metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊6Q8, C4.13D60, C60.11D4, C12.23D20, C20.23D12, C4⋊C4⋊5D15, C6.43(Q8×D5), C2.7(Q8×D15), C60⋊5C4⋊12C2, C5⋊3(C4.D12), (C2×C4).44D30, C6.36(C2×D20), C2.10(C2×D60), C10.43(S3×Q8), C30.96(C2×Q8), C3⋊3(D10⋊2Q8), (C2×C12).40D10, (C2×C20).139D6, C30.264(C2×D4), C10.37(C2×D12), C15⋊29(C22⋊Q8), (C2×Dic30)⋊10C2, D30⋊3C4.3C2, (C2×C60).21C22, C30.220(C4○D4), C6.99(D4⋊2D5), (C2×C30).294C23, C2.13(D4⋊2D15), C10.99(D4⋊2S3), C22.52(C22×D15), (C2×Dic15).12C22, (C22×D15).84C22, (C5×C4⋊C4)⋊8S3, (C3×C4⋊C4)⋊8D5, (C15×C4⋊C4)⋊8C2, (C2×C4×D15).2C2, (C2×C6).290(C22×D5), (C2×C10).289(C22×S3), SmallGroup(480,862)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊6Q8
G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=cac-1=a-1, ad=da, cbc-1=a13b, bd=db, dcd-1=c-1 >
Subgroups: 900 in 148 conjugacy classes, 57 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, D10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C22⋊Q8, Dic10, C4×D5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, Dic15, C60, C60, D30, D30, C2×C30, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C4.D12, Dic30, C4×D15, C2×Dic15, C2×Dic15, C2×C60, C2×C60, C22×D15, D10⋊2Q8, C60⋊5C4, D30⋊3C4, C15×C4⋊C4, C2×Dic30, C2×C4×D15, D30⋊6Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, D12, C22×S3, D15, C22⋊Q8, D20, C22×D5, C2×D12, D4⋊2S3, S3×Q8, D30, C2×D20, D4⋊2D5, Q8×D5, C4.D12, D60, C22×D15, D10⋊2Q8, C2×D60, D4⋊2D15, Q8×D15, D30⋊6Q8
►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)
(1 207)(2 206)(3 205)(4 204)(5 203)(6 202)(7 201)(8 200)(9 199)(10 198)(11 197)(12 196)(13 195)(14 194)(15 193)(16 192)(17 191)(18 190)(19 189)(20 188)(21 187)(22 186)(23 185)(24 184)(25 183)(26 182)(27 181)(28 210)(29 209)(30 208)(31 90)(32 89)(33 88)(34 87)(35 86)(36 85)(37 84)(38 83)(39 82)(40 81)(41 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(91 211)(92 240)(93 239)(94 238)(95 237)(96 236)(97 235)(98 234)(99 233)(100 232)(101 231)(102 230)(103 229)(104 228)(105 227)(106 226)(107 225)(108 224)(109 223)(110 222)(111 221)(112 220)(113 219)(114 218)(115 217)(116 216)(117 215)(118 214)(119 213)(120 212)(121 178)(122 177)(123 176)(124 175)(125 174)(126 173)(127 172)(128 171)(129 170)(130 169)(131 168)(132 167)(133 166)(134 165)(135 164)(136 163)(137 162)(138 161)(139 160)(140 159)(141 158)(142 157)(143 156)(144 155)(145 154)(146 153)(147 152)(148 151)(149 180)(150 179)
(1 143 208 172)(2 142 209 171)(3 141 210 170)(4 140 181 169)(5 139 182 168)(6 138 183 167)(7 137 184 166)(8 136 185 165)(9 135 186 164)(10 134 187 163)(11 133 188 162)(12 132 189 161)(13 131 190 160)(14 130 191 159)(15 129 192 158)(16 128 193 157)(17 127 194 156)(18 126 195 155)(19 125 196 154)(20 124 197 153)(21 123 198 152)(22 122 199 151)(23 121 200 180)(24 150 201 179)(25 149 202 178)(26 148 203 177)(27 147 204 176)(28 146 205 175)(29 145 206 174)(30 144 207 173)(31 97 66 216)(32 96 67 215)(33 95 68 214)(34 94 69 213)(35 93 70 212)(36 92 71 211)(37 91 72 240)(38 120 73 239)(39 119 74 238)(40 118 75 237)(41 117 76 236)(42 116 77 235)(43 115 78 234)(44 114 79 233)(45 113 80 232)(46 112 81 231)(47 111 82 230)(48 110 83 229)(49 109 84 228)(50 108 85 227)(51 107 86 226)(52 106 87 225)(53 105 88 224)(54 104 89 223)(55 103 90 222)(56 102 61 221)(57 101 62 220)(58 100 63 219)(59 99 64 218)(60 98 65 217)
