metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊8Q8, Dic15.39D4, C6.7(Q8×D5), C6.55(D4×D5), C10.7(S3×Q8), C5⋊1(D6⋊Q8), C10.57(S3×D4), C15⋊7(C22⋊Q8), C30.26(C2×Q8), Dic3⋊C4⋊11D5, C3⋊1(D10⋊Q8), (C2×C20).185D6, C30.119(C2×D4), D30⋊4C4.4C2, C6.28(C4○D20), C30.42(C4○D4), C10.D4⋊11S3, (C2×C12).183D10, C2.10(D15⋊Q8), (C2×C30).67C23, Dic15⋊5C4⋊10C2, (C2×Dic5).19D6, C10.31(C4○D12), (C2×C60).161C22, (C2×Dic3).20D10, C2.11(D10⋊D6), (C6×Dic5).38C22, C2.20(D6.D10), (C10×Dic3).39C22, (C22×D15).97C22, (C2×Dic15).191C22, (C2×C15⋊Q8)⋊5C2, (C2×C4×D15).7C2, (C2×C4).174(S3×D5), C22.153(C2×S3×D5), (C5×Dic3⋊C4)⋊11C2, (C2×C6).79(C22×D5), (C3×C10.D4)⋊11C2, (C2×C10).79(C22×S3), SmallGroup(480,453)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊8Q8
G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=cac-1=a-1, dad-1=a19, cbc-1=a28b, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 844 in 148 conjugacy classes, 48 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, 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, C3×Dic5, Dic15, C60, D30, D30, C2×C30, C10.D4, C10.D4, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, D6⋊Q8, C15⋊Q8, C6×Dic5, C10×Dic3, C4×D15, C2×Dic15, C2×C60, C22×D15, D10⋊Q8, D30⋊4C4, Dic15⋊5C4, C3×C10.D4, C5×Dic3⋊C4, C2×C15⋊Q8, C2×C4×D15, D30⋊8Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, C22×S3, C22⋊Q8, C22×D5, C4○D12, S3×D4, S3×Q8, S3×D5, C4○D20, D4×D5, Q8×D5, D6⋊Q8, C2×S3×D5, D10⋊Q8, D15⋊Q8, D6.D10, D10⋊D6, D30⋊8Q8
►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)
(1 144)(2 143)(3 142)(4 141)(5 140)(6 139)(7 138)(8 137)(9 136)(10 135)(11 134)(12 133)(13 132)(14 131)(15 130)(16 129)(17 128)(18 127)(19 126)(20 125)(21 124)(22 123)(23 122)(24 121)(25 150)(26 149)(27 148)(28 147)(29 146)(30 145)(31 169)(32 168)(33 167)(34 166)(35 165)(36 164)(37 163)(38 162)(39 161)(40 160)(41 159)(42 158)(43 157)(44 156)(45 155)(46 154)(47 153)(48 152)(49 151)(50 180)(51 179)(52 178)(53 177)(54 176)(55 175)(56 174)(57 173)(58 172)(59 171)(60 170)(61 110)(62 109)(63 108)(64 107)(65 106)(66 105)(67 104)(68 103)(69 102)(70 101)(71 100)(72 99)(73 98)(74 97)(75 96)(76 95)(77 94)(78 93)(79 92)(80 91)(81 120)(82 119)(83 118)(84 117)(85 116)(86 115)(87 114)(88 113)(89 112)(90 111)(181 231)(182 230)(183 229)(184 228)(185 227)(186 226)(187 225)(188 224)(189 223)(190 222)(191 221)(192 220)(193 219)(194 218)(195 217)(196 216)(197 215)(198 214)(199 213)(200 212)(201 211)(202 240)(203 239)(204 238)(205 237)(206 236)(207 235)(208 234)(209 233)(210 232)
(1 231 130 197)(2 230 131 196)(3 229 132 195)(4 228 133 194)(5 227 134 193)(6 226 135 192)(7 225 136 191)(8 224 137 190)(9 223 138 189)(10 222 139 188)(11 221 140 187)(12 220 141 186)(13 219 142 185)(14 218 143 184)(15 217 144 183)(16 216 145 182)(17 215 146 181)(18 214 147 210)(19 213 148 209)(20 212 149 208)(21 211 150 207)(22 240 121 206)(23 239 122 205)(24 238 123 204)(25 237 124 203)(26 236 125 202)(27 235 126 201)(28 234 127 200)(29 233 128 199)(30 232 129 198)(31 108 162 72)(32 107 163 71)(33 106 164 70)(34 105 165 69)(35 104 166 68)(36 103 167 67)(37 102 168 66)(38 101 169 65)(39 100 170 64)(40 99 171 63)(41 98 172 62)(42 97 173 61)(43 96 174 90)(44 95 175 89)(45 94 176 88)(46 93 177 87)(47 92 178 86)(48 91 179 85)(49 120 180 84)(50 119 151 83)(51 118 152 82)(52 117 153 81)(53 116 154 80)(54 115 155 79)(55 114 156 78)(56 113 157 77)(57 112 158 76)(58 111 159 75)(59 110 160 74)(60 109 161 73)
(1 83 130 119)(2 72 131 108)(3 61 132 97)(4 80 133 116)(5 69 134 105)(6 88 135 94)(7 77 136 113)(8 66 137 102)(9 85 138 91)(10 74 139 110)(11 63 140 99)(12 82 141 118)(13 71 142 107)(14 90 143 96)(15 79 144 115)(16 68 145 104)(17 87 146 93)(18 76 147 112)(19 65 148 101)(20 84 149 120)(21 73 150 109)(22 62 121 98)(23 81 122 117)(24 70 123 106)(25 89 124 95)(26 78 125 114)(27 67 126 103)(28 86 127 92)(29 75 128 111)(30 64 129 100)(31 230 162 196)(32 219 163 185)(33 238 164 204)(34 227 165 193)(35 216 166 182)(36 235 167 201)(37 224 168 190)(38 213 169 209)(39 232 170 198)(40 221 171 187)(41 240 172 206)(42 229 173 195)(43 218 174 184)(44 237 175 203)(45 226 176 192)(46 215 177 181)(47 234 178 200)(48 223 179 189)(49 212 180 208)(50 231 151 197)(51 220 152 186)(52 239 153 205)(53 228 154 194)(54 217 155 183)(55 236 156 202)(56 225 157 191)(57 214 158 210)(58 233 159 199)(59 222 160 188)(60 211 161 207)
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), (1,144)(2,143)(3,142)(4,141)(5,140)(6,139)(7,138)(8,137)(9,136)(10,135)(11,134)(12,133)(13,132)(14,131)(15,130)(16,129)(17,128)(18,127)(19,126)(20,125)(21,124)(22,123)(23,122)(24,121)(25,150)(26,149)(27,148)(28,147)(29,146)(30,145)(31,169)(32,168)(33,167)(34,166)(35,165)(36,164)(37,163)(38,162)(39,161)(40,160)(41,159)(42,158)(43,157)(44,156)(45,155)(46,154)(47,153)(48,152)(49,151)(50,180)(51,179)(52,178)(53,177)(54,176)(55,175)(56,174)(57,173)(58,172)(59,171)(60,170)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111)(181,231)(182,230)(183,229)(184,228)(185,227)(186,226)(187,225)(188,224)(189,223)(190,222)(191,221)(192,220)(193,219)(194,218)(195,217)(196,216)(197,215)(198,214)(199,213)(200,212)(201,211)(202,240)(203,239)(204,238)(205,237)(206,236)(207,235)(208,234)(209,233)(210,232), (1,231,130,197)(2,230,131,196)(3,229,132,195)(4,228,133,194)(5,227,134,193)(6,226,135,192)(7,225,136,191)(8,224,137,190)(9,223,138,189)(10,222,139,188)(11,221,140,187)(12,220,141,186)(13,219,142,185)(14,218,143,184)(15,217,144,183)(16,216,145,182)(17,215,146,181)(18,214,147,210)(19,213,148,209)(20,212,149,208)(21,211,150,207)(22,240,121,206)(23,239,122,205)(24,238,123,204)(25,237,124,203)(26,236,125,202)(27,235,126,201)(28,234,127,200)(29,233,128,199)(30,232,129,198)(31,108,162,72)(32,107,163,71)(33,106,164,70)(34,105,165,69)(35,104,166,68)(36,103,167,67)(37,102,168,66)(38,101,169,65)(39,100,170,64)(40,99,171,63)(41,98,172,62)(42,97,173,61)(43,96,174,90)(44,95,175,89)(45,94,176,88)(46,93,177,87)(47,92,178,86)(48,91,179,85)(49,120,180,84)(50,119,151,83)(51,118,152,82)(52,117,153,81)(53,116,154,80)(54,115,155,79)(55,114,156,78)(56,113,157,77)(57,112,158,76)(58,111,159,75)(59,110,160,74)(60,109,161,73), (1,83,130,119)(2,72,131,108)(3,61,132,97)(4,80,133,116)(5,69,134,105)(6,88,135,94)(7,77,136,113)(8,66,137,102)(9,85,138,91)(10,74,139,110)(11,63,140,99)(12,82,141,118)(13,71,142,107)(14,90,143,96)(15,79,144,115)(16,68,145,104)(17,87,146,93)(18,76,147,112)(19,65,148,101)(20,84,149,120)(21,73,150,109)(22,62,121,98)(23,81,122,117)(24,70,123,106)(25,89,124,95)(26,78,125,114)(27,67,126,103)(28,86,127,92)(29,75,128,111)(30,64,129,100)(31,230,162,196)(32,219,163,185)(33,238,164,204)(34,227,165,193)(35,216,166,182)(36,235,167,201)(37,224,168,190)(38,213,169,209)(39,232,170,198)(40,221,171,187)(41,240,172,206)(42,229,173,195)(43,218,174,184)(44,237,175,203)(45,226,176,192)(46,215,177,181)(47,234,178,200)(48,223,179,189)(49,212,180,208)(50,231,151,197)(51,220,152,186)(52,239,153,205)(53,228,154,194)(54,217,155,183)(55,236,156,202)(56,225,157,191)(57,214,158,210)(58,233,159,199)(59,222,160,188)(60,211,161,207)>;
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), (1,144)(2,143)(3,142)(4,141)(5,140)(6,139)(7,138)(8,137)(9,136)(10,135)(11,134)(12,133)(13,132)(14,131)(15,130)(16,129)(17,128)(18,127)(19,126)(20,125)(21,124)(22,123)(23,122)(24,121)(25,150)(26,149)(27,148)(28,147)(29,146)(30,145)(31,169)(32,168)(33,167)(34,166)(35,165)(36,164)(37,163)(38,162)(39,161)(40,160)(41,159)(42,158)(43,157)(44,156)(45,155)(46,154)(47,153)(48,152)(49,151)(50,180)(51,179)(52,178)(53,177)(54,176)(55,175)(56,174)(57,173)(58,172)(59,171)(60,170)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111)(181,231)(182,230)(183,229)(184,228)(185,227)(186,226)(187,225)(188,224)(189,223)(190,222)(191,221)(192,220)(193,219)(194,218)(195,217)(196,216)(197,215)(198,214)(199,213)(200,212)(201,211)(202,240)(203,239)(204,238)(205,237)(206,236)(207,235)(208,234)(209,233)(210,232), (1,231,130,197)(2,230,131,196)(3,229,132,195)(4,228,133,194)(5,227,134,193)(6,226,135,192)(7,225,136,191)(8,224,137,190)(9,223,138,189)(10,222,139,188)(11,221,140,187)(12,220,141,186)(13,219,142,185)(14,218,143,184)(15,217,144,183)(16,216,145,182)(17,215,146,181)(18,214,147,210)(19,213,148,209)(20,212,149,208)(21,211,150,207)(22,240,121,206)(23,239,122,205)(24,238,123,204)(25,237,124,203)(26,236,125,202)(27,235,126,201)(28,234,127,200)(29,233,128,199)(30,232,129,198)(31,108,162,72)(32,107,163,71)(33,106,164,70)(34,105,165,69)(35,104,166,68)(36,103,167,67)(37,102,168,66)(38,101,169,65)(39,100,170,64)(40,99,171,63)(41,98,172,62)(42,97,173,61)(43,96,174,90)(44,95,175,89)(45,94,176,88)(46,93,177,87)(47,92,178,86)(48,91,179,85)(49,120,180,84)(50,119,151,83)(51,118,152,82)(52,117,153,81)(53,116,154,80)(54,115,155,79)(55,114,156,78)(56,113,157,77)(57,112,158,76)(58,111,159,75)(59,110,160,74)(60,109,161,73), (1,83,130,119)(2,72,131,108)(3,61,132,97)(4,80,133,116)(5,69,134,105)(6,88,135,94)(7,77,136,113)(8,66,137,102)(9,85,138,91)(10,74,139,110)(11,63,140,99)(12,82,141,118)(13,71,142,107)(14,90,143,96)(15,79,144,115)(16,68,145,104)(17,87,146,93)(18,76,147,112)(19,65,148,101)(20,84,149,120)(21,73,150,109)(22,62,121,98)(23,81,122,117)(24,70,123,106)(25,89,124,95)(26,78,125,114)(27,67,126,103)(28,86,127,92)(29,75,128,111)(30,64,129,100)(31,230,162,196)(32,219,163,185)(33,238,164,204)(34,227,165,193)(35,216,166,182)(36,235,167,201)(37,224,168,190)(38,213,169,209)(39,232,170,198)(40,221,171,187)(41,240,172,206)(42,229,173,195)(43,218,174,184)(44,237,175,203)(45,226,176,192)(46,215,177,181)(47,234,178,200)(48,223,179,189)(49,212,180,208)(50,231,151,197)(51,220,152,186)(52,239,153,205)(53,228,154,194)(54,217,155,183)(55,236,156,202)(56,225,157,191)(57,214,158,210)(58,233,159,199)(59,222,160,188)(60,211,161,207) );
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)], [(1,144),(2,143),(3,142),(4,141),(5,140),(6,139),(7,138),(8,137),(9,136),(10,135),(11,134),(12,133),(13,132),(14,131),(15,130),(16,129),(17,128),(18,127),(19,126),(20,125),(21,124),(22,123),(23,122),(24,121),(25,150),(26,149),(27,148),(28,147),(29,146),(30,145),(31,169),(32,168),(33,167),(34,166),(35,165),(36,164),(37,163),(38,162),(39,161),(40,160),(41,159),(42,158),(43,157),(44,156),(45,155),(46,154),(47,153),(48,152),(49,151),(50,180),(51,179),(52,178),(53,177),(54,176),(55,175),(56,174),(57,173),(58,172),(59,171),(60,170),(61,110),(62,109),(63,108),(64,107),(65,106),(66,105),(67,104),(68,103),(69,102),(70,101),(71,100),(72,99),(73,98),(74,97),(75,96),(76,95),(77,94),(78,93),(79,92),(80,91),(81,120),(82,119),(83,118),(84,117),(85,116),(86,115),(87,114),(88,113),(89,112),(90,111),(181,231),(182,230),(183,229),(184,228),(185,227),(186,226),(187,225),(188,224),(189,223),(190,222),(191,221),(192,220),(193,219),(194,218),(195,217),(196,216),(197,215),(198,214),(199,213),(200,212),(201,211),(202,240),(203,239),(204,238),(205,237),(206,236),(207,235),(208,234),(209,233),(210,232)], [(1,231,130,197),(2,230,131,196),(3,229,132,195),(4,228,133,194),(5,227,134,193),(6,226,135,192),(7,225,136,191),(8,224,137,190),(9,223,138,189),(10,222,139,188),(11,221,140,187),(12,220,141,186),(13,219,142,185),(14,218,143,184),(15,217,144,183),(16,216,145,182),(17,215,146,181),(18,214,147,210),(19,213,148,209),(20,212,149,208),(21,211,150,207),(22,240,121,206),(23,239,122,205),(24,238,123,204),(25,237,124,203),(26,236,125,202),(27,235,126,201),(28,234,127,200),(29,233,128,199),(30,232,129,198),(31,108,162,72),(32,107,163,71),(33,106,164,70),(34,105,165,69),(35,104,166,68),(36,103,167,67),(37,102,168,66),(38,101,169,65),(39,100,170,64),(40,99,171,63),(41,98,172,62),(42,97,173,61),(43,96,174,90),(44,95,175,89),(45,94,176,88),(46,93,177,87),(47,92,178,86),(48,91,179,85),(49,120,180,84),(50,119,151,83),(51,118,152,82),(52,117,153,81),(53,116,154,80),(54,115,155,79),(55,114,156,78),(56,113,157,77),(57,112,158,76),(58,111,159,75),(59,110,160,74),(60,109,161,73)], [(1,83,130,119),(2,72,131,108),(3,61,132,97),(4,80,133,116),(5,69,134,105),(6,88,135,94),(7,77,136,113),(8,66,137,102),(9,85,138,91),(10,74,139,110),(11,63,140,99),(12,82,141,118),(13,71,142,107),(14,90,143,96),(15,79,144,115),(16,68,145,104),(17,87,146,93),(18,76,147,112),(19,65,148,101),(20,84,149,120),(21,73,150,109),(22,62,121,98),(23,81,122,117),(24,70,123,106),(25,89,124,95),(26,78,125,114),(27,67,126,103),(28,86,127,92),(29,75,128,111),(30,64,129,100),(31,230,162,196),(32,219,163,185),(33,238,164,204),(34,227,165,193),(35,216,166,182),(36,235,167,201),(37,224,168,190),(38,213,169,209),(39,232,170,198),(40,221,171,187),(41,240,172,206),(42,229,173,195),(43,218,174,184),(44,237,175,203),(45,226,176,192),(46,215,177,181),(47,234,178,200),(48,223,179,189),(49,212,180,208),(50,231,151,197),(51,220,152,186),(52,239,153,205),(53,228,154,194),(54,217,155,183),(55,236,156,202),(56,225,157,191),(57,214,158,210),(58,233,159,199),(59,222,160,188),(60,211,161,207)]])
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 | 2 | 2 | 12 | 12 | 20 | 20 | 30 | 30 | 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 | 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 | S3 | D4 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C4○D12 | C4○D20 | S3×D4 | S3×Q8 | S3×D5 | D4×D5 | Q8×D5 | C2×S3×D5 | D15⋊Q8 | D6.D10 | D10⋊D6 |
| kernel | D30⋊8Q8 | D30⋊4C4 | Dic15⋊5C4 | C3×C10.D4 | C5×Dic3⋊C4 | C2×C15⋊Q8 | C2×C4×D15 | C10.D4 | Dic15 | D30 | Dic3⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C10 | C6 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 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⋊8Q8 ►in GL6(𝔽61)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 43 | 1 | 0 | 0 | 0 | 0 |
| 43 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 15 | 46 | 0 | 0 | 0 | 0 |
| 11 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 56 |
| 0 | 0 | 0 | 0 | 15 | 47 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 30 | 53 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 41 | 44 |
| 0 | 0 | 0 | 0 | 20 | 20 |
G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,18,0,0,0,0,0,0,60,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[43,43,0,0,0,0,1,18,0,0,0,0,0,0,60,60,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[15,11,0,0,0,0,46,46,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,14,15,0,0,0,0,56,47],[8,30,0,0,0,0,8,53,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,41,20,0,0,0,0,44,20] >;
D30⋊8Q8 in GAP, Magma, Sage, TeX
D_{30}\rtimes_8Q_8 % in TeX
G:=Group("D30:8Q8"); // GroupNames label
G:=SmallGroup(480,453);
// by ID
G=gap.SmallGroup(480,453);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,120,422,219,58,1356,18822]);
// 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^19,c*b*c^-1=a^28*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations