metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊5Q8, Dic15.34D4, C4⋊C4⋊4D15, C6.42(Q8×D5), C2.6(Q8×D15), C5⋊5(D6⋊Q8), C2.14(D4×D15), (C2×C20).39D6, (C2×C4).13D30, C6.107(D4×D5), C30.95(C2×Q8), C10.42(S3×Q8), (C2×Dic30)⋊7C2, C3⋊5(D10⋊Q8), C10.109(S3×D4), C30.314(C2×D4), C15⋊28(C22⋊Q8), D30⋊3C4.2C2, (C2×C12).210D10, C30.4Q8⋊34C2, C30.174(C4○D4), C6.101(C4○D20), (C2×C60).180C22, (C2×C30).293C23, C10.101(C4○D12), C2.15(D60⋊11C2), C22.51(C22×D15), (C2×Dic15).11C22, (C22×D15).83C22, (C5×C4⋊C4)⋊7S3, (C3×C4⋊C4)⋊7D5, (C15×C4⋊C4)⋊7C2, (C2×C4×D15).11C2, (C2×C6).289(C22×D5), (C2×C10).288(C22×S3), SmallGroup(480,861)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊5Q8
G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=cac-1=dad-1=a-1, cbc-1=a13b, dbd-1=a28b, dcd-1=c-1 >
Subgroups: 900 in 148 conjugacy classes, 51 normal (47 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, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, D10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×C12, C22×S3, D15, C30, C22⋊Q8, Dic10, C4×D5, C2×Dic5, C2×C20, C22×D5, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, Dic15, Dic15, C60, D30, D30, C2×C30, C10.D4, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, D6⋊Q8, Dic30, C4×D15, C2×Dic15, C2×C60, C22×D15, D10⋊Q8, C30.4Q8, D30⋊3C4, C15×C4⋊C4, C2×Dic30, C2×C4×D15, D30⋊5Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, C22×S3, D15, C22⋊Q8, C22×D5, C4○D12, S3×D4, S3×Q8, D30, C4○D20, D4×D5, Q8×D5, D6⋊Q8, C22×D15, D10⋊Q8, D60⋊11C2, D4×D15, Q8×D15, D30⋊5Q8
►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)
(1 233)(2 232)(3 231)(4 230)(5 229)(6 228)(7 227)(8 226)(9 225)(10 224)(11 223)(12 222)(13 221)(14 220)(15 219)(16 218)(17 217)(18 216)(19 215)(20 214)(21 213)(22 212)(23 211)(24 240)(25 239)(26 238)(27 237)(28 236)(29 235)(30 234)(31 208)(32 207)(33 206)(34 205)(35 204)(36 203)(37 202)(38 201)(39 200)(40 199)(41 198)(42 197)(43 196)(44 195)(45 194)(46 193)(47 192)(48 191)(49 190)(50 189)(51 188)(52 187)(53 186)(54 185)(55 184)(56 183)(57 182)(58 181)(59 210)(60 209)(61 156)(62 155)(63 154)(64 153)(65 152)(66 151)(67 180)(68 179)(69 178)(70 177)(71 176)(72 175)(73 174)(74 173)(75 172)(76 171)(77 170)(78 169)(79 168)(80 167)(81 166)(82 165)(83 164)(84 163)(85 162)(86 161)(87 160)(88 159)(89 158)(90 157)(91 125)(92 124)(93 123)(94 122)(95 121)(96 150)(97 149)(98 148)(99 147)(100 146)(101 145)(102 144)(103 143)(104 142)(105 141)(106 140)(107 139)(108 138)(109 137)(110 136)(111 135)(112 134)(113 133)(114 132)(115 131)(116 130)(117 129)(118 128)(119 127)(120 126)
(1 49 219 191)(2 48 220 190)(3 47 221 189)(4 46 222 188)(5 45 223 187)(6 44 224 186)(7 43 225 185)(8 42 226 184)(9 41 227 183)(10 40 228 182)(11 39 229 181)(12 38 230 210)(13 37 231 209)(14 36 232 208)(15 35 233 207)(16 34 234 206)(17 33 235 205)(18 32 236 204)(19 31 237 203)(20 60 238 202)(21 59 239 201)(22 58 240 200)(23 57 211 199)(24 56 212 198)(25 55 213 197)(26 54 214 196)(27 53 215 195)(28 52 216 194)(29 51 217 193)(30 50 218 192)(61 96 178 145)(62 95 179 144)(63 94 180 143)(64 93 151 142)(65 92 152 141)(66 91 153 140)(67 120 154 139)(68 119 155 138)(69 118 156 137)(70 117 157 136)(71 116 158 135)(72 115 159 134)(73 114 160 133)(74 113 161 132)(75 112 162 131)(76 111 163 130)(77 110 164 129)(78 109 165 128)(79 108 166 127)(80 107 167 126)(81 106 168 125)(82 105 169 124)(83 104 170 123)(84 103 171 122)(85 102 172 121)(86 101 173 150)(87 100 174 149)(88 99 175 148)(89 98 176 147)(90 97 177 146)
(1 175 219 88)(2 174 220 87)(3 173 221 86)(4 172 222 85)(5 171 223 84)(6 170 224 83)(7 169 225 82)(8 168 226 81)(9 167 227 80)(10 166 228 79)(11 165 229 78)(12 164 230 77)(13 163 231 76)(14 162 232 75)(15 161 233 74)(16 160 234 73)(17 159 235 72)(18 158 236 71)(19 157 237 70)(20 156 238 69)(21 155 239 68)(22 154 240 67)(23 153 211 66)(24 152 212 65)(25 151 213 64)(26 180 214 63)(27 179 215 62)(28 178 216 61)(29 177 217 90)(30 176 218 89)(31 117 203 136)(32 116 204 135)(33 115 205 134)(34 114 206 133)(35 113 207 132)(36 112 208 131)(37 111 209 130)(38 110 210 129)(39 109 181 128)(40 108 182 127)(41 107 183 126)(42 106 184 125)(43 105 185 124)(44 104 186 123)(45 103 187 122)(46 102 188 121)(47 101 189 150)(48 100 190 149)(49 99 191 148)(50 98 192 147)(51 97 193 146)(52 96 194 145)(53 95 195 144)(54 94 196 143)(55 93 197 142)(56 92 198 141)(57 91 199 140)(58 120 200 139)(59 119 201 138)(60 118 202 137)
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), (1,233)(2,232)(3,231)(4,230)(5,229)(6,228)(7,227)(8,226)(9,225)(10,224)(11,223)(12,222)(13,221)(14,220)(15,219)(16,218)(17,217)(18,216)(19,215)(20,214)(21,213)(22,212)(23,211)(24,240)(25,239)(26,238)(27,237)(28,236)(29,235)(30,234)(31,208)(32,207)(33,206)(34,205)(35,204)(36,203)(37,202)(38,201)(39,200)(40,199)(41,198)(42,197)(43,196)(44,195)(45,194)(46,193)(47,192)(48,191)(49,190)(50,189)(51,188)(52,187)(53,186)(54,185)(55,184)(56,183)(57,182)(58,181)(59,210)(60,209)(61,156)(62,155)(63,154)(64,153)(65,152)(66,151)(67,180)(68,179)(69,178)(70,177)(71,176)(72,175)(73,174)(74,173)(75,172)(76,171)(77,170)(78,169)(79,168)(80,167)(81,166)(82,165)(83,164)(84,163)(85,162)(86,161)(87,160)(88,159)(89,158)(90,157)(91,125)(92,124)(93,123)(94,122)(95,121)(96,150)(97,149)(98,148)(99,147)(100,146)(101,145)(102,144)(103,143)(104,142)(105,141)(106,140)(107,139)(108,138)(109,137)(110,136)(111,135)(112,134)(113,133)(114,132)(115,131)(116,130)(117,129)(118,128)(119,127)(120,126), (1,49,219,191)(2,48,220,190)(3,47,221,189)(4,46,222,188)(5,45,223,187)(6,44,224,186)(7,43,225,185)(8,42,226,184)(9,41,227,183)(10,40,228,182)(11,39,229,181)(12,38,230,210)(13,37,231,209)(14,36,232,208)(15,35,233,207)(16,34,234,206)(17,33,235,205)(18,32,236,204)(19,31,237,203)(20,60,238,202)(21,59,239,201)(22,58,240,200)(23,57,211,199)(24,56,212,198)(25,55,213,197)(26,54,214,196)(27,53,215,195)(28,52,216,194)(29,51,217,193)(30,50,218,192)(61,96,178,145)(62,95,179,144)(63,94,180,143)(64,93,151,142)(65,92,152,141)(66,91,153,140)(67,120,154,139)(68,119,155,138)(69,118,156,137)(70,117,157,136)(71,116,158,135)(72,115,159,134)(73,114,160,133)(74,113,161,132)(75,112,162,131)(76,111,163,130)(77,110,164,129)(78,109,165,128)(79,108,166,127)(80,107,167,126)(81,106,168,125)(82,105,169,124)(83,104,170,123)(84,103,171,122)(85,102,172,121)(86,101,173,150)(87,100,174,149)(88,99,175,148)(89,98,176,147)(90,97,177,146), (1,175,219,88)(2,174,220,87)(3,173,221,86)(4,172,222,85)(5,171,223,84)(6,170,224,83)(7,169,225,82)(8,168,226,81)(9,167,227,80)(10,166,228,79)(11,165,229,78)(12,164,230,77)(13,163,231,76)(14,162,232,75)(15,161,233,74)(16,160,234,73)(17,159,235,72)(18,158,236,71)(19,157,237,70)(20,156,238,69)(21,155,239,68)(22,154,240,67)(23,153,211,66)(24,152,212,65)(25,151,213,64)(26,180,214,63)(27,179,215,62)(28,178,216,61)(29,177,217,90)(30,176,218,89)(31,117,203,136)(32,116,204,135)(33,115,205,134)(34,114,206,133)(35,113,207,132)(36,112,208,131)(37,111,209,130)(38,110,210,129)(39,109,181,128)(40,108,182,127)(41,107,183,126)(42,106,184,125)(43,105,185,124)(44,104,186,123)(45,103,187,122)(46,102,188,121)(47,101,189,150)(48,100,190,149)(49,99,191,148)(50,98,192,147)(51,97,193,146)(52,96,194,145)(53,95,195,144)(54,94,196,143)(55,93,197,142)(56,92,198,141)(57,91,199,140)(58,120,200,139)(59,119,201,138)(60,118,202,137)>;
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), (1,233)(2,232)(3,231)(4,230)(5,229)(6,228)(7,227)(8,226)(9,225)(10,224)(11,223)(12,222)(13,221)(14,220)(15,219)(16,218)(17,217)(18,216)(19,215)(20,214)(21,213)(22,212)(23,211)(24,240)(25,239)(26,238)(27,237)(28,236)(29,235)(30,234)(31,208)(32,207)(33,206)(34,205)(35,204)(36,203)(37,202)(38,201)(39,200)(40,199)(41,198)(42,197)(43,196)(44,195)(45,194)(46,193)(47,192)(48,191)(49,190)(50,189)(51,188)(52,187)(53,186)(54,185)(55,184)(56,183)(57,182)(58,181)(59,210)(60,209)(61,156)(62,155)(63,154)(64,153)(65,152)(66,151)(67,180)(68,179)(69,178)(70,177)(71,176)(72,175)(73,174)(74,173)(75,172)(76,171)(77,170)(78,169)(79,168)(80,167)(81,166)(82,165)(83,164)(84,163)(85,162)(86,161)(87,160)(88,159)(89,158)(90,157)(91,125)(92,124)(93,123)(94,122)(95,121)(96,150)(97,149)(98,148)(99,147)(100,146)(101,145)(102,144)(103,143)(104,142)(105,141)(106,140)(107,139)(108,138)(109,137)(110,136)(111,135)(112,134)(113,133)(114,132)(115,131)(116,130)(117,129)(118,128)(119,127)(120,126), (1,49,219,191)(2,48,220,190)(3,47,221,189)(4,46,222,188)(5,45,223,187)(6,44,224,186)(7,43,225,185)(8,42,226,184)(9,41,227,183)(10,40,228,182)(11,39,229,181)(12,38,230,210)(13,37,231,209)(14,36,232,208)(15,35,233,207)(16,34,234,206)(17,33,235,205)(18,32,236,204)(19,31,237,203)(20,60,238,202)(21,59,239,201)(22,58,240,200)(23,57,211,199)(24,56,212,198)(25,55,213,197)(26,54,214,196)(27,53,215,195)(28,52,216,194)(29,51,217,193)(30,50,218,192)(61,96,178,145)(62,95,179,144)(63,94,180,143)(64,93,151,142)(65,92,152,141)(66,91,153,140)(67,120,154,139)(68,119,155,138)(69,118,156,137)(70,117,157,136)(71,116,158,135)(72,115,159,134)(73,114,160,133)(74,113,161,132)(75,112,162,131)(76,111,163,130)(77,110,164,129)(78,109,165,128)(79,108,166,127)(80,107,167,126)(81,106,168,125)(82,105,169,124)(83,104,170,123)(84,103,171,122)(85,102,172,121)(86,101,173,150)(87,100,174,149)(88,99,175,148)(89,98,176,147)(90,97,177,146), (1,175,219,88)(2,174,220,87)(3,173,221,86)(4,172,222,85)(5,171,223,84)(6,170,224,83)(7,169,225,82)(8,168,226,81)(9,167,227,80)(10,166,228,79)(11,165,229,78)(12,164,230,77)(13,163,231,76)(14,162,232,75)(15,161,233,74)(16,160,234,73)(17,159,235,72)(18,158,236,71)(19,157,237,70)(20,156,238,69)(21,155,239,68)(22,154,240,67)(23,153,211,66)(24,152,212,65)(25,151,213,64)(26,180,214,63)(27,179,215,62)(28,178,216,61)(29,177,217,90)(30,176,218,89)(31,117,203,136)(32,116,204,135)(33,115,205,134)(34,114,206,133)(35,113,207,132)(36,112,208,131)(37,111,209,130)(38,110,210,129)(39,109,181,128)(40,108,182,127)(41,107,183,126)(42,106,184,125)(43,105,185,124)(44,104,186,123)(45,103,187,122)(46,102,188,121)(47,101,189,150)(48,100,190,149)(49,99,191,148)(50,98,192,147)(51,97,193,146)(52,96,194,145)(53,95,195,144)(54,94,196,143)(55,93,197,142)(56,92,198,141)(57,91,199,140)(58,120,200,139)(59,119,201,138)(60,118,202,137) );
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)], [(1,233),(2,232),(3,231),(4,230),(5,229),(6,228),(7,227),(8,226),(9,225),(10,224),(11,223),(12,222),(13,221),(14,220),(15,219),(16,218),(17,217),(18,216),(19,215),(20,214),(21,213),(22,212),(23,211),(24,240),(25,239),(26,238),(27,237),(28,236),(29,235),(30,234),(31,208),(32,207),(33,206),(34,205),(35,204),(36,203),(37,202),(38,201),(39,200),(40,199),(41,198),(42,197),(43,196),(44,195),(45,194),(46,193),(47,192),(48,191),(49,190),(50,189),(51,188),(52,187),(53,186),(54,185),(55,184),(56,183),(57,182),(58,181),(59,210),(60,209),(61,156),(62,155),(63,154),(64,153),(65,152),(66,151),(67,180),(68,179),(69,178),(70,177),(71,176),(72,175),(73,174),(74,173),(75,172),(76,171),(77,170),(78,169),(79,168),(80,167),(81,166),(82,165),(83,164),(84,163),(85,162),(86,161),(87,160),(88,159),(89,158),(90,157),(91,125),(92,124),(93,123),(94,122),(95,121),(96,150),(97,149),(98,148),(99,147),(100,146),(101,145),(102,144),(103,143),(104,142),(105,141),(106,140),(107,139),(108,138),(109,137),(110,136),(111,135),(112,134),(113,133),(114,132),(115,131),(116,130),(117,129),(118,128),(119,127),(120,126)], [(1,49,219,191),(2,48,220,190),(3,47,221,189),(4,46,222,188),(5,45,223,187),(6,44,224,186),(7,43,225,185),(8,42,226,184),(9,41,227,183),(10,40,228,182),(11,39,229,181),(12,38,230,210),(13,37,231,209),(14,36,232,208),(15,35,233,207),(16,34,234,206),(17,33,235,205),(18,32,236,204),(19,31,237,203),(20,60,238,202),(21,59,239,201),(22,58,240,200),(23,57,211,199),(24,56,212,198),(25,55,213,197),(26,54,214,196),(27,53,215,195),(28,52,216,194),(29,51,217,193),(30,50,218,192),(61,96,178,145),(62,95,179,144),(63,94,180,143),(64,93,151,142),(65,92,152,141),(66,91,153,140),(67,120,154,139),(68,119,155,138),(69,118,156,137),(70,117,157,136),(71,116,158,135),(72,115,159,134),(73,114,160,133),(74,113,161,132),(75,112,162,131),(76,111,163,130),(77,110,164,129),(78,109,165,128),(79,108,166,127),(80,107,167,126),(81,106,168,125),(82,105,169,124),(83,104,170,123),(84,103,171,122),(85,102,172,121),(86,101,173,150),(87,100,174,149),(88,99,175,148),(89,98,176,147),(90,97,177,146)], [(1,175,219,88),(2,174,220,87),(3,173,221,86),(4,172,222,85),(5,171,223,84),(6,170,224,83),(7,169,225,82),(8,168,226,81),(9,167,227,80),(10,166,228,79),(11,165,229,78),(12,164,230,77),(13,163,231,76),(14,162,232,75),(15,161,233,74),(16,160,234,73),(17,159,235,72),(18,158,236,71),(19,157,237,70),(20,156,238,69),(21,155,239,68),(22,154,240,67),(23,153,211,66),(24,152,212,65),(25,151,213,64),(26,180,214,63),(27,179,215,62),(28,178,216,61),(29,177,217,90),(30,176,218,89),(31,117,203,136),(32,116,204,135),(33,115,205,134),(34,114,206,133),(35,113,207,132),(36,112,208,131),(37,111,209,130),(38,110,210,129),(39,109,181,128),(40,108,182,127),(41,107,183,126),(42,106,184,125),(43,105,185,124),(44,104,186,123),(45,103,187,122),(46,102,188,121),(47,101,189,150),(48,100,190,149),(49,99,191,148),(50,98,192,147),(51,97,193,146),(52,96,194,145),(53,95,195,144),(54,94,196,143),(55,93,197,142),(56,92,198,141),(57,91,199,140),(58,120,200,139),(59,119,201,138),(60,118,202,137)]])
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 | D15 | C4○D12 | D30 | C4○D20 | D60⋊11C2 | S3×D4 | S3×Q8 | D4×D5 | Q8×D5 | D4×D15 | Q8×D15 |
| kernel | D30⋊5Q8 | C30.4Q8 | D30⋊3C4 | C15×C4⋊C4 | C2×Dic30 | C2×C4×D15 | C5×C4⋊C4 | Dic15 | D30 | C3×C4⋊C4 | C2×C20 | C30 | C2×C12 | C4⋊C4 | C10 | C2×C4 | C6 | C2 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 2 | 6 | 4 | 4 | 12 | 8 | 16 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D30⋊5Q8 ►in GL6(𝔽61)
| 4 | 15 | 0 | 0 | 0 | 0 |
| 52 | 58 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 15 | 0 | 0 | 0 | 0 |
| 60 | 57 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 24 | 0 | 0 |
| 0 | 0 | 30 | 47 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 2 | 26 | 0 | 0 | 0 | 0 |
| 35 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 14 | 0 | 0 |
| 0 | 0 | 36 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 55 |
| 0 | 0 | 0 | 0 | 41 | 60 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 58 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 47 | 0 | 0 |
| 0 | 0 | 25 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 58 |
| 0 | 0 | 0 | 0 | 7 | 9 |
G:=sub<GL(6,GF(61))| [4,52,0,0,0,0,15,58,0,0,0,0,0,0,0,1,0,0,0,0,60,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,60,0,0,0,0,15,57,0,0,0,0,0,0,14,30,0,0,0,0,24,47,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[2,35,0,0,0,0,26,59,0,0,0,0,0,0,31,36,0,0,0,0,14,30,0,0,0,0,0,0,1,41,0,0,0,0,55,60],[50,58,0,0,0,0,0,11,0,0,0,0,0,0,30,25,0,0,0,0,47,31,0,0,0,0,0,0,52,7,0,0,0,0,58,9] >;
D30⋊5Q8 in GAP, Magma, Sage, TeX
D_{30}\rtimes_5Q_8 % in TeX
G:=Group("D30:5Q8"); // GroupNames label
G:=SmallGroup(480,861);
// by ID
G=gap.SmallGroup(480,861);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,590,219,100,2693,18822]);
// 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=d*a*d^-1=a^-1,c*b*c^-1=a^13*b,d*b*d^-1=a^28*b,d*c*d^-1=c^-1>;
// generators/relations