(1 211 208 92)(2 212 209 93)(3 213 210 94)(4 214 181 95)(5 215 182 96)(6 216 183 97)(7 217 184 98)(8 218 185 99)(9 219 186 100)(10 220 187 101)(11 221 188 102)(12 222 189 103)(13 223 190 104)(14 224 191 105)(15 225 192 106)(16 226 193 107)(17 227 194 108)(18 228 195 109)(19 229 196 110)(20 230 197 111)(21 231 198 112)(22 232 199 113)(23 233 200 114)(24 234 201 115)(25 235 202 116)(26 236 203 117)(27 237 204 118)(28 238 205 119)(29 239 206 120)(30 240 207 91)(31 138 66 167)(32 139 67 168)(33 140 68 169)(34 141 69 170)(35 142 70 171)(36 143 71 172)(37 144 72 173)(38 145 73 174)(39 146 74 175)(40 147 75 176)(41 148 76 177)(42 149 77 178)(43 150 78 179)(44 121 79 180)(45 122 80 151)(46 123 81 152)(47 124 82 153)(48 125 83 154)(49 126 84 155)(50 127 85 156)(51 128 86 157)(52 129 87 158)(53 130 88 159)(54 131 89 160)(55 132 90 161)(56 133 61 162)(57 134 62 163)(58 135 63 164)(59 136 64 165)(60 137 65 166)
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), (1,207)(2,206)(3,205)(4,204)(5,203)(6,202)(7,201)(8,200)(9,199)(10,198)(11,197)(12,196)(13,195)(14,194)(15,193)(16,192)(17,191)(18,190)(19,189)(20,188)(21,187)(22,186)(23,185)(24,184)(25,183)(26,182)(27,181)(28,210)(29,209)(30,208)(31,90)(32,89)(33,88)(34,87)(35,86)(36,85)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(91,211)(92,240)(93,239)(94,238)(95,237)(96,236)(97,235)(98,234)(99,233)(100,232)(101,231)(102,230)(103,229)(104,228)(105,227)(106,226)(107,225)(108,224)(109,223)(110,222)(111,221)(112,220)(113,219)(114,218)(115,217)(116,216)(117,215)(118,214)(119,213)(120,212)(121,178)(122,177)(123,176)(124,175)(125,174)(126,173)(127,172)(128,171)(129,170)(130,169)(131,168)(132,167)(133,166)(134,165)(135,164)(136,163)(137,162)(138,161)(139,160)(140,159)(141,158)(142,157)(143,156)(144,155)(145,154)(146,153)(147,152)(148,151)(149,180)(150,179), (1,143,208,172)(2,142,209,171)(3,141,210,170)(4,140,181,169)(5,139,182,168)(6,138,183,167)(7,137,184,166)(8,136,185,165)(9,135,186,164)(10,134,187,163)(11,133,188,162)(12,132,189,161)(13,131,190,160)(14,130,191,159)(15,129,192,158)(16,128,193,157)(17,127,194,156)(18,126,195,155)(19,125,196,154)(20,124,197,153)(21,123,198,152)(22,122,199,151)(23,121,200,180)(24,150,201,179)(25,149,202,178)(26,148,203,177)(27,147,204,176)(28,146,205,175)(29,145,206,174)(30,144,207,173)(31,97,66,216)(32,96,67,215)(33,95,68,214)(34,94,69,213)(35,93,70,212)(36,92,71,211)(37,91,72,240)(38,120,73,239)(39,119,74,238)(40,118,75,237)(41,117,76,236)(42,116,77,235)(43,115,78,234)(44,114,79,233)(45,113,80,232)(46,112,81,231)(47,111,82,230)(48,110,83,229)(49,109,84,228)(50,108,85,227)(51,107,86,226)(52,106,87,225)(53,105,88,224)(54,104,89,