14(Q8⋊2D5), (C2×Dic3).35D10, Dic3.9(C5⋊D4), C2.15(C12.28D10), (C6×Dic5).68C22, (C2×Dic15).92C22, (C10×Dic3).71C22, (C22×D15).38C22, (C2×C4).50(S3×D5), C2.14(S3×C5⋊D4), C6.33(C2×C5⋊D4), C22.180(C2×S3×D5), (C2×D30.C2).6C2, (C3×C10.D4)⋊22C2, (C2×C6).126(C22×D5), (C2×C10).126(C22×S3), SmallGroup(480,500)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊3Q8
G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a11, cbc-1=a15b, dbd-1=a10b, dcd-1=c-1 >
Subgroups: 780 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, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, D30, D30, C2×C30, C10.D4, C10.D4, C4⋊Dic5, D10⋊C4, C2×C4×D5, Q8×C10, D6⋊Q8, D30.C2, C6×Dic5, C5×Dic6, C10×Dic3, C2×Dic15, C2×C60, C22×D15, D10⋊3Q8, D30⋊4C4, Dic15⋊5C4, C6.Dic10, C3×C10.D4, D30⋊3C4, C2×D30.C2, C10×Dic6, D30⋊3Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, C22×S3, C22⋊Q8, C5⋊D4, C22×D5, C4○D12, S3×D4, S3×Q8, S3×D5, Q8×D5, Q8⋊2D5, C2×C5⋊D4, D6⋊Q8, C2×S3×D5, D10⋊3Q8, D15⋊Q8, C12.28D10, S3×C5⋊D4, D30⋊3Q8
►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)
(1 190)(2 189)(3 188)(4 187)(5 186)(6 185)(7 184)(8 183)(9 182)(10 181)(11 210)(12 209)(13 208)(14 207)(15 206)(16 205)(17 204)(18 203)(19 202)(20 201)(21 200)(22 199)(23 198)(24 197)(25 196)(26 195)(27 194)(28 193)(29 192)(30 191)(31 232)(32 231)(33 230)(34 229)(35 228)(36 227)(37 226)(38 225)(39 224)(40 223)(41 222)(42 221)(43 220)(44 219)(45 218)(46 217)(47 216)(48 215)(49 214)(50 213)(51 212)(52 211)(53 240)(54 239)(55 238)(56 237)(57 236)(58 235)(59 234)(60 233)(61 156)(62 155)(63 154)(64 153)(65 152)(66 151)(67 180)(68 179)(69 178)(70 177)(71 176)(72 175)(73 174)(74 173)(75 172)(76 171)(77 170)(78 169)(79 168)(80 167)(81 166)(82 165)(83 164)(84 163)(85 162)(86 161)(87 160)(88 159)(89 158)(90 157)(91 133)(92 132)(93 131)(94 130)(95 129)(96 128)(97 127)(98 126)(99 125)(100 124)(101 123)(102 122)(103 121)(104 150)(105 149)(106 148)(107 147)(108 146)(109 145)(110 144)(111 143)(112 142)(113 141)(114 140)(115 139)(116 138)(117 137)(118 136)(119 135)(120 134)
(1 129 191 111)(2 130 192 112)(3 131 193 113)(4 132 194 114)(5 133 195 115)(6 134 196 116)(7 135 197 117)(8 136 198 118)(9 137 199 119)(10 138 200 120)(11 139 201 91)(12 140 202 92)(13 141 203 93)(14 142 204 94)(15 143 205 95)(16 144 206 96)(17 145 207 97)(18 146 208 98)(19 147 209 99)(20 148 210 100)(21 149 181 101)(22 150 182 102)(23 121 183 103)(24 122 184 104)(25 123 185 105)(26 124 186 106)(27 125 187 107)(28 126 188 108)(29 127 189 109)(30 128 190 110)(31 161 240 79)(32 162 211 80)(33 163 212 81)(34 164 213 82)(35 165 214 83)(36 166 215 84)(37 167 216 85)(38 168 217 86)(39 169 218 87)(40 170 219 88)(41 171 220 89)(42 172 221 90)(43 173 222 61)(44 174 223 62)(45 175 224 63)(46 176 225 64)(47 177 226 65)(48 178 227 66)(49 179 228 67)(50 180 229 68)(51 151 230 69)(52 152 231 70)(53 153 232 71)(54 154 233 72)(55 155 234 73)(56 156 235 74)(57 157 236 75)(58 158 237 76)(59 159 238 77)(60 160 239 78)
(1 165 191 83)(2 176 192 64)(3 157 193 75)(4 168 194 86)(5 179 195 67)(6 160 196 78)(7 171 197 89)(8 152 198 70)(9 163 199 81)(10 174 200 62)(11 155 201 73)(12 166 202 84)(13 177 203 65)(14 158 204 76)(15 169 205 87)(16 180 206 68)(17 161 207 79)(18 172 208 90)(19 153 209 71)(20 164 210 82)(21 175 181 63)(22 156 182 74)(23 167 183 85)(24 178 184 66)(25 159 185 77)(26 170 186 88)(27 151 187 69)(28 162 188 80)(29 173 189 61)(30 154 190 72)(31 97 240 145)(32 108 211 126)(33 119 212 137)(34 100 213 148)(35 111 214 129)(36 92 215 140)(37 103 216 121)(38 114 217 132)(39 95 218 143)(40 106 219 124)(41 117 220 135)(42 98 221 146)(43 109 222 127)(44 120 223 138)(45 101 224 149)(46 112 225 130)(47 93 226 141)(48 104 227 122)(49 115 228 133)(50 96 229 144)(51 107 230 125)(52 118 231 136)(53 99 232 147)(54 110 233 128)(55 91 234 139)(56 102 235 150)(57 113 236 131)(58 94 237 142)(59 105 238 123)(60 116 239 134)
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), (1,190)(2,189)(3,188)(4,187)(5,186)(6,185)(7,184)(8,183)(9,182)(10,181)(11,210)(12,209)(13,208)(14,207)(15,206)(16,205)(17,204)(18,203)(19,202)(20,201)(21,200)(22,199)(23,198)(24,197)(25,196)(26,195)(27,194)(28,193)(29,192)(30,191)(31,232)(32,231)(33,230)(34,229)(35,228)(36,227)(37,226)(38,225)(39,224)(40,223)(41,222)(42,221)(43,220)(44,219)(45,218)(46,217)(47,216)(48,215)(49,214)(50,213)(51,212)(52,211)(53,240)(54,239)(55,238)(56,237)(57,236)(58,235)(59,234)(60,233)(61,156)(62,155)(63,154)(64,153)(65,152)(66,151)(67,180)(68,179)(69,178)(70,177)(71,176)(72,175)(73,174)(74,173)(75,172)(76,171)(77,170)(78,169)(79,168)(80,167)(81,166)(82,165)(83,164)(84,163)(85,162)(86,161)(87,160)(88,159)(89,158)(90,157)(91,133)(92,132)(93,131)(94,130)(95,129)(96,128)(97,127)(98,126)(99,125)(100,124)(101,123)(102,122)(103,121)(104,150)(105,149)(106,148)(107,147)(108,146)(109,145)(110,144)(111,143)(112,142)(113,141)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134), (1,129,191,111)(2,130,192,112)(3,131,193,113)(4,132,194,114)(5,133,195,115)(6,134,196,116)(7,135,197,117)(8,136,198,118)(9,137,199,119)(10,138,200,120)(11,139,201,91)(12,140,202,92)(13,141,203,93)(14,142,204,94)(15,143,205,95)(16,144,206,96)(17,145,207,97)(18,146,208,98)(19,147,209,99)(20,148,210,100)(21,149,181,101)(22,150,182,102)(23,121,183,103)(24,122,184,104)(25,123,185,105)(26,124,186,106)(27,125,187,107)(28,126,188,108)(29,127,189,109)(30,128,190,110)(31,161,240,79)(32,162,211,80)(33,163,212,81)(34,164,213,82)(35,165,214,83)(36,166,215,84)(37,167,216,85)(38,168,217,86)(39,169,218,87)(40,170,219,88)(41,171,220,89)(42,172,221,90)(43,173,222,61)(44,174,223,62)(45,175,224,63)(46,176,225,64)(47,177,226,65)(48,178,227,66)(49,179,228,67)(50,180,229,68)(51,151,230,69)(52,152,231,70)(53,153,232,71)(54,154,233,72)(55,155,234,73)(56,156,235,74)(57,157,236,75)(58,158,237,76)(59,159,238,77)(60,160,239,78), (1,165,191,83)(2,176,192,64)(3,157,193,75)(4,168,194,86)(5,179,195,67)(6,160,196,78)(7,171,197,89)(8,152,198,70)(9,163,199,81)(10,174,200,62)(11,155,201,73)(12,166,202,84)(13,177,203,65)(14,158,204,76)(15,169,205,87)(16,180,206,68)(17,161,207,79)(18,172,208,90)(19,153,209,71)(20,164,210,82)(21,175,181,63)(22,156,182,74)(23,167,183,85)(24,178,184,66)(25,159,185,77)(26,170,186,88)(27,151,187,69)(28,162,188,80)(29,173,189,61)(30,154,190,72)(31,97,240,145)(32,108,211,126)(33,119,212,137)(34,100,213,148)(35,111,214,129)(36,92,215,140)(37,103,216,121)(38,114,217,132)(39,95,218,143)(40,106,219,124)(41,117,220,135)(42,98,221,146)(43,109,222,127)(44,120,223,138)(45,101,224,149)(46,112,225,130)(47,93,226,141)(48,104,227,122)(49,115,228,133)(50,96,229,144)(51,107,230,125)(52,118,231,136)(53,99,232,147)(54,110,233,128)(55,91,234,139)(56,102,235,150)(57,113,236,131)(58,94,237,142)(59,105,238,123)(60,116,239,134)>;
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), (1,190)(2,189)(3,188)(4,187)(5,186)(6,185)(7,184)(8,183)(9,182)(10,181)(11,210)(12,209)(13,208)(14,207)(15,206)(16,205)(17,204)(18,203)(19,202)(20,201)(21,200)(22,199)(23,198)(24,197)(25,196)(26,195)(27,194)(28,193)(29,192)(30,191)(31,232)(32,231)(33,230)(34,229)(35,228)(36,227)(37,226)(38,225)(39,224)(40,223)(41,222)(42,221)(43,220)(44,219)(45,218)(46,217)(47,216)(48,215)(49,214)(50,213)(51,212)(52,211)(53,240)(54,239)(55,238)(56,237)(57,236)(58,235)(59,234)(60,233)(61,156)(62,155)(63,154)(64,153)(65,152)(66,151)(67,180)(68,179)(69,178)(70,177)(71,176)(72,175)(73,174)(74,173)(75,172)(76,171)(77,170)(78,169)(79,168)(80,167)(81,166)(82,165)(83,164)(84,163)(85,162)(86,161)(87,160)(88,159)(89,158)(90,157)(91,133)(92,132)(93,131)(94,130)(95,129)(96,128)(97,127)(98,126)(99,125)(100,124)(101,123)(102,122)(103,121)(104,150)(105,149)(106,148)(107,147)(108,146)(109,145)(110,144)(111,143)(112,142)(113,141)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134), (1,129,191,111)(2,130,192,112)(3,131,193,113)(4,132,194,114)(5,133,195,115)(6,134,196,116)(7,135,197,117)(8,136,198,118)(9,137,199,119)(10,138,200,120)(11,139,201,91)(12,140,202,92)(13,141,203,93)(14,142,204,94)(15,143,205,95)(16,144,206,96)(17,145,207,97)(18,146,208,98)(19,147,209,99)(20,148,210,100)(21,149,181,101)(22,150,182,102)(23,121,183,103)(24,122,184,104)(25,123,185,105)(26,124,186,106)(27,125,187,107)(28,126,188,108)(29,127,189,109)(30,128,190,110)(31,161,240,79)(32,162,211,80)(33,163,212,81)(34,164,213,82)(35,165,214,83)(36,166,215,84)(37,167,216,85)(38,168,217,86)(39,169,218,87)(40,170,219,88)(41,171,220,89)(42,172,221,90)(43,173,222,61)(44,174,223,62)(45,175,224,63)(46,176,225,64)(47,177,226,65)(48,178,227,66)(49,179,228,67)(50,180,229,68)(51,151,230,69)(52,152,231,70)(53,153,232,71)(54,154,233,72)(55,155,234,73)(56,156,235,74)(57,157,236,75)(58,158,237,76)(59,159,238,77)(60,160,239,78), (1,165,191,83)(2,176,192,64)(3,157,193,75)(4,168,194,86)(5,179,195,67)(6,160,196,78)(7,171,197,89)(8,152,198,70)(9,163,199,81)(10,174,200,62)(11,155,201,73)(12,166,202,84)(13,177,203,65)(14,158,204,76)(15,169,205,87)(16,180,206,68)(17,161,207,79)(18,172,208,90)(19,153,209,71)(20,164,210,82)(21,175,181,63)(22,156,182,74)(23,167,183,85)(24,178,184,66)(25,159,185,77)(26,170,186,88)(27,151,187,69)(28,162,188,80)(29,173,189,61)(30,154,190,72)(31,97,240,145)(32,108,211,126)(33,119,212,137)(34,100,213,148)(35,111,214,129)(36,92,215,140)(37,103,216,121)(38,114,217,132)(39,95,218,143)(40,106,219,124)(41,117,220,135)(42,98,221,146)(43,109,222,127)(44,120,223,138)(45,101,224,149)(46,112,225,130)(47,93,226,141)(48,104,227,122)(49,115,228,133)(50,96,229,144)(51,107,230,125)(52,118,231,136)(53,99,232,147)(54,110,233,128)(55,91,234,139)(56,102,235,150)(57,113,236,131)(58,94,237,142)(59,105,238,123)(60,116,239,134) );
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)], [(1,190),(2,189),(3,188),(4,187),(5,186),(6,185),(7,184),(8,183),(9,182),(10,181),(11,210),(12,209),(13,208),(14,207),(15,206),(16,205),(17,204),(18,203),(19,202),(20,201),(21,200),(22,199),(23,198),(24,197),(25,196),(26,195),(27,194),(28,193),(29,192),(30,191),(31,232),(32,231),(33,230),(34,229),(35,228),(36,227),(37,226),(38,225),(39,224),(40,223),(41,222),(42,221),(43,220),(44,219),(45,218),(46,217),(47,216),(48,215),(49,214),(50,213),(51,212),(52,211),(53,240),(54,239),(55,238),(56,237),(57,236),(58,235),(59,234),(60,233),(61,156),(62,155),(63,154),(64,153),(65,152),(66,151),(67,180),(68,179),(69,178),(70,177),(71,176),(72,175),(73,174),(74,173),(75,172),(76,171),(77,170),(78,169),(79,168),(80,167),(81,166),(82,165),(83,164),(84,163),(85,162),(86,161),(87,160),(88,159),(89,158),(90,157),(91,133),(92,132),(93,131),(94,130),(95,129),(96,128),(97,127),(98,126),(99,125),(100,124),(101,123),(102,122),(103,121),(104,150),(105,149),(106,148),(107,147),(108,146),(109,145),(110,144),(111,143),(112,142),(113,141),(114,140),(115,139),(116,138),(117,137),(118,136),(119,135),(120,134)], [(1,129,191,111),(2,130,192,112),(3,131,193,113),(4,132,194,114),(5,133,195,115),(6,134,196,116),(7,135,197,117),(8,136,198,118),(9,137,199,119),(10,138,200,120),(11,139,201,91),(12,140,202,92),(13,141,203,93),(14,142,204,94),(15,143,205,95),(16,144,206,96),(17,145,207,97),(18,146,208,98),(19,147,209,99),(20,148,210,100),(21,149,181,101),(22,150,182,102),(23,121,183,103),(24,122,184,104),(25,123,185,105),(26,124,186,106),(27,125,187,107),(28,126,188,108),(29,127,189,109),(30,128,190,110),(31,161,240,79),(32,162,211,80),(33,163,212,81),(34,164,213,82),(35,165,214,83),(36,166,215,84),(37,167,216,85),(38,168,217,86),(39,169,218,87),(40,170,219,88),(41,171,220,89),(42,172,221,90),(43,173,222,61),(44,174,223,62),(45,175,224,63),(46,176,225,64),(47,177,226,65),(48,178,227,66),(49,179,228,67),(50,180,229,68),(51,151,230,69),(52,152,231,70),(53,153,232,71),(54,154,233,72),(55,155,234,73),(56,156,235,74),(57,157,236,75),(58,158,237,76),(59,159,238,77),(60,160,239,78)], [(1,165,191,83),(2,176,192,64),(3,157,193,75),(4,168,194,86),(5,179,195,67),(6,160,196,78),(7,171,197,89),(8,152,198,70),(9,163,199,81),(10,174,200,62),(11,155,201,73),(12,166,202,84),(13,177,203,65),(14,158,204,76),(15,169,205,87),(16,180,206,68),(17,161,207,79),(18,172,208,90),(19,153,209,71),(20,164,210,82),(21,175,181,63),(22,156,182,74),(23,167,183,85),(24,178,184,66),(25,159,185,77),(26,170,186,88),(27,151,187,69),(28,162,188,80),(29,173,189,61),(30,154,190,72),(31,97,240,145),(32,108,211,126),(33,119,212,137),(34,100,213,148),(35,111,214,129),(36,92,215,140),(37,103,216,121),(38,114,217,132),(39,95,218,143),(40,106,219,124),(41,117,220,135),(42,98,221,146),(43,109,222,127),(44,120,223,138),(45,101,224,149),(46,112,225,130),(47,93,226,141),(48,104,227,122),(49,115,228,133),(50,96,229,144),(51,107,230,125),(52,118,231,136),(53,99,232,147),(54,110,233,128),(55,91,234,139),(56,102,235,150),(57,113,236,131),(58,94,237,142),(59,105,238,123),(60,116,239,134)]])
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 | C5⋊D4 | C4○D12 | S3×D4 | S3×Q8 | S3×D5 | Q8×D5 | Q8⋊2D5 | C2×S3×D5 | D15⋊Q8 | C12.28D10 | S3×C5⋊D4 |
| kernel | D30⋊3Q8 | D30⋊4C4 | Dic15⋊5C4 | C6.Dic10 | C3×C10.D4 | D30⋊3C4 | C2×D30.C2 | C10×Dic6 | C10.D4 | C5×Dic3 | D30 | C2×Dic6 | 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⋊3Q8 ►in GL4(𝔽61) generated by
| 1 | 19 | 0 | 0 |
| 60 | 43 | 0 | 0 |
| 0 | 0 | 1 | 60 |
| 0 | 0 | 1 | 0 |
| 0 | 43 | 0 | 0 |
| 44 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 60 | 1 |
| 30 | 60 | 0 | 0 |
| 45 | 31 | 0 | 0 |
| 0 | 0 | 38 | 46 |
| 0 | 0 | 15 | 23 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 11 | 0 |
G:=sub<GL(4,GF(61))| [1,60,0,0,19,43,0,0,0,0,1,1,0,0,60,0],[0,44,0,0,43,0,0,0,0,0,60,60,0,0,0,1],[30,45,0,0,60,31,0,0,0,0,38,15,0,0,46,23],[1,0,0,0,0,1,0,0,0,0,0,11,0,0,11,0] >;
D30⋊3Q8 in GAP, Magma, Sage, TeX
D_{30}\rtimes_3Q_8 % in TeX
G:=Group("D30:3Q8"); // GroupNames label
G:=SmallGroup(480,500);
// by ID
G=gap.SmallGroup(480,500);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,219,100,1356,18822]);
// 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,c*b*c^-1=a^15*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations