metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊2Q8, Dic5.19D12, C4⋊Dic3⋊7D5, C6.20(D4×D5), C6.14(Q8×D5), C5⋊1(C4.D12), C2.21(D5×D12), C30.44(C2×D4), C30.41(C2×Q8), C10.14(S3×Q8), C3⋊2(D10⋊Q8), C10.20(C2×D12), (C2×C12).16D10, (C2×C20).225D6, C15⋊13(C22⋊Q8), C6.72(C4○D20), C10.D4⋊17S3, C2.16(D15⋊Q8), C30.Q8⋊22C2, (C3×Dic5).10D4, (C2×Dic5).32D6, D30⋊3C4.10C2, D30⋊4C4.12C2, C30.119(C4○D4), (C2×C30).109C23, (C2×C60).318C22, (C2×Dic3).34D10, C10.46(D4⋊2S3), (C6×Dic5).63C22, C2.18(Dic5.D6), (C2×Dic15).88C22, (C10×Dic3).67C22, (C22×D15).36C22, (C2×C15⋊Q8)⋊7C2, (C2×C4).45(S3×D5), (C5×C4⋊Dic3)⋊18C2, C22.175(C2×S3×D5), (C2×D30.C2).5C2, (C3×C10.D4)⋊20C2, (C2×C6).121(C22×D5), (C2×C10).121(C22×S3), SmallGroup(480,495)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊2Q8
G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=cac-1=a-1, dad-1=a19, cbc-1=a13b, dbd-1=a18b, 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, C12, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, 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, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, D30, D30, C2×C30, C10.D4, C10.D4, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C4.D12, D30.C2, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C22×D15, D10⋊Q8, D30⋊4C4, C30.Q8, C3×C10.D4, C5×C4⋊Dic3, D30⋊3C4, C2×D30.C2, C2×C15⋊Q8, D30⋊2Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, D12, C22×S3, C22⋊Q8, C22×D5, C2×D12, D4⋊2S3, S3×Q8, S3×D5, C4○D20, D4×D5, Q8×D5, C4.D12, C2×S3×D5, D10⋊Q8, D15⋊Q8, D5×D12, Dic5.D6, D30⋊2Q8
►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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 45)(58 60)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)(81 84)(82 83)(91 119)(92 118)(93 117)(94 116)(95 115)(96 114)(97 113)(98 112)(99 111)(100 110)(101 109)(102 108)(103 107)(104 106)(121 144)(122 143)(123 142)(124 141)(125 140)(126 139)(127 138)(128 137)(129 136)(130 135)(131 134)(132 133)(145 150)(146 149)(147 148)(151 177)(152 176)(153 175)(154 174)(155 173)(156 172)(157 171)(158 170)(159 169)(160 168)(161 167)(162 166)(163 165)(178 180)(181 190)(182 189)(183 188)(184 187)(185 186)(191 210)(192 209)(193 208)(194 207)(195 206)(196 205)(197 204)(198 203)(199 202)(200 201)(211 221)(212 220)(213 219)(214 218)(215 217)(222 240)(223 239)(224 238)(225 237)(226 236)(227 235)(228 234)(229 233)(230 232)
(1 37 68 239)(2 36 69 238)(3 35 70 237)(4 34 71 236)(5 33 72 235)(6 32 73 234)(7 31 74 233)(8 60 75 232)(9 59 76 231)(10 58 77 230)(11 57 78 229)(12 56 79 228)(13 55 80 227)(14 54 81 226)(15 53 82 225)(16 52 83 224)(17 51 84 223)(18 50 85 222)(19 49 86 221)(20 48 87 220)(21 47 88 219)(22 46 89 218)(23 45 90 217)(24 44 61 216)(25 43 62 215)(26 42 63 214)(27 41 64 213)(28 40 65 212)(29 39 66 211)(30 38 67 240)(91 193 165 140)(92 192 166 139)(93 191 167 138)(94 190 168 137)(95 189 169 136)(96 188 170 135)(97 187 171 134)(98 186 172 133)(99 185 173 132)(100 184 174 131)(101 183 175 130)(102 182 176 129)(103 181 177 128)(104 210 178 127)(105 209 179 126)(106 208 180 125)(107 207 151 124)(108 206 152 123)(109 205 153 122)(110 204 154 121)(111 203 155 150)(112 202 156 149)(113 201 157 148)(114 200 158 147)(115 199 159 146)(116 198 160 145)(117 197 161 144)(118 196 162 143)(119 195 163 142)(120 194 164 141)
(1 133 68 186)(2 122 69 205)(3 141 70 194)(4 130 71 183)(5 149 72 202)(6 138 73 191)(7 127 74 210)(8 146 75 199)(9 135 76 188)(10 124 77 207)(11 143 78 196)(12 132 79 185)(13 121 80 204)(14 140 81 193)(15 129 82 182)(16 148 83 201)(17 137 84 190)(18 126 85 209)(19 145 86 198)(20 134 87 187)(21 123 88 206)(22 142 89 195)(23 131 90 184)(24 150 61 203)(25 139 62 192)(26 128 63 181)(27 147 64 200)(28 136 65 189)(29 125 66 208)(30 144 67 197)(31 178 233 104)(32 167 234 93)(33 156 235 112)(34 175 236 101)(35 164 237 120)(36 153 238 109)(37 172 239 98)(38 161 240 117)(39 180 211 106)(40 169 212 95)(41 158 213 114)(42 177 214 103)(43 166 215 92)(44 155 216 111)(45 174 217 100)(46 163 218 119)(47 152 219 108)(48 171 220 97)(49 160 221 116)(50 179 222 105)(51 168 223 94)(52 157 224 113)(53 176 225 102)(54 165 226 91)(55 154 227 110)(56 173 228 99)(57 162 229 118)(58 151 230 107)(59 170 231 96)(60 159 232 115)
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,45)(58,60)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)(91,119)(92,118)(93,117)(94,116)(95,115)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)(121,144)(122,143)(123,142)(124,141)(125,140)(126,139)(127,138)(128,137)(129,136)(130,135)(131,134)(132,133)(145,150)(146,149)(147,148)(151,177)(152,176)(153,175)(154,174)(155,173)(156,172)(157,171)(158,170)(159,169)(160,168)(161,167)(162,166)(163,165)(178,180)(181,190)(182,189)(183,188)(184,187)(185,186)(191,210)(192,209)(193,208)(194,207)(195,206)(196,205)(197,204)(198,203)(199,202)(200,201)(211,221)(212,220)(213,219)(214,218)(215,217)(222,240)(223,239)(224,238)(225,237)(226,236)(227,235)(228,234)(229,233)(230,232), (1,37,68,239)(2,36,69,238)(3,35,70,237)(4,34,71,236)(5,33,72,235)(6,32,73,234)(7,31,74,233)(8,60,75,232)(9,59,76,231)(10,58,77,230)(11,57,78,229)(12,56,79,228)(13,55,80,227)(14,54,81,226)(15,53,82,225)(16,52,83,224)(17,51,84,223)(18,50,85,222)(19,49,86,221)(20,48,87,220)(21,47,88,219)(22,46,89,218)(23,45,90,217)(24,44,61,216)(25,43,62,215)(26,42,63,214)(27,41,64,213)(28,40,65,212)(29,39,66,211)(30,38,67,240)(91,193,165,140)(92,192,166,139)(93,191,167,138)(94,190,168,137)(95,189,169,136)(96,188,170,135)(97,187,171,134)(98,186,172,133)(99,185,173,132)(100,184,174,131)(101,183,175,130)(102,182,176,129)(103,181,177,128)(104,210,178,127)(105,209,179,126)(106,208,180,125)(107,207,151,124)(108,206,152,123)(109,205,153,122)(110,204,154,121)(111,203,155,150)(112,202,156,149)(113,201,157,148)(114,200,158,147)(115,199,159,146)(116,198,160,145)(117,197,161,144)(118,196,162,143)(119,195,163,142)(120,194,164,141), (1,133,68,186)(2,122,69,205)(3,141,70,194)(4,130,71,183)(5,149,72,202)(6,138,73,191)(7,127,74,210)(8,146,75,199)(9,135,76,188)(10,124,77,207)(11,143,78,196)(12,132,79,185)(13,121,80,204)(14,140,81,193)(15,129,82,182)(16,148,83,201)(17,137,84,190)(18,126,85,209)(19,145,86,198)(20,134,87,187)(21,123,88,206)(22,142,89,195)(23,131,90,184)(24,150,61,203)(25,139,62,192)(26,128,63,181)(27,147,64,200)(28,136,65,189)(29,125,66,208)(30,144,67,197)(31,178,233,104)(32,167,234,93)(33,156,235,112)(34,175,236,101)(35,164,237,120)(36,153,238,109)(37,172,239,98)(38,161,240,117)(39,180,211,106)(40,169,212,95)(41,158,213,114)(42,177,214,103)(43,166,215,92)(44,155,216,111)(45,174,217,100)(46,163,218,119)(47,152,219,108)(48,171,220,97)(49,160,221,116)(50,179,222,105)(51,168,223,94)(52,157,224,113)(53,176,225,102)(54,165,226,91)(55,154,227,110)(56,173,228,99)(57,162,229,118)(58,151,230,107)(59,170,231,96)(60,159,232,115)>;
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,45)(58,60)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)(91,119)(92,118)(93,117)(94,116)(95,115)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)(121,144)(122,143)(123,142)(124,141)(125,140)(126,139)(127,138)(128,137)(129,136)(130,135)(131,134)(132,133)(145,150)(146,149)(147,148)(151,177)(152,176)(153,175)(154,174)(155,173)(156,172)(157,171)(158,170)(159,169)(160,168)(161,167)(162,166)(163,165)(178,180)(181,190)(182,189)(183,188)(184,187)(185,186)(191,210)(192,209)(193,208)(194,207)(195,206)(196,205)(197,204)(198,203)(199,202)(200,201)(211,221)(212,220)(213,219)(214,218)(215,217)(222,240)(223,239)(224,238)(225,237)(226,236)(227,235)(228,234)(229,233)(230,232), (1,37,68,239)(2,36,69,238)(3,35,70,237)(4,34,71,236)(5,33,72,235)(6,32,73,234)(7,31,74,233)(8,60,75,232)(9,59,76,231)(10,58,77,230)(11,57,78,229)(12,56,79,228)(13,55,80,227)(14,54,81,226)(15,53,82,225)(16,52,83,224)(17,51,84,223)(18,50,85,222)(19,49,86,221)(20,48,87,220)(21,47,88,219)(22,46,89,218)(23,45,90,217)(24,44,61,216)(25,43,62,215)(26,42,63,214)(27,41,64,213)(28,40,65,212)(29,39,66,211)(30,38,67,240)(91,193,165,140)(92,192,166,139)(93,191,167,138)(94,190,168,137)(95,189,169,136)(96,188,170,135)(97,187,171,134)(98,186,172,133)(99,185,173,132)(100,184,174,131)(101,183,175,130)(102,182,176,129)(103,181,177,128)(104,210,178,127)(105,209,179,126)(106,208,180,125)(107,207,151,124)(108,206,152,123)(109,205,153,122)(110,204,154,121)(111,203,155,150)(112,202,156,149)(113,201,157,148)(114,200,158,147)(115,199,159,146)(116,198,160,145)(117,197,161,144)(118,196,162,143)(119,195,163,142)(120,194,164,141), (1,133,68,186)(2,122,69,205)(3,141,70,194)(4,130,71,183)(5,149,72,202)(6,138,73,191)(7,127,74,210)(8,146,75,199)(9,135,76,188)(10,124,77,207)(11,143,78,196)(12,132,79,185)(13,121,80,204)(14,140,81,193)(15,129,82,182)(16,148,83,201)(17,137,84,190)(18,126,85,209)(19,145,86,198)(20,134,87,187)(21,123,88,206)(22,142,89,195)(23,131,90,184)(24,150,61,203)(25,139,62,192)(26,128,63,181)(27,147,64,200)(28,136,65,189)(29,125,66,208)(30,144,67,197)(31,178,233,104)(32,167,234,93)(33,156,235,112)(34,175,236,101)(35,164,237,120)(36,153,238,109)(37,172,239,98)(38,161,240,117)(39,180,211,106)(40,169,212,95)(41,158,213,114)(42,177,214,103)(43,166,215,92)(44,155,216,111)(45,174,217,100)(46,163,218,119)(47,152,219,108)(48,171,220,97)(49,160,221,116)(50,179,222,105)(51,168,223,94)(52,157,224,113)(53,176,225,102)(54,165,226,91)(55,154,227,110)(56,173,228,99)(57,162,229,118)(58,151,230,107)(59,170,231,96)(60,159,232,115) );
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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,57),(32,56),(33,55),(34,54),(35,53),(36,52),(37,51),(38,50),(39,49),(40,48),(41,47),(42,46),(43,45),(58,60),(61,74),(62,73),(63,72),(64,71),(65,70),(66,69),(67,68),(75,90),(76,89),(77,88),(78,87),(79,86),(80,85),(81,84),(82,83),(91,119),(92,118),(93,117),(94,116),(95,115),(96,114),(97,113),(98,112),(99,111),(100,110),(101,109),(102,108),(103,107),(104,106),(121,144),(122,143),(123,142),(124,141),(125,140),(126,139),(127,138),(128,137),(129,136),(130,135),(131,134),(132,133),(145,150),(146,149),(147,148),(151,177),(152,176),(153,175),(154,174),(155,173),(156,172),(157,171),(158,170),(159,169),(160,168),(161,167),(162,166),(163,165),(178,180),(181,190),(182,189),(183,188),(184,187),(185,186),(191,210),(192,209),(193,208),(194,207),(195,206),(196,205),(197,204),(198,203),(199,202),(200,201),(211,221),(212,220),(213,219),(214,218),(215,217),(222,240),(223,239),(224,238),(225,237),(226,236),(227,235),(228,234),(229,233),(230,232)], [(1,37,68,239),(2,36,69,238),(3,35,70,237),(4,34,71,236),(5,33,72,235),(6,32,73,234),(7,31,74,233),(8,60,75,232),(9,59,76,231),(10,58,77,230),(11,57,78,229),(12,56,79,228),(13,55,80,227),(14,54,81,226),(15,53,82,225),(16,52,83,224),(17,51,84,223),(18,50,85,222),(19,49,86,221),(20,48,87,220),(21,47,88,219),(22,46,89,218),(23,45,90,217),(24,44,61,216),(25,43,62,215),(26,42,63,214),(27,41,64,213),(28,40,65,212),(29,39,66,211),(30,38,67,240),(91,193,165,140),(92,192,166,139),(93,191,167,138),(94,190,168,137),(95,189,169,136),(96,188,170,135),(97,187,171,134),(98,186,172,133),(99,185,173,132),(100,184,174,131),(101,183,175,130),(102,182,176,129),(103,181,177,128),(104,210,178,127),(105,209,179,126),(106,208,180,125),(107,207,151,124),(108,206,152,123),(109,205,153,122),(110,204,154,121),(111,203,155,150),(112,202,156,149),(113,201,157,148),(114,200,158,147),(115,199,159,146),(116,198,160,145),(117,197,161,144),(118,196,162,143),(119,195,163,142),(120,194,164,141)], [(1,133,68,186),(2,122,69,205),(3,141,70,194),(4,130,71,183),(5,149,72,202),(6,138,73,191),(7,127,74,210),(8,146,75,199),(9,135,76,188),(10,124,77,207),(11,143,78,196),(12,132,79,185),(13,121,80,204),(14,140,81,193),(15,129,82,182),(16,148,83,201),(17,137,84,190),(18,126,85,209),(19,145,86,198),(20,134,87,187),(21,123,88,206),(22,142,89,195),(23,131,90,184),(24,150,61,203),(25,139,62,192),(26,128,63,181),(27,147,64,200),(28,136,65,189),(29,125,66,208),(30,144,67,197),(31,178,233,104),(32,167,234,93),(33,156,235,112),(34,175,236,101),(35,164,237,120),(36,153,238,109),(37,172,239,98),(38,161,240,117),(39,180,211,106),(40,169,212,95),(41,158,213,114),(42,177,214,103),(43,166,215,92),(44,155,216,111),(45,174,217,100),(46,163,218,119),(47,152,219,108),(48,171,220,97),(49,160,221,116),(50,179,222,105),(51,168,223,94),(52,157,224,113),(53,176,225,102),(54,165,226,91),(55,154,227,110),(56,173,228,99),(57,162,229,118),(58,151,230,107),(59,170,231,96),(60,159,232,115)]])
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 | D12 | C4○D20 | D4⋊2S3 | S3×Q8 | S3×D5 | D4×D5 | Q8×D5 | C2×S3×D5 | D15⋊Q8 | D5×D12 | Dic5.D6 |
| kernel | D30⋊2Q8 | D30⋊4C4 | C30.Q8 | C3×C10.D4 | C5×C4⋊Dic3 | D30⋊3C4 | C2×D30.C2 | C2×C15⋊Q8 | C10.D4 | C3×Dic5 | D30 | C4⋊Dic3 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | Dic5 | C6 | 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 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D30⋊2Q8 ►in GL4(𝔽61) generated by
| 1 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 43 | 1 |
| 0 | 0 | 42 | 1 |
| 1 | 1 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 44 |
| 0 | 0 | 43 | 0 |
| 38 | 15 | 0 | 0 |
| 38 | 23 | 0 | 0 |
| 0 | 0 | 8 | 14 |
| 0 | 0 | 52 | 53 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 46 | 11 |
| 0 | 0 | 46 | 15 |
G:=sub<GL(4,GF(61))| [1,60,0,0,1,0,0,0,0,0,43,42,0,0,1,1],[1,0,0,0,1,60,0,0,0,0,0,43,0,0,44,0],[38,38,0,0,15,23,0,0,0,0,8,52,0,0,14,53],[60,0,0,0,0,60,0,0,0,0,46,46,0,0,11,15] >;
D30⋊2Q8 in GAP, Magma, Sage, TeX
D_{30}\rtimes_2Q_8 % in TeX
G:=Group("D30:2Q8"); // GroupNames label
G:=SmallGroup(480,495);
// by ID
G=gap.SmallGroup(480,495);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,422,135,58,1356,18822]);
// 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^19,c*b*c^-1=a^13*b,d*b*d^-1=a^18*b,d*c*d^-1=c^-1>;
// generators/relations