223)(55,103,90,222)(56,102,61,221)(57,101,62,220)(58,100,63,219)(59,99,64,218)(60,98,65,217), (1,211,208,92)(2,212,209,93)(3,213,210,94)(4,214,181,95)(5,215,182,96)(6,216,183,97)(7,217,184,98)(8,218,185,99)(9,219,186,100)(10,220,187,101)(11,221,188,102)(12,222,189,103)(13,223,190,104)(14,224,191,105)(15,225,192,106)(16,226,193,107)(17,227,194,108)(18,228,195,109)(19,229,196,110)(20,230,197,111)(21,231,198,112)(22,232,199,113)(23,233,200,114)(24,234,201,115)(25,235,202,116)(26,236,203,117)(27,237,204,118)(28,238,205,119)(29,239,206,120)(30,240,207,91)(31,138,66,167)(32,139,67,168)(33,140,68,169)(34,141,69,170)(35,142,70,171)(36,143,71,172)(37,144,72,173)(38,145,73,174)(39,146,74,175)(40,147,75,176)(41,148,76,177)(42,149,77,178)(43,150,78,179)(44,121,79,180)(45,122,80,151)(46,123,81,152)(47,124,82,153)(48,125,83,154)(49,126,84,155)(50,127,85,156)(51,128,86,157)(52,129,87,158)(53,130,88,159)(54,131,89,160)(55,132,90,161)(56,133,61,162)(57,134,62,163)(58,135,63,164)(59,136,64,165)(60,137,65,166)>;
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), (1,207)(2,206)(3,205)(4,204)(5,203)(6,202)(7,201)(8,200)(9,199)(10,198)(11,197)(12,196)(13,195)(14,194)(15,193)(16,192)(17,191)(18,190)(19,189)(20,188)(21,187)(22,186)(23,185)(24,184)(25,183)(26,182)(27,181)(28,210)(29,209)(30,208)(31,90)(32,89)(33,88)(34,87)(35,86)(36,85)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(91,211)(92,240)(93,239)(94,238)(95,237)(96,236)(97,235)(98,234)(99,233)(100,232)(101,231)(102,230)(103,229)(104,228)(105,227)(106,226)(107,225)(108,224)(109,223)(110,222)(111,221)(112,220)(113,219)(114,218)(115,217)(116,216)(117,215)(118,214)(119,213)(120,212)(121,178)(122,177)(123,176)(124,175)(125,174)(126,173)(127,172)(128,171)(129,170)(130,169)(131,168)(132,167)(133,166)(134,165)(135,164)(136,163)(137,162)(138,161)(139,160)(140,159)(141,158)(142,157)(143,156)(144,155)(145,154)(146,153)(147,152)(148,151)(149,180)(150,179), (1,143,208,172)(2,142,209,171)(3,141,210,170)(4,140,181,169)(5,139,182,168)(6,138,183,167)(7,137,184,166)(8,136,185,165)(9,135,186,164)(10,134,187,163)(11,133,188,162)(12,132,189,161)(13,131,190,160)(14,130,191,159)(15,129,192,158)(16,128,193,157)(17,127,194,156)(18,126,195,155)(19,125,196,154)(20,124,197,153)(21,123,198,152)(22,122,199,151)(23,121,200,180)(24,150,201,179)(25,149,202,178)(26,148,203,177)(27,147,204,176)(28,146,205,175)(29,145,206,174)(30,144,207,173)(31,97,66,216)(32,96,67,215)(33,95,68,214)(34,94,69,213)(35,93,70,212)(36,92,71,211)(37,91,72,240)(38,120,73,239)(39,119,74,238)(40,118,75,237)(41,117,76,236)(42,116,77,235)(43,115,78,234)(44,114,79,233)(45,113,80,232)(46,112,81,231)(47,111,82,230)(48,110,83,229)(49,109,84,228)(50,108,85,227)(51,107,86,226)(52,106,87,225)(53,105,88,224)(54,104,89,223)(55,103,90,222)(56,102,61,221)(57,101,62,220)(58,100,63,219)(59,99,64,218)(60,98,65,217), (1,211,208,92)(2,212,209,93)(3,213,210,94)(4,214,181,95)(5,215,182,96)(6,216,183,97)(7,217,184,98)(8,218,185,99)(9,219,186,100)(10,220,187,101)(11,221,188,102)(12,222,189,103)(13,223,190,104)(14,224,191,105)(15,225,192,106)(16,226,193,107)(17,227,194,108)(18,228,195,109)(19,229,196,110)(20,230,197,111)(21,231,198,112)(22,232,199,113)(23,233,200,114)(24,234,201,115)(25,235,202,116)(26,236,203,117)(27,237,204,118)(28,238,205,119)(29,239,206,120)(30,240,207,91)(31,138,66,167)(32,139,67,168)(33,140,68,169)(34,141,69,170)(35,142,70,171)(36,143,71,172)(37,144,72,173)(38,145,73,174)(39,146,74,175)(40,147,75,176)(41,148,76,177)(42,149,77,178)(43,150,78,179)(44,121,79,180)(45,122,80,151)(46,123,81,152)(47,124,82,153)(48,125,83,154)(49,126,84,155)(50,127,85,156)(51,128,86,157)(52,129,87,158)(53,130,88,159)(54,131,89,160)(55,132,90,161)(56,133,61,162)(57,134,62,163)(58,135,63,164)(59,136,64,165)(60,137,65,166) );
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)], [(1,207),(2,206),(3,205),(4,204),(5,203),(6,202),(7,201),(8,200),(9,199),(10,198),(11,197),(12,196),(13,195),(14,194),(15,193),(16,192),(17,191),(18,190),(19,189),(20,188),(21,187),(22,186),(23,185),(24,184),(25,183),(26,182),(27,181),(28,210),(29,209),(30,208),(31,90),(32,89),(33,88),(34,87),(35,86),(36,85),(37,84),(38,83),(39,82),(40,81),(41,80),(42,79),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73),(49,72),(50,71),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61),(91,211),(92,240),(93,239),(94,238),(95,237),(96,236),(97,235),(98,234),(99,233),(100,232),(101,231),(102,230),(103,229),(104,228),(105,227),(106,226),(107,225),(108,224),(109,223),(110,222),(111,221),(112,220),(113,219),(114,218),(115,217),(116,216),(117,215),(118,214),(119,213),(120,212),(121,178),(122,177),(123,176),(124,175),(125,174),(126,173),(127,172),(128,171),(129,170),(130,169),(131,168),(132,167),(133,166),(134,165),(135,164),(136,163),(137,162),(138,161),(139,160),(140,159),(141,158),(142,157),(143,156),(144,155),(145,154),(146,153),(147,152),(148,151),(149,180),(150,179)], [(1,143,208,172),(2,142,209,171),(3,141,210,170),(4,140,181,169),(5,139,182,168),(6,138,183,167),(7,137,184,166),(8,136,185,165),(9,135,186,164),(10,134,187,163),(11,133,188,162),(12,132,189,161),(13,131,190,160),(14,130,191,159),(15,129,192,158),(16,128,193,157),(17,127,194,156),(18,126,195,155),(19,125,196,154),(20,124,197,153),(21,123,198,152),(22,122,199,151),(23,121,200,180),(24,150,201,179),(25,149,202,178),(26,148,203,177),(27,147,204,176),(28,146,205,175),(29,145,206,174),(30,144,207,173),(31,97,66,216),(32,96,67,215),(33,95,68,214),(34,94,69,213),(35,93,70,212),(36,92,71,211),(37,91,72,240),(38,120,73,239),(39,119,74,238),(40,118,75,237),(41,117,76,236),(42,116,77,235),(43,115,78,234),(44,114,79,233),(45,113,80,232),(46,112,81,231),(47,111,82,230),(48,110,83,229),(49,109,84,228),(50,108,85,227),(51,107,86,226),(52,106,87,225),(53,105,88,224),(54,104,89,223),(55,103,90,222),(56,102,61,221),(57,101,62,220),(58,100,63,219),(59,99,64,218),(60,98,65,217)], [(1,211,208,92),(2,212,209,93),(3,213,210,94),(4,214,181,95),(5,215,182,96),(6,216,183,97),(7,217,184,98),(8,218,185,99),(9,219,186,100),(10,220,187,101),(11,221,188,102),(12,222,189,103),(13,223,190,104),(14,224,191,105),(15,225,192,106),(16,226,193,107),(17,227,194,108),(18,228,195,109),(19,229,196,110),(20,230,197,111),(21,231,198,112),(22,232,199,113),(23,233,200,114),(24,234,201,115),(25,235,202,116),(26,236,203,117),(27,237,204,118),(28,238,205,119),(29,239,206,120),(30,240,207,91),(31,138,66,167),(32,139,67,168),(33,140,68,169),(34,141,69,170),(35,142,70,171),(36,143,71,172),(37,144,72,173),(38,145,73,174),(39,146,74,175),(40,147,75,176),(41,148,76,177),(42,149,77,178),(43,150,78,179),(44,121,79,180),(45,122,80,151),(46,123,81,152),(47,124,82,153),(48,125,83,154),(49,126,84,155),(50,127,85,156),(51,128,86,157),(52,129,87,158),(53,130,88,159),(54,131,89,160),(55,132,90,161),(56,133,61,162),(57,134,62,163),(58,135,63,164),(59,136,64,165),(60,137,65,166)]])
84 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 | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| 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 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 2 | 2 | 4 | 4 | 30 | 30 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | + | + | - | - | - | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D5 | D6 | C4○D4 | D10 | D12 | D15 | D20 | D30 | D60 | D4⋊2S3 | S3×Q8 | D4⋊2D5 | Q8×D5 | D4⋊2D15 | Q8×D15 |
| kernel | D30⋊6Q8 | C60⋊5C4 | D30⋊3C4 | C15×C4⋊C4 | C2×Dic30 | C2×C4×D15 | C5×C4⋊C4 | C60 | D30 | C3×C4⋊C4 | C2×C20 | C30 | C2×C12 | C20 | C4⋊C4 | C12 | C2×C4 | C4 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 2 | 6 | 4 | 4 | 8 | 12 | 16 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D30⋊6Q8 ►in GL6(𝔽61)
| 56 | 9 | 0 | 0 | 0 | 0 |
| 52 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 58 | 15 | 0 | 0 |
| 0 | 0 | 52 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 18 | 1 | 0 | 0 | 0 | 0 |
| 43 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 58 | 15 | 0 | 0 |
| 0 | 0 | 32 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 60 | 60 |
| 8 | 45 | 0 | 0 | 0 | 0 |
| 23 | 53 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 42 | 0 | 0 |
| 0 | 0 | 19 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 59 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 11 | 11 |
G:=sub<GL(6,GF(61))| [56,52,0,0,0,0,9,16,0,0,0,0,0,0,58,52,0,0,0,0,15,4,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[18,43,0,0,0,0,1,43,0,0,0,0,0,0,58,32,0,0,0,0,15,3,0,0,0,0,0,0,1,60,0,0,0,0,0,60],[8,23,0,0,0,0,45,53,0,0,0,0,0,0,39,19,0,0,0,0,42,22,0,0,0,0,0,0,60,1,0,0,0,0,59,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,11,0,0,0,0,0,11] >;
D30⋊6Q8 in GAP, Magma, Sage, TeX
D_{30}\rtimes_6Q_8 % in TeX
G:=Group("D30:6Q8"); // GroupNames label
G:=SmallGroup(480,862);
// by ID
G=gap.SmallGroup(480,862);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,142,2693,18822]);
// 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,a*d=d*a,c*b*c^-1=a^13*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations