metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊7D4, C6.85(D4×D5), C5⋊6(Dic3⋊D4), C3⋊2(C20⋊2D4), (C5×Dic3)⋊4D4, C10.87(S3×D4), C23.D5⋊8S3, C15⋊19(C4⋊D4), D6⋊Dic5⋊32C2, C30.237(C2×D4), C23.21(S3×D5), Dic3⋊3(C5⋊D4), C6.Dic10⋊35C2, (C2×Dic5).62D6, (C22×C10).51D6, (C22×C6).36D10, C30.151(C4○D4), C10.84(C4○D12), C6.57(D4⋊2D5), (C2×C30).199C23, (C22×S3).28D10, C2.39(D10⋊D6), (C2×Dic3).123D10, (C22×C30).61C22, C2.27(Dic3.D10), (C6×Dic5).115C22, (C22×D15).65C22, (C10×Dic3).115C22, (C2×Dic15).138C22, (C2×C3⋊D4)⋊2D5, (C10×C3⋊D4)⋊2C2, C6.64(C2×C5⋊D4), C2.39(S3×C5⋊D4), (C2×C15⋊7D4)⋊14C2, (C2×C5⋊D12)⋊11C2, C22.231(C2×S3×D5), (C2×D30.C2)⋊16C2, (S3×C2×C10).50C22, (C3×C23.D5)⋊10C2, (C2×C6).211(C22×D5), (C2×C10).211(C22×S3), SmallGroup(480,633)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊7D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=dad=a11, cbc-1=a10b, dbd=a25b, dcd=c-1 >
Subgroups: 1068 in 188 conjugacy classes, 50 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, D15, C30, C30, C4⋊D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, S3×C10, D30, D30, C2×C30, C2×C30, C4⋊Dic5, C23.D5, C23.D5, C2×C4×D5, C2×C5⋊D4, D4×C10, Dic3⋊D4, D30.C2, C5⋊D12, C6×Dic5, C10×Dic3, C5×C3⋊D4, C2×Dic15, C15⋊7D4, S3×C2×C10, C22×D15, C22×C30, C20⋊2D4, D6⋊Dic5, C6.Dic10, C3×C23.D5, C2×D30.C2, C2×C5⋊D12, C10×C3⋊D4, C2×C15⋊7D4, D30⋊7D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C4⋊D4, C5⋊D4, C22×D5, C4○D12, S3×D4, S3×D5, D4×D5, D4⋊2D5, C2×C5⋊D4, Dic3⋊D4, C2×S3×D5, C20⋊2D4, Dic3.D10, S3×C5⋊D4, D10⋊D6, D30⋊7D4
►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)
(1 239)(2 238)(3 237)(4 236)(5 235)(6 234)(7 233)(8 232)(9 231)(10 230)(11 229)(12 228)(13 227)(14 226)(15 225)(16 224)(17 223)(18 222)(19 221)(20 220)(21 219)(22 218)(23 217)(24 216)(25 215)(26 214)(27 213)(28 212)(29 211)(30 240)(31 189)(32 188)(33 187)(34 186)(35 185)(36 184)(37 183)(38 182)(39 181)(40 210)(41 209)(42 208)(43 207)(44 206)(45 205)(46 204)(47 203)(48 202)(49 201)(50 200)(51 199)(52 198)(53 197)(54 196)(55 195)(56 194)(57 193)(58 192)(59 191)(60 190)(61 176)(62 175)(63 174)(64 173)(65 172)(66 171)(67 170)(68 169)(69 168)(70 167)(71 166)(72 165)(73 164)(74 163)(75 162)(76 161)(77 160)(78 159)(79 158)(80 157)(81 156)(82 155)(83 154)(84 153)(85 152)(86 151)(87 180)(88 179)(89 178)(90 177)(91 142)(92 141)(93 140)(94 139)(95 138)(96 137)(97 136)(98 135)(99 134)(100 133)(101 132)(102 131)(103 130)(104 129)(105 128)(106 127)(107 126)(108 125)(109 124)(110 123)(111 122)(112 121)(113 150)(114 149)(115 148)(116 147)(117 146)(118 145)(119 144)(120 143)
(1 131 240 103)(2 142 211 114)(3 123 212 95)(4 134 213 106)(5 145 214 117)(6 126 215 98)(7 137 216 109)(8 148 217 120)(9 129 218 101)(10 140 219 112)(11 121 220 93)(12 132 221 104)(13 143 222 115)(14 124 223 96)(15 135 224 107)(16 146 225 118)(17 127 226 99)(18 138 227 110)(19 149 228 91)(20 130 229 102)(21 141 230 113)(22 122 231 94)(23 133 232 105)(24 144 233 116)(25 125 234 97)(26 136 235 108)(27 147 236 119)(28 128 237 100)(29 139 238 111)(30 150 239 92)(31 156 185 87)(32 167 186 68)(33 178 187 79)(34 159 188 90)(35 170 189 71)(36 151 190 82)(37 162 191 63)(38 173 192 74)(39 154 193 85)(40 165 194 66)(41 176 195 77)(42 157 196 88)(43 168 197 69)(44 179 198 80)(45 160 199 61)(46 171 200 72)(47 152 201 83)(48 163 202 64)(49 174 203 75)(50 155 204 86)(51 166 205 67)(52 177 206 78)(53 158 207 89)(54 169 208 70)(55 180 209 81)(56 161 210 62)(57 172 181 73)(58 153 182 84)(59 164 183 65)(60 175 184 76)
(1 210)(2 191)(3 202)(4 183)(5 194)(6 205)(7 186)(8 197)(9 208)(10 189)(11 200)(12 181)(13 192)(14 203)(15 184)(16 195)(17 206)(18 187)(19 198)(20 209)(21 190)(22 201)(23 182)(24 193)(25 204)(26 185)(27 196)(28 207)(29 188)(30 199)(31 235)(32 216)(33 227)(34 238)(35 219)(36 230)(37 211)(38 222)(39 233)(40 214)(41 225)(42 236)(43 217)(44 228)(45 239)(46 220)(47 231)(48 212)(49 223)(50 234)(51 215)(52 226)(53 237)(54 218)(55 229)(56 240)(57 221)(58 232)(59 213)(60 224)(61 92)(62 103)(63 114)(64 95)(65 106)(66 117)(67 98)(68 109)(69 120)(70 101)(71 112)(72 93)(73 104)(74 115)(75 96)(76 107)(77 118)(78 99)(79 110)(80 91)(81 102)(82 113)(83 94)(84 105)(85 116)(86 97)(87 108)(88 119)(89 100)(90 111)(121 171)(122 152)(123 163)(124 174)(125 155)(126 166)(127 177)(128 158)(129 169)(130 180)(131 161)(132 172)(133 153)(134 164)(135 175)(136 156)(137 167)(138 178)(139 159)(140 170)(141 151)(142 162)(143 173)(144 154)(145 165)(146 176)(147 157)(148 168)(149 179)(150 160)
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), (1,239)(2,238)(3,237)(4,236)(5,235)(6,234)(7,233)(8,232)(9,231)(10,230)(11,229)(12,228)(13,227)(14,226)(15,225)(16,224)(17,223)(18,222)(19,221)(20,220)(21,219)(22,218)(23,217)(24,216)(25,215)(26,214)(27,213)(28,212)(29,211)(30,240)(31,189)(32,188)(33,187)(34,186)(35,185)(36,184)(37,183)(38,182)(39,181)(40,210)(41,209)(42,208)(43,207)(44,206)(45,205)(46,204)(47,203)(48,202)(49,201)(50,200)(51,199)(52,198)(53,197)(54,196)(55,195)(56,194)(57,193)(58,192)(59,191)(60,190)(61,176)(62,175)(63,174)(64,173)(65,172)(66,171)(67,170)(68,169)(69,168)(70,167)(71,166)(72,165)(73,164)(74,163)(75,162)(76,161)(77,160)(78,159)(79,158)(80,157)(81,156)(82,155)(83,154)(84,153)(85,152)(86,151)(87,180)(88,179)(89,178)(90,177)(91,142)(92,141)(93,140)(94,139)(95,138)(96,137)(97,136)(98,135)(99,134)(100,133)(101,132)(102,131)(103,130)(104,129)(105,128)(106,127)(107,126)(108,125)(109,124)(110,123)(111,122)(112,121)(113,150)(114,149)(115,148)(116,147)(117,146)(118,145)(119,144)(120,143), (1,131,240,103)(2,142,211,114)(3,123,212,95)(4,134,213,106)(5,145,214,117)(6,126,215,98)(7,137,216,109)(8,148,217,120)(9,129,218,101)(10,140,219,112)(11,121,220,93)(12,132,221,104)(13,143,222,115)(14,124,223,96)(15,135,224,107)(16,146,225,118)(17,127,226,99)(18,138,227,110)(19,149,228,91)(20,130,229,102)(21,141,230,113)(22,122,231,94)(23,133,232,105)(24,144,233,116)(25,125,234,97)(26,136,235,108)(27,147,236,119)(28,128,237,100)(29,139,238,111)(30,150,239,92)(31,156,185,87)(32,167,186,68)(33,178,187,79)(34,159,188,90)(35,170,189,71)(36,151,190,82)(37,162,191,63)(38,173,192,74)(39,154,193,85)(40,165,194,66)(41,176,195,77)(42,157,196,88)(43,168,197,69)(44,179,198,80)(45,160,199,61)(46,171,200,72)(47,152,201,83)(48,163,202,64)(49,174,203,75)(50,155,204,86)(51,166,205,67)(52,177,206,78)(53,158,207,89)(54,169,208,70)(55,180,209,81)(56,161,210,62)(57,172,181,73)(58,153,182,84)(59,164,183,65)(60,175,184,76), (1,210)(2,191)(3,202)(4,183)(5,194)(6,205)(7,186)(8,197)(9,208)(10,189)(11,200)(12,181)(13,192)(14,203)(15,184)(16,195)(17,206)(18,187)(19,198)(20,209)(21,190)(22,201)(23,182)(24,193)(25,204)(26,185)(27,196)(28,207)(29,188)(30,199)(31,235)(32,216)(33,227)(34,238)(35,219)(36,230)(37,211)(38,222)(39,233)(40,214)(41,225)(42,236)(43,217)(44,228)(45,239)(46,220)(47,231)(48,212)(49,223)(50,234)(51,215)(52,226)(53,237)(54,218)(55,229)(56,240)(57,221)(58,232)(59,213)(60,224)(61,92)(62,103)(63,114)(64,95)(65,106)(66,117)(67,98)(68,109)(69,120)(70,101)(71,112)(72,93)(73,104)(74,115)(75,96)(76,107)(77,118)(78,99)(79,110)(80,91)(81,102)(82,113)(83,94)(84,105)(85,116)(86,97)(87,108)(88,119)(89,100)(90,111)(121,171)(122,152)(123,163)(124,174)(125,155)(126,166)(127,177)(128,158)(129,169)(130,180)(131,161)(132,172)(133,153)(134,164)(135,175)(136,156)(137,167)(138,178)(139,159)(140,170)(141,151)(142,162)(143,173)(144,154)(145,165)(146,176)(147,157)(148,168)(149,179)(150,160)>;
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), (1,239)(2,238)(3,237)(4,236)(5,235)(6,234)(7,233)(8,232)(9,231)(10,230)(11,229)(12,228)(13,227)(14,226)(15,225)(16,224)(17,223)(18,222)(19,221)(20,220)(21,219)(22,218)(23,217)(24,216)(25,215)(26,214)(27,213)(28,212)(29,211)(30,240)(31,189)(32,188)(33,187)(34,186)(35,185)(36,184)(37,183)(38,182)(39,181)(40,210)(41,209)(42,208)(43,207)(44,206)(45,205)(46,204)(47,203)(48,202)(49,201)(50,200)(51,199)(52,198)(53,197)(54,196)(55,195)(56,194)(57,193)(58,192)(59,191)(60,190)(61,176)(62,175)(63,174)(64,173)(65,172)(66,171)(67,170)(68,169)(69,168)(70,167)(71,166)(72,165)(73,164)(74,163)(75,162)(76,161)(77,160)(78,159)(79,158)(80,157)(81,156)(82,155)(83,154)(84,153)(85,152)(86,151)(87,180)(88,179)(89,178)(90,177)(91,142)(92,141)(93,140)(94,139)(95,138)(96,137)(97,136)(98,135)(99,134)(100,133)(101,132)(102,131)(103,130)(104,129)(105,128)(106,127)(107,126)(108,125)(109,124)(110,123)(111,122)(112,121)(113,150)(114,149)(115,148)(116,147)(117,146)(118,145)(119,144)(120,143), (1,131,240,103)(2,142,211,114)(3,123,212,95)(4,134,213,106)(5,145,214,117)(6,126,215,98)(7,137,216,109)(8,148,217,120)(9,129,218,101)(10,140,219,112)(11,121,220,93)(12,132,221,104)(13,143,222,115)(14,124,223,96)(15,135,224,107)(16,146,225,118)(17,127,226,99)(18,138,227,110)(19,149,228,91)(20,130,229,102)(21,141,230,113)(22,122,231,94)(23,133,232,105)(24,144,233,116)(25,125,234,97)(26,136,235,108)(27,147,236,119)(28,128,237,100)(29,139,238,111)(30,150,239,92)(31,156,185,87)(32,167,186,68)(33,178,187,79)(34,159,188,90)(35,170,189,71)(36,151,190,82)(37,162,191,63)(38,173,192,74)(39,154,193,85)(40,165,194,66)(41,176,195,77)(42,157,196,88)(43,168,197,69)(44,179,198,80)(45,160,199,61)(46,171,200,72)(47,152,201,83)(48,163,202,64)(49,174,203,75)(50,155,204,86)(51,166,205,67)(52,177,206,78)(53,158,207,89)(54,169,208,70)(55,180,209,81)(56,161,210,62)(57,172,181,73)(58,153,182,84)(59,164,183,65)(60,175,184,76), (1,210)(2,191)(3,202)(4,183)(5,194)(6,205)(7,186)(8,197)(9,208)(10,189)(11,200)(12,181)(13,192)(14,203)(15,184)(16,195)(17,206)(18,187)(19,198)(20,209)(21,190)(22,201)(23,182)(24,193)(25,204)(26,185)(27,196)(28,207)(29,188)(30,199)(31,235)(32,216)(33,227)(34,238)(35,219)(36,230)(37,211)(38,222)(39,233)(40,214)(41,225)(42,236)(43,217)(44,228)(45,239)(46,220)(47,231)(48,212)(49,223)(50,234)(51,215)(52,226)(53,237)(54,218)(55,229)(56,240)(57,221)(58,232)(59,213)(60,224)(61,92)(62,103)(63,114)(64,95)(65,106)(66,117)(67,98)(68,109)(69,120)(70,101)(71,112)(72,93)(73,104)(74,115)(75,96)(76,107)(77,118)(78,99)(79,110)(80,91)(81,102)(82,113)(83,94)(84,105)(85,116)(86,97)(87,108)(88,119)(89,100)(90,111)(121,171)(122,152)(123,163)(124,174)(125,155)(126,166)(127,177)(128,158)(129,169)(130,180)(131,161)(132,172)(133,153)(134,164)(135,175)(136,156)(137,167)(138,178)(139,159)(140,170)(141,151)(142,162)(143,173)(144,154)(145,165)(146,176)(147,157)(148,168)(149,179)(150,160) );
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)], [(1,239),(2,238),(3,237),(4,236),(5,235),(6,234),(7,233),(8,232),(9,231),(10,230),(11,229),(12,228),(13,227),(14,226),(15,225),(16,224),(17,223),(18,222),(19,221),(20,220),(21,219),(22,218),(23,217),(24,216),(25,215),(26,214),(27,213),(28,212),(29,211),(30,240),(31,189),(32,188),(33,187),(34,186),(35,185),(36,184),(37,183),(38,182),(39,181),(40,210),(41,209),(42,208),(43,207),(44,206),(45,205),(46,204),(47,203),(48,202),(49,201),(50,200),(51,199),(52,198),(53,197),(54,196),(55,195),(56,194),(57,193),(58,192),(59,191),(60,190),(61,176),(62,175),(63,174),(64,173),(65,172),(66,171),(67,170),(68,169),(69,168),(70,167),(71,166),(72,165),(73,164),(74,163),(75,162),(76,161),(77,160),(78,159),(79,158),(80,157),(81,156),(82,155),(83,154),(84,153),(85,152),(86,151),(87,180),(88,179),(89,178),(90,177),(91,142),(92,141),(93,140),(94,139),(95,138),(96,137),(97,136),(98,135),(99,134),(100,133),(101,132),(102,131),(103,130),(104,129),(105,128),(106,127),(107,126),(108,125),(109,124),(110,123),(111,122),(112,121),(113,150),(114,149),(115,148),(116,147),(117,146),(118,145),(119,144),(120,143)], [(1,131,240,103),(2,142,211,114),(3,123,212,95),(4,134,213,106),(5,145,214,117),(6,126,215,98),(7,137,216,109),(8,148,217,120),(9,129,218,101),(10,140,219,112),(11,121,220,93),(12,132,221,104),(13,143,222,115),(14,124,223,96),(15,135,224,107),(16,146,225,118),(17,127,226,99),(18,138,227,110),(19,149,228,91),(20,130,229,102),(21,141,230,113),(22,122,231,94),(23,133,232,105),(24,144,233,116),(25,125,234,97),(26,136,235,108),(27,147,236,119),(28,128,237,100),(29,139,238,111),(30,150,239,92),(31,156,185,87),(32,167,186,68),(33,178,187,79),(34,159,188,90),(35,170,189,71),(36,151,190,82),(37,162,191,63),(38,173,192,74),(39,154,193,85),(40,165,194,66),(41,176,195,77),(42,157,196,88),(43,168,197,69),(44,179,198,80),(45,160,199,61),(46,171,200,72),(47,152,201,83),(48,163,202,64),(49,174,203,75),(50,155,204,86),(51,166,205,67),(52,177,206,78),(53,158,207,89),(54,169,208,70),(55,180,209,81),(56,161,210,62),(57,172,181,73),(58,153,182,84),(59,164,183,65),(60,175,184,76)], [(1,210),(2,191),(3,202),(4,183),(5,194),(6,205),(7,186),(8,197),(9,208),(10,189),(11,200),(12,181),(13,192),(14,203),(15,184),(16,195),(17,206),(18,187),(19,198),(20,209),(21,190),(22,201),(23,182),(24,193),(25,204),(26,185),(27,196),(28,207),(29,188),(30,199),(31,235),(32,216),(33,227),(34,238),(35,219),(36,230),(37,211),(38,222),(39,233),(40,214),(41,225),(42,236),(43,217),(44,228),(45,239),(46,220),(47,231),(48,212),(49,223),(50,234),(51,215),(52,226),(53,237),(54,218),(55,229),(56,240),(57,221),(58,232),(59,213),(60,224),(61,92),(62,103),(63,114),(64,95),(65,106),(66,117),(67,98),(68,109),(69,120),(70,101),(71,112),(72,93),(73,104),(74,115),(75,96),(76,107),(77,118),(78,99),(79,110),(80,91),(81,102),(82,113),(83,94),(84,105),(85,116),(86,97),(87,108),(88,119),(89,100),(90,111),(121,171),(122,152),(123,163),(124,174),(125,155),(126,166),(127,177),(128,158),(129,169),(130,180),(131,161),(132,172),(133,153),(134,164),(135,175),(136,156),(137,167),(138,178),(139,159),(140,170),(141,151),(142,162),(143,173),(144,154),(145,165),(146,176),(147,157),(148,168),(149,179),(150,160)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 4 | 12 | 30 | 30 | 2 | 6 | 6 | 10 | 10 | 20 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 20 | 20 | 20 | 20 | 4 | 4 | 12 | 12 | 12 | 12 | 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 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | C4○D12 | S3×D4 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | Dic3.D10 | S3×C5⋊D4 | D10⋊D6 |
| kernel | D30⋊7D4 | D6⋊Dic5 | C6.Dic10 | C3×C23.D5 | C2×D30.C2 | C2×C5⋊D12 | C10×C3⋊D4 | C2×C15⋊7D4 | C23.D5 | C5×Dic3 | D30 | C2×C3⋊D4 | C2×Dic5 | C22×C10 | C30 | C2×Dic3 | C22×S3 | C22×C6 | Dic3 | C10 | C10 | C23 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 8 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D30⋊7D4 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 60 | 0 | 0 |
| 0 | 0 | 19 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 41 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 | 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 | 0 | 50 |
| 0 | 0 | 0 | 0 | 50 | 0 |
| 1 | 6 | 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 | 15 | 23 |
| 0 | 0 | 0 | 0 | 38 | 46 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,18,19,0,0,0,0,60,60,0,0,0,0,0,0,1,1,0,0,0,0,60,0],[60,41,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,17,0,0,0,0,0,0,0,60,60,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,50,0,0,0,0,50,0],[1,0,0,0,0,0,6,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,38,0,0,0,0,23,46] >;
D30⋊7D4 in GAP, Magma, Sage, TeX
D_{30}\rtimes_7D_4 % in TeX
G:=Group("D30:7D4"); // GroupNames label
G:=SmallGroup(480,633);
// by ID
G=gap.SmallGroup(480,633);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=a^10*b,d*b*d=a^25*b,d*c*d=c^-1>;
// generators/relations