metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊10Q8, C60.92D4, C12.22D20, C6.11(Q8×D5), C4⋊Dic3⋊12D5, C5⋊2(D6⋊3Q8), C6.59(C2×D20), C15⋊9(C22⋊Q8), C10.11(S3×Q8), C30.33(C2×Q8), C3⋊4(D10⋊2Q8), C30.123(C2×D4), (C2×C20).122D6, (C2×Dic10)⋊11S3, (C6×Dic10)⋊11C2, D30⋊4C4.8C2, C30.51(C4○D4), (C2×C12).123D10, C20.38(C3⋊D4), C4.25(C3⋊D20), C2.14(D15⋊Q8), (C2×C30).80C23, C6.Dic10⋊10C2, (C2×Dic5).28D6, C6.10(D4⋊2D5), (C2×C60).199C22, (C2×Dic3).28D10, C2.14(D12⋊D5), C10.29(Q8⋊3S3), (C6×Dic5).47C22, (C10×Dic3).47C22, (C2×Dic15).197C22, (C22×D15).100C22, (C5×C4⋊Dic3)⋊9C2, (C2×C4×D15).15C2, (C2×C4).209(S3×D5), C2.18(C2×C3⋊D20), C10.14(C2×C3⋊D4), C22.165(C2×S3×D5), (C2×C6).92(C22×D5), (C2×C10).92(C22×S3), SmallGroup(480,466)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊10Q8
G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a19, bc=cb, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 796 in 148 conjugacy classes, 54 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, Q8, C23, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, 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, C4⋊Dic3, D6⋊C4, S3×C2×C4, C6×Q8, C5×Dic3, C3×Dic5, Dic15, C60, D30, D30, C2×C30, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, D6⋊3Q8, C3×Dic10, C6×Dic5, C10×Dic3, C4×D15, C2×Dic15, C2×C60, C22×D15, D10⋊2Q8, D30⋊4C4, C6.Dic10, C5×C4⋊Dic3, C6×Dic10, C2×C4×D15, D30⋊10Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, C3⋊D4, C22×S3, C22⋊Q8, D20, C22×D5, S3×Q8, Q8⋊3S3, C2×C3⋊D4, S3×D5, C2×D20, D4⋊2D5, Q8×D5, D6⋊3Q8, C3⋊D20, C2×S3×D5, D10⋊2Q8, D12⋊D5, D15⋊Q8, C2×C3⋊D20, D30⋊10Q8
►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)
(1 160)(2 159)(3 158)(4 157)(5 156)(6 155)(7 154)(8 153)(9 152)(10 151)(11 180)(12 179)(13 178)(14 177)(15 176)(16 175)(17 174)(18 173)(19 172)(20 171)(21 170)(22 169)(23 168)(24 167)(25 166)(26 165)(27 164)(28 163)(29 162)(30 161)(31 112)(32 111)(33 110)(34 109)(35 108)(36 107)(37 106)(38 105)(39 104)(40 103)(41 102)(42 101)(43 100)(44 99)(45 98)(46 97)(47 96)(48 95)(49 94)(50 93)(51 92)(52 91)(53 120)(54 119)(55 118)(56 117)(57 116)(58 115)(59 114)(60 113)(61 143)(62 142)(63 141)(64 140)(65 139)(66 138)(67 137)(68 136)(69 135)(70 134)(71 133)(72 132)(73 131)(74 130)(75 129)(76 128)(77 127)(78 126)(79 125)(80 124)(81 123)(82 122)(83 121)(84 150)(85 149)(86 148)(87 147)(88 146)(89 145)(90 144)(181 224)(182 223)(183 222)(184 221)(185 220)(186 219)(187 218)(188 217)(189 216)(190 215)(191 214)(192 213)(193 212)(194 211)(195 240)(196 239)(197 238)(198 237)(199 236)(200 235)(201 234)(202 233)(203 232)(204 231)(205 230)(206 229)(207 228)(208 227)(209 226)(210 225)
