metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊1Q8, C6.13(Q8×D5), C5⋊1(D6⋊3Q8), (C2×C20).16D6, C6.132(D4×D5), C10.13(S3×Q8), C30.38(C2×Q8), Dic3⋊C4⋊18D5, (C2×Dic10)⋊6S3, C3⋊6(D10⋊Q8), C6.8(C4○D20), C30.132(C2×D4), C15⋊11(C22⋊Q8), (C3×Dic5).8D4, (C6×Dic10)⋊16C2, D30⋊3C4.7C2, C30.58(C4○D4), (C2×C12).225D10, C2.15(D15⋊Q8), C30.Q8⋊17C2, Dic15⋊5C4⋊17C2, (C2×Dic5).29D6, D30⋊4C4.11C2, (C2×C60).256C22, (C2×C30).101C23, (C2×Dic3).31D10, C2.15(D60⋊C2), C10.13(Q8⋊3S3), Dic5.21(C3⋊D4), (C6×Dic5).58C22, (C10×Dic3).61C22, (C2×Dic15).81C22, (C22×D15).32C22, (C2×C4).40(S3×D5), C2.16(D5×C3⋊D4), C10.34(C2×C3⋊D4), C22.170(C2×S3×D5), (C5×Dic3⋊C4)⋊18C2, (C2×D30.C2).4C2, (C2×C6).113(C22×D5), (C2×C10).113(C22×S3), SmallGroup(480,487)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊Q8
G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a19, cbc-1=a15b, dbd-1=a18b, dcd-1=c-1 >
Subgroups: 796 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, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, D15, C30, C22⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, S3×C2×C4, C6×Q8, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, D30, D30, C2×C30, C10.D4, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, D6⋊3Q8, D30.C2, C3×Dic10, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C22×D15, D10⋊Q8, D30⋊4C4, C30.Q8, Dic15⋊5C4, C5×Dic3⋊C4, D30⋊3C4, C2×D30.C2, C6×Dic10, D30⋊Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, C3⋊D4, C22×S3, C22⋊Q8, C22×D5, S3×Q8, Q8⋊3S3, C2×C3⋊D4, S3×D5, C4○D20, D4×D5, Q8×D5, D6⋊3Q8, C2×S3×D5, D10⋊Q8, D60⋊C2, D15⋊Q8, D5×C3⋊D4, D30⋊Q8
►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)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(61 71)(62 70)(63 69)(64 68)(65 67)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(101 102)(113 120)(114 119)(115 118)(116 117)(121 130)(122 129)(123 128)(124 127)(125 126)(131 150)(132 149)(133 148)(134 147)(135 146)(136 145)(137 144)(138 143)(139 142)(140 141)(151 159)(152 158)(153 157)(154 156)(160 180)(161 179)(162 178)(163 177)(164 176)(165 175)(166 174)(167 173)(168 172)(169 171)(181 203)(182 202)(183 201)(184 200)(185 199)(186 198)(187 197)(188 196)(189 195)(190 194)(191 193)(204 210)(205 209)(206 208)(211 240)(212 239)(213 238)(214 237)(215 236)(216 235)(217 234)(218 233)(219 232)(220 231)(221 230)(222 229)(223 228)(224 227)(225 226)
(1 74 117 200)(2 75 118 201)(3 76 119 202)(4 77 120 203)(5 78 91 204)(6 79 92 205)(7 80 93 206)(8 81 94 207)(9 82 95 208)(10 83 96 209)(11 84 97 210)(12 85 98 181)(13 86 99 182)(14 87 100 183)(15 88 101 184)(16 89 102 185)(17 90 103 186)(18 61 104 187)(19 62 105 188)(20 63 106 189)(21 64 107 190)(22 65 108 191)(23 66 109 192)(24 67 110 193)(25 68 111 194)(26 69 112 195)(27 70 113 196)(28 71 114 197)(29 72 115 198)(30 73 116 199)(31 133 155 233)(32 134 156 234)(33 135 157 235)(34 136 158 236)(35 137 159 237)(36 138 160 238)(37 139 161 239)(38 140 162 240)(39 141 163 211)(40 142 164 212)(41 143 165 213)(42 144 166 214)(43 145 167 215)(44 146 168 216)(45 147 169 217)(46 148 170 218)(47 149 171 219)(48 150 172 220)(49 121 173 221)(50 122 174 222)(51 123 175 223)(52 124 176 224)(53 125 177 225)(54 126 178 226)(55 127 179 227)(56 128 180 228)(57 129 151 229)(58 130 152 230)(59 131 153 231)(60 132 154 232)
(1 226 117 126)(2 215 118 145)(3 234 119 134)(4 223 120 123)(5 212 91 142)(6 231 92 131)(7 220 93 150)(8 239 94 139)(9 228 95 128)(10 217 96 147)(11 236 97 136)(12 225 98 125)(13 214 99 144)(14 233 100 133)(15 222 101 122)(16 211 102 141)(17 230 103 130)(18 219 104 149)(19 238 105 138)(20 227 106 127)(21 216 107 146)(22 235 108 135)(23 224 109 124)(24 213 110 143)(25 232 111 132)(26 221 112 121)(27 240 113 140)(28 229 114 129)(29 218 115 148)(30 237 116 137)(31 87 155 183)(32 76 156 202)(33 65 157 191)(34 84 158 210)(35 73 159 199)(36 62 160 188)(37 81 161 207)(38 70 162 196)(39 89 163 185)(40 78 164 204)(41 67 165 193)(42 86 166 182)(43 75 167 201)(44 64 168 190)(45 83 169 209)(46 72 170 198)(47 61 171 187)(48 80 172 206)(49 69 173 195)(50 88 174 184)(51 77 175 203)(52 66 176 192)(53 85 177 181)(54 74 178 200)(55 63 179 189)(56 82 180 208)(57 71 151 197)(58 90 152 186)(59 79 153 205)(60 68 154 194)
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)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(61,71)(62,70)(63,69)(64,68)(65,67)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)(113,120)(114,119)(115,118)(116,117)(121,130)(122,129)(123,128)(124,127)(125,126)(131,150)(132,149)(133,148)(134,147)(135,146)(136,145)(137,144)(138,143)(139,142)(140,141)(151,159)(152,158)(153,157)(154,156)(160,180)(161,179)(162,178)(163,177)(164,176)(165,175)(166,174)(167,173)(168,172)(169,171)(181,203)(182,202)(183,201)(184,200)(185,199)(186,198)(187,197)(188,196)(189,195)(190,194)(191,193)(204,210)(205,209)(206,208)(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)(217,234)(218,233)(219,232)(220,231)(221,230)(222,229)(223,228)(224,227)(225,226), (1,74,117,200)(2,75,118,201)(3,76,119,202)(4,77,120,203)(5,78,91,204)(6,79,92,205)(7,80,93,206)(8,81,94,207)(9,82,95,208)(10,83,96,209)(11,84,97,210)(12,85,98,181)(13,86,99,182)(14,87,100,183)(15,88,101,184)(16,89,102,185)(17,90,103,186)(18,61,104,187)(19,62,105,188)(20,63,106,189)(21,64,107,190)(22,65,108,191)(23,66,109,192)(24,67,110,193)(25,68,111,194)(26,69,112,195)(27,70,113,196)(28,71,114,197)(29,72,115,198)(30,73,116,199)(31,133,155,233)(32,134,156,234)(33,135,157,235)(34,136,158,236)(35,137,159,237)(36,138,160,238)(37,139,161,239)(38,140,162,240)(39,141,163,211)(40,142,164,212)(41,143,165,213)(42,144,166,214)(43,145,167,215)(44,146,168,216)(45,147,169,217)(46,148,170,218)(47,149,171,219)(48,150,172,220)(49,121,173,221)(50,122,174,222)(51,123,175,223)(52,124,176,224)(53,125,177,225)(54,126,178,226)(55,127,179,227)(56,128,180,228)(57,129,151,229)(58,130,152,230)(59,131,153,231)(60,132,154,232), (1,226,117,126)(2,215,118,145)(3,234,119,134)(4,223,120,123)(5,212,91,142)(6,231,92,131)(7,220,93,150)(8,239,94,139)(9,228,95,128)(10,217,96,147)(11,236,97,136)(12,225,98,125)(13,214,99,144)(14,233,100,133)(15,222,101,122)(16,211,102,141)(17,230,103,130)(18,219,104,149)(19,238,105,138)(20,227,106,127)(21,216,107,146)(22,235,108,135)(23,224,109,124)(24,213,110,143)(25,232,111,132)(26,221,112,121)(27,240,113,140)(28,229,114,129)(29,218,115,148)(30,237,116,137)(31,87,155,183)(32,76,156,202)(33,65,157,191)(34,84,158,210)(35,73,159,199)(36,62,160,188)(37,81,161,207)(38,70,162,196)(39,89,163,185)(40,78,164,204)(41,67,165,193)(42,86,166,182)(43,75,167,201)(44,64,168,190)(45,83,169,209)(46,72,170,198)(47,61,171,187)(48,80,172,206)(49,69,173,195)(50,88,174,184)(51,77,175,203)(52,66,176,192)(53,85,177,181)(54,74,178,200)(55,63,179,189)(56,82,180,208)(57,71,151,197)(58,90,152,186)(59,79,153,205)(60,68,154,194)>;
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)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(61,71)(62,70)(63,69)(64,68)(65,67)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)(113,120)(114,119)(115,118)(116,117)(121,130)(122,129)(123,128)(124,127)(125,126)(131,150)(132,149)(133,148)(134,147)(135,146)(136,145)(137,144)(138,143)(139,142)(140,141)(151,159)(152,158)(153,157)(154,156)(160,180)(161,179)(162,178)(163,177)(164,176)(165,175)(166,174)(167,173)(168,172)(169,171)(181,203)(182,202)(183,201)(184,200)(185,199)(186,198)(187,197)(188,196)(189,195)(190,194)(191,193)(204,210)(205,209)(206,208)(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)(217,234)(218,233)(219,232)(220,231)(221,230)(222,229)(223,228)(224,227)(225,226), (1,74,117,200)(2,75,118,201)(3,76,119,202)(4,77,120,203)(5,78,91,204)(6,79,92,205)(7,80,93,206)(8,81,94,207)(9,82,95,208)(10,83,96,209)(11,84,97,210)(12,85,98,181)(13,86,99,182)(14,87,100,183)(15,88,101,184)(16,89,102,185)(17,90,103,186)(18,61,104,187)(19,62,105,188)(20,63,106,189)(21,64,107,190)(22,65,108,191)(23,66,109,192)(24,67,110,193)(25,68,111,194)(26,69,112,195)(27,70,113,196)(28,71,114,197)(29,72,115,198)(30,73,116,199)(31,133,155,233)(32,134,156,234)(33,135,157,235)(34,136,158,236)(35,137,159,237)(36,138,160,238)(37,139,161,239)(38,140,162,240)(39,141,163,211)(40,142,164,212)(41,143,165,213)(42,144,166,214)(43,145,167,215)(44,146,168,216)(45,147,169,217)(46,148,170,218)(47,149,171,219)(48,150,172,220)(49,121,173,221)(50,122,174,222)(51,123,175,223)(52,124,176,224)(53,125,177,225)(54,126,178,226)(55,127,179,227)(56,128,180,228)(57,129,151,229)(58,130,152,230)(59,131,153,231)(60,132,154,232), (1,226,117,126)(2,215,118,145)(3,234,119,134)(4,223,120,123)(5,212,91,142)(6,231,92,131)(7,220,93,150)(8,239,94,139)(9,228,95,128)(10,217,96,147)(11,236,97,136)(12,225,98,125)(13,214,99,144)(14,233,100,133)(15,222,101,122)(16,211,102,141)(17,230,103,130)(18,219,104,149)(19,238,105,138)(20,227,106,127)(21,216,107,146)(22,235,108,135)(23,224,109,124)(24,213,110,143)(25,232,111,132)(26,221,112,121)(27,240,113,140)(28,229,114,129)(29,218,115,148)(30,237,116,137)(31,87,155,183)(32,76,156,202)(33,65,157,191)(34,84,158,210)(35,73,159,199)(36,62,160,188)(37,81,161,207)(38,70,162,196)(39,89,163,185)(40,78,164,204)(41,67,165,193)(42,86,166,182)(43,75,167,201)(44,64,168,190)(45,83,169,209)(46,72,170,198)(47,61,171,187)(48,80,172,206)(49,69,173,195)(50,88,174,184)(51,77,175,203)(52,66,176,192)(53,85,177,181)(54,74,178,200)(55,63,179,189)(56,82,180,208)(57,71,151,197)(58,90,152,186)(59,79,153,205)(60,68,154,194) );
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),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(61,71),(62,70),(63,69),(64,68),(65,67),(72,90),(73,89),(74,88),(75,87),(76,86),(77,85),(78,84),(79,83),(80,82),(91,112),(92,111),(93,110),(94,109),(95,108),(96,107),(97,106),(98,105),(99,104),(100,103),(101,102),(113,120),(114,119),(115,118),(116,117),(121,130),(122,129),(123,128),(124,127),(125,126),(131,150),(132,149),(133,148),(134,147),(135,146),(136,145),(137,144),(138,143),(139,142),(140,141),(151,159),(152,158),(153,157),(154,156),(160,180),(161,179),(162,178),(163,177),(164,176),(165,175),(166,174),(167,173),(168,172),(169,171),(181,203),(182,202),(183,201),(184,200),(185,199),(186,198),(187,197),(188,196),(189,195),(190,194),(191,193),(204,210),(205,209),(206,208),(211,240),(212,239),(213,238),(214,237),(215,236),(216,235),(217,234),(218,233),(219,232),(220,231),(221,230),(222,229),(223,228),(224,227),(225,226)], [(1,74,117,200),(2,75,118,201),(3,76,119,202),(4,77,120,203),(5,78,91,204),(6,79,92,205),(7,80,93,206),(8,81,94,207),(9,82,95,208),(10,83,96,209),(11,84,97,210),(12,85,98,181),(13,86,99,182),(14,87,100,183),(15,88,101,184),(16,89,102,185),(17,90,103,186),(18,61,104,187),(19,62,105,188),(20,63,106,189),(21,64,107,190),(22,65,108,191),(23,66,109,192),(24,67,110,193),(25,68,111,194),(26,69,112,195),(27,70,113,196),(28,71,114,197),(29,72,115,198),(30,73,116,199),(31,133,155,233),(32,134,156,234),(33,135,157,235),(34,136,158,236),(35,137,159,237),(36,138,160,238),(37,139,161,239),(38,140,162,240),(39,141,163,211),(40,142,164,212),(41,143,165,213),(42,144,166,214),(43,145,167,215),(44,146,168,216),(45,147,169,217),(46,148,170,218),(47,149,171,219),(48,150,172,220),(49,121,173,221),(50,122,174,222),(51,123,175,223),(52,124,176,224),(53,125,177,225),(54,126,178,226),(55,127,179,227),(56,128,180,228),(57,129,151,229),(58,130,152,230),(59,131,153,231),(60,132,154,232)], [(1,226,117,126),(2,215,118,145),(3,234,119,134),(4,223,120,123),(5,212,91,142),(6,231,92,131),(7,220,93,150),(8,239,94,139),(9,228,95,128),(10,217,96,147),(11,236,97,136),(12,225,98,125),(13,214,99,144),(14,233,100,133),(15,222,101,122),(16,211,102,141),(17,230,103,130),(18,219,104,149),(19,238,105,138),(20,227,106,127),(21,216,107,146),(22,235,108,135),(23,224,109,124),(24,213,110,143),(25,232,111,132),(26,221,112,121),(27,240,113,140),(28,229,114,129),(29,218,115,148),(30,237,116,137),(31,87,155,183),(32,76,156,202),(33,65,157,191),(34,84,158,210),(35,73,159,199),(36,62,160,188),(37,81,161,207),(38,70,162,196),(39,89,163,185),(40,78,164,204),(41,67,165,193),(42,86,166,182),(43,75,167,201),(44,64,168,190),(45,83,169,209),(46,72,170,198),(47,61,171,187),(48,80,172,206),(49,69,173,195),(50,88,174,184),(51,77,175,203),(52,66,176,192),(53,85,177,181),(54,74,178,200),(55,63,179,189),(56,82,180,208),(57,71,151,197),(58,90,152,186),(59,79,153,205),(60,68,154,194)]])
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 | C3⋊D4 | C4○D20 | S3×Q8 | Q8⋊3S3 | S3×D5 | D4×D5 | Q8×D5 | C2×S3×D5 | D60⋊C2 | D15⋊Q8 | D5×C3⋊D4 |
| kernel | D30⋊Q8 | D30⋊4C4 | C30.Q8 | Dic15⋊5C4 | C5×Dic3⋊C4 | D30⋊3C4 | C2×D30.C2 | C6×Dic10 | C2×Dic10 | C3×Dic5 | D30 | Dic3⋊C4 | 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⋊Q8 ►in GL6(𝔽61)
| 1 | 20 | 0 | 0 | 0 | 0 |
| 9 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 19 | 0 | 0 |
| 0 | 0 | 60 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 20 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 43 | 0 | 0 |
| 0 | 0 | 44 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 22 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 11 | 0 | 0 |
| 0 | 0 | 54 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 55 | 5 |
| 0 | 0 | 0 | 0 | 54 | 6 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 35 | 0 | 0 |
| 0 | 0 | 4 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(61))| [1,9,0,0,0,0,20,59,0,0,0,0,0,0,43,60,0,0,0,0,19,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,20,60,0,0,0,0,0,0,0,44,0,0,0,0,43,0,0,0,0,0,0,0,60,22,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,25,54,0,0,0,0,11,36,0,0,0,0,0,0,55,54,0,0,0,0,5,6],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,15,4,0,0,0,0,35,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D30⋊Q8 in GAP, Magma, Sage, TeX
D_{30}\rtimes Q_8 % in TeX
G:=Group("D30:Q8"); // GroupNames label
G:=SmallGroup(480,487);
// by ID
G=gap.SmallGroup(480,487);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,254,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^19,c*b*c^-1=a^15*b,d*b*d^-1=a^18*b,d*c*d^-1=c^-1>;
// generators/relations