(1 114 161 60)(2 115 162 31)(3 116 163 32)(4 117 164 33)(5 118 165 34)(6 119 166 35)(7 120 167 36)(8 91 168 37)(9 92 169 38)(10 93 170 39)(11 94 171 40)(12 95 172 41)(13 96 173 42)(14 97 174 43)(15 98 175 44)(16 99 176 45)(17 100 177 46)(18 101 178 47)(19 102 179 48)(20 103 180 49)(21 104 151 50)(22 105 152 51)(23 106 153 52)(24 107 154 53)(25 108 155 54)(26 109 156 55)(27 110 157 56)(28 111 158 57)(29 112 159 58)(30 113 160 59)(61 206 135 221)(62 207 136 222)(63 208 137 223)(64 209 138 224)(65 210 139 225)(66 181 140 226)(67 182 141 227)(68 183 142 228)(69 184 143 229)(70 185 144 230)(71 186 145 231)(72 187 146 232)(73 188 147 233)(74 189 148 234)(75 190 149 235)(76 191 150 236)(77 192 121 237)(78 193 122 238)(79 194 123 239)(80 195 124 240)(81 196 125 211)(82 197 126 212)(83 198 127 213)(84 199 128 214)(85 200 129 215)(86 201 130 216)(87 202 131 217)(88 203 132 218)(89 204 133 219)(90 205 134 220)
(1 218 161 203)(2 237 162 192)(3 226 163 181)(4 215 164 200)(5 234 165 189)(6 223 166 208)(7 212 167 197)(8 231 168 186)(9 220 169 205)(10 239 170 194)(11 228 171 183)(12 217 172 202)(13 236 173 191)(14 225 174 210)(15 214 175 199)(16 233 176 188)(17 222 177 207)(18 211 178 196)(19 230 179 185)(20 219 180 204)(21 238 151 193)(22 227 152 182)(23 216 153 201)(24 235 154 190)(25 224 155 209)(26 213 156 198)(27 232 157 187)(28 221 158 206)(29 240 159 195)(30 229 160 184)(31 77 115 121)(32 66 116 140)(33 85 117 129)(34 74 118 148)(35 63 119 137)(36 82 120 126)(37 71 91 145)(38 90 92 134)(39 79 93 123)(40 68 94 142)(41 87 95 131)(42 76 96 150)(43 65 97 139)(44 84 98 128)(45 73 99 147)(46 62 100 136)(47 81 101 125)(48 70 102 144)(49 89 103 133)(50 78 104 122)(51 67 105 141)(52 86 106 130)(53 75 107 149)(54 64 108 138)(55 83 109 127)(56 72 110 146)(57 61 111 135)(58 80 112 124)(59 69 113 143)(60 88 114 132)
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), (1,160)(2,159)(3,158)(4,157)(5,156)(6,155)(7,154)(8,153)(9,152)(10,151)(11,180)(12,179)(13,178)(14,177)(15,176)(16,175)(17,174)(18,173)(19,172)(20,171)(21,170)(22,169)(23,168)(24,167)(25,166)(26,165)(27,164)(28,163)(29,162)(30,161)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113)(61,143)(62,142)(63,141)(64,140)(65,139)(66,138)(67,137)(68,136)(69,135)(70,134)(71,133)(72,132)(73,131)(74,130)(75,129)(76,128)(77,127)(78,126)(79,125)(80,124)(81,123)(82,122)(83,121)(84,150)(85,149)(86,148)(87,147)(88,146)(89,145)(90,144)(181,224)(182,223)(183,222)(184,221)(185,220)(186,219)(187,218)(188,217)(189,216)(190,215)(191,214)(192,213)(193,212)(194,211)(195,240)(196,239)(197,238)(198,237)(199,236)(200,235)(201,234)(202,233)(203,232)(204,231)(205,230)(206,229)(207,228)(208,227)(209,226)(210,225), (1,114,161,60)(2,115,162,31)(3,116,163,32)(4,117,164,33)(5,118,165,34)(6,119,166,35)(7,120,167,36)(8,91,168,37)(9,92,169,38)(10,93,170,39)(11,94,171,40)(12,95,172,41)(13,96,173,42)(14,97,174,43)(15,98,175,44)(16,99,176,45)(17,100,177,46)(18,101,178,47)(19,102,179,48)(20,103,180,49)(21,104,151,50)(22,105,152,51)(23,106,153,52)(24,107,154,53)(25,108,155,54)(26,109,156,55)(27,110,157,56)(28,111,158,57)(29,112,159,58)(30,113,160,59)(61,206,135,221)(62,207,136,222)(63,208,137,223)(64,209,138,224)(65,210,139,225)(66,181,140,226)(67,182,141,227)(68,183,142,228)(69,184,143,229)(70,185,144,230)(71,186,145,231)(72,187,146,232)(73,188,147,233)(74,189,148,234)(75,190,149,235)(76,191,150,236)(77,192,121,237)(78,193,122,238)(79,194,123,239)(80,195,124,240)(81,196,125,211)(82,197,126,212)(83,198,127,213)(84,199,128,214)(85,200,129,215)(86,201,130,216)(87,202,131,217)(88,203,132,218)(89,204,133,219)(90,205,134,220), (1,218,161,203)(2,237,162,192)(3,226,163,181)(4,215,164,200)(5,234,165,189)(6,223,166,208)(7,212,167,197)(8,231,168,186)(9,220,169,205)(10,239,170,194)(11,228,171,183)(12,217,172,202)(13,236,173,191)(14,225,174,210)(15,214,175,199)(16,233,176,188)(17,222,177,207)(18,211,178,196)(19,230,179,185)(20,219,180,204)(21,238,151,193)(22,227,152,182)(23,216,153,201)(24,235,154,190)(25,224,155,209)(26,213,156,198)(27,232,157,187)(28,221,158,206)(29,240,159,195)(30,229,160,184)(31,77,115,121)(32,66,116,140)(33,85,117,129)(34,74,118,148)(35,63,119,137)(36,82,120,126)(37,71,91,145)(38,90,92,134)(39,79,93,123)(40,68,94,142)(41,87,95,131)(42,76,96,150)(43,65,97,139)(44,84,98,128)(45,73,99,147)(46,62,100,136)(47,81,101,125)(48,70,102,144)(49,89,103,133)(50,78,104,122)(51,67,105,141)(52,86,106,130)(53,75,107,149)(54,64,108,138)(55,83,109,127)(56,72,110,146)(57,61,111,135)(58,80,112,124)(59,69,113,143)(60,88,114,132)>;
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), (1,160)(2,159)(3,158)(4,157)(5,156)(6,155)(7,154)(8,153)(9,152)(10,151)(11,180)(12,179)(13,178)(14,177)(15,176)(16,175)(17,174)(18,173)(19,172)(20,171)(21,170)(22,169)(23,168)(24,167)(25,166)(26,165)(27,164)(28,163)(29,162)(30,161)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113)(61,143)(62,142)(63,141)(64,140)(65,139)(66,138)(67,137)(68,136)(69,135)(70,134)(71,133)(72,132)(73,131)(74,130)(75,129)(76,128)(77,127)(78,126)(79,125)(80,124)(81,123)(82,122)(83,121)(84,150)(85,149)(86,148)(87,147)(88,146)(89,145)(90,144)(181,224)(182,223)(183,222)(184,221)(185,220)(186,219)(187,218)(188,217)(189,216)(190,215)(191,214)(192,213)(193,212)(194,211)(195,240)(196,239)(197,238)(198,237)(199,236)(200,235)(201,234)(202,233)(203,232)(204,231)(205,230)(206,229)(207,228)(208,227)(209,226)(210,225), (1,114,161,60)(2,115,162,31)(3,116,163,32)(4,117,164,33)(5,118,165,34)(6,119,166,35)(7,120,167,36)(8,91,168,37)(9,92,169,38)(10,93,170,39)(11,94,171,40)(12,95,172,41)(13,96,173,42)(14,97,174,43)(15,98,175,44)(16,99,176,45)(17,100,177,46)(18,101,178,47)(19,102,179,48)(20,103,180,49)(21,104,151,50)(22,105,152,51)(23,106,153,52)(24,107,154,53)(25,108,155,54)(26,109,156,55)(27,110,157,56)(28,111,158,57)(29,112,159,58)(30,113,160,59)(61,206,135,221)(62,207,136,222)(63,208,137,223)(64,209,138,224)(65,210,139,225)(66,181,140,226)(67,182,141,227)(68,183,142,228)(69,184,143,229)(70,185,144,230)(71,186,145,231)(72,187,146,232)(73,188,147,233)(74,189,148,234)(75,190,149,235)(76,191,150,236)(77,192,121,237)(78,193,122,238)(79,194,123,239)(80,195,124,240)(81,196,125,211)(82,197,126,212)(83,198,127,213)(84,199,128,214)(85,200,129,215)(86,201,130,216)(87,202,131,217)(88,203,132,218)(89,204,133,219)(90,205,134,220), (1,218,161,203)(2,237,162,192)(3,226,163,181)(4,215,164,200)(5,234,165,189)(6,223,166,208)(7,212,167,197)(8,231,168,186)(9,220,169,205)(10,239,170,194)(11,228,171,183)(12,217,172,202)(13,236,173,191)(14,225,174,210)(15,214,175,199)(16,233,176,188)(17,222,177,207)(18,211,178,196)(19,230,179,185)(20,219,180,204)(21,238,151,193)(22,227,152,182)(23,216,153,201)(24,235,154,190)(25,224,155,209)(26,213,156,198)(27,232,157,187)(28,221,158,206)(29,240,159,195)(30,229,160,184)(31,77,115,121)(32,66,116,140)(33,85,117,129)(34,74,118,148)(35,63,119,137)(36,82,120,126)(37,71,91,145)(38,90,92,134)(39,79,93,123)(40,68,94,142)(41,87,95,131)(42,76,96,150)(43,65,97,139)(44,84,98,128)(45,73,99,147)(46,62,100,136)(47,81,101,125)(48,70,102,144)(49,89,103,133)(50,78,104,122)(51,67,105,141)(52,86,106,130)(53,75,107,149)(54,64,108,138)(55,83,109,127)(56,72,110,146)(57,61,111,135)(58,80,112,124)(59,69,113,143)(60,88,114,132) );
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)], [(1,160),(2,159),(3,158),(4,157),(5,156),(6,155),(7,154),(8,153),(9,152),(10,151),(11,180),(12,179),(13,178),(14,177),(15,176),(16,175),(17,174),(18,173),(19,172),(20,171),(21,170),(22,169),(23,168),(24,167),(25,166),(26,165),(27,164),(28,163),(29,162),(30,161),(31,112),(32,111),(33,110),(34,109),(35,108),(36,107),(37,106),(38,105),(39,104),(40,103),(41,102),(42,101),(43,100),(44,99),(45,98),(46,97),(47,96),(48,95),(49,94),(50,93),(51,92),(52,91),(53,120),(54,119),(55,118),(56,117),(57,116),(58,115),(59,114),(60,113),(61,143),(62,142),(63,141),(64,140),(65,139),(66,138),(67,137),(68,136),(69,135),(70,134),(71,133),(72,132),(73,131),(74,130),(75,129),(76,128),(77,127),(78,126),(79,125),(80,124),(81,123),(82,122),(83,121),(84,150),(85,149),(86,148),(87,147),(88,146),(89,145),(90,144),(181,224),(182,223),(183,222),(184,221),(185,220),(186,219),(187,218),(188,217),(189,216),(190,215),(191,214),(192,213),(193,212),(194,211),(195,240),(196,239),(197,238),(198,237),(199,236),(200,235),(201,234),(202,233),(203,232),(204,231),(205,230),(206,229),(207,228),(208,227),(209,226),(210,225)], [(1,114,161,60),(2,115,162,31),(3,116,163,32),(4,117,164,33),(5,118,165,34),(6,119,166,35),(7,120,167,36),(8,91,168,37),(9,92,169,38),(10,93,170,39),(11,94,171,40),(12,95,172,41),(13,96,173,42),(14,97,174,43),(15,98,175,44),(16,99,176,45),(17,100,177,46),(18,101,178,47),(19,102,179,48),(20,103,180,49),(21,104,151,50),(22,105,152,51),(23,106,153,52),(24,107,154,53),(25,108,155,54),(26,109,156,55),(27,110,157,56),(28,111,158,57),(29,112,159,58),(30,113,160,59),(61,206,135,221),(62,207,136,222),(63,208,137,223),(64,209,138,224),(65,210,139,225),(66,181,140,226),(67,182,141,227),(68,183,142,228),(69,184,143,229),(70,185,144,230),(71,186,145,231),(72,187,146,232),(73,188,147,233),(74,189,148,234),(75,190,149,235),(76,191,150,236),(77,192,121,237),(78,193,122,238),(79,194,123,239),(80,195,124,240),(81,196,125,211),(82,197,126,212),(83,198,127,213),(84,199,128,214),(85,200,129,215),(86,201,130,216),(87,202,131,217),(88,203,132,218),(89,204,133,219),(90,205,134,220)], [(1,218,161,203),(2,237,162,192),(3,226,163,181),(4,215,164,200),(5,234,165,189),(6,223,166,208),(7,212,167,197),(8,231,168,186),(9,220,169,205),(10,239,170,194),(11,228,171,183),(12,217,172,202),(13,236,173,191),(14,225,174,210),(15,214,175,199),(16,233,176,188),(17,222,177,207),(18,211,178,196),(19,230,179,185),(20,219,180,204),(21,238,151,193),(22,227,152,182),(23,216,153,201),(24,235,154,190),(25,224,155,209),(26,213,156,198),(27,232,157,187),(28,221,158,206),(29,240,159,195),(30,229,160,184),(31,77,115,121),(32,66,116,140),(33,85,117,129),(34,74,118,148),(35,63,119,137),(36,82,120,126),(37,71,91,145),(38,90,92,134),(39,79,93,123),(40,68,94,142),(41,87,95,131),(42,76,96,150),(43,65,97,139),(44,84,98,128),(45,73,99,147),(46,62,100,136),(47,81,101,125),(48,70,102,144),(49,89,103,133),(50,78,104,122),(51,67,105,141),(52,86,106,130),(53,75,107,149),(54,64,108,138),(55,83,109,127),(56,72,110,146),(57,61,111,135),(58,80,112,124),(59,69,113,143),(60,88,114,132)]])
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 | 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 | S3 | D4 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | D20 | S3×Q8 | Q8⋊3S3 | S3×D5 | D4⋊2D5 | Q8×D5 | C3⋊D20 | C2×S3×D5 | D12⋊D5 | D15⋊Q8 |
| kernel | D30⋊10Q8 | D30⋊4C4 | C6.Dic10 | C5×C4⋊Dic3 | C6×Dic10 | C2×C4×D15 | C2×Dic10 | C60 | D30 | C4⋊Dic3 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C20 | C12 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of D30⋊10Q8 ►in GL6(𝔽61)
| 18 | 1 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 23 | 53 | 0 | 0 | 0 | 0 |
| 5 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 56 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 0 | 55 | 11 |
| 47 | 59 | 0 | 0 | 0 | 0 |
| 6 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 58 |
| 0 | 0 | 0 | 0 | 14 | 38 |
G:=sub<GL(6,GF(61))| [18,60,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[23,5,0,0,0,0,53,38,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,56,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,50,55,0,0,0,0,0,11],[47,6,0,0,0,0,59,14,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,14,0,0,0,0,58,38] >;
D30⋊10Q8 in GAP, Magma, Sage, TeX
D_{30}\rtimes_{10}Q_8 % in TeX
G:=Group("D30:10Q8"); // GroupNames label
G:=SmallGroup(480,466);
// by ID
G=gap.SmallGroup(480,466);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,64,422,219,100,1356,18822]);
// 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^19,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations