metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.7D4, D6⋊C4⋊18D5, C4⋊Dic5⋊8S3, C6.44(D4×D5), C10.45(S3×D4), (C2×C20).27D6, C30.56(C2×D4), D6⋊Dic5⋊12C2, C5⋊4(D6.D4), D30⋊3C4⋊16C2, C6.12(C4○D20), (C2×C12).229D10, Dic15⋊5C4⋊24C2, (C2×Dic5).43D6, C30.124(C4○D4), C10.73(C4○D12), C6.48(D4⋊2D5), C3⋊2(D10.12D4), C2.20(C20⋊D6), (C2×C60).260C22, (C2×C30).128C23, (C22×S3).13D10, C2.18(D60⋊C2), C10.17(Q8⋊3S3), (C2×Dic3).109D10, C15⋊12(C22.D4), (C6×Dic5).80C22, C2.19(Dic3.D10), (C10×Dic3).81C22, (C22×D15).43C22, (C2×Dic15).101C22, (C5×D6⋊C4)⋊18C2, (C2×C4).58(S3×D5), (C2×D30.C2)⋊8C2, (C3×C4⋊Dic5)⋊19C2, (C2×C5⋊D12).8C2, C22.188(C2×S3×D5), (S3×C2×C10).26C22, (C2×C6).140(C22×D5), (C2×C10).140(C22×S3), SmallGroup(480,514)
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, ac=ca, dad=a11, cbc-1=a15b, dbd=a25b, dcd=a15c-1 >
Subgroups: 908 in 156 conjugacy classes, 46 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, 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, C2×C12, C2×C12, C22×S3, C22×S3, C5×S3, D15, C30, C22.D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, D30, D30, C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, D6.D4, D30.C2, C5⋊D12, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, D10.12D4, D6⋊Dic5, Dic15⋊5C4, C3×C4⋊Dic5, C5×D6⋊C4, D30⋊3C4, C2×D30.C2, C2×C5⋊D12, D30.7D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C22.D4, C22×D5, C4○D12, S3×D4, Q8⋊3S3, S3×D5, C4○D20, D4×D5, D4⋊2D5, D6.D4, C2×S3×D5, D10.12D4, D60⋊C2, C20⋊D6, Dic3.D10, 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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 85)(62 84)(63 83)(64 82)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(86 90)(87 89)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(101 102)(113 120)(114 119)(115 118)(116 117)(121 138)(122 137)(123 136)(124 135)(125 134)(126 133)(127 132)(128 131)(129 130)(139 150)(140 149)(141 148)(142 147)(143 146)(144 145)(151 162)(152 161)(153 160)(154 159)(155 158)(156 157)(163 180)(164 179)(165 178)(166 177)(167 176)(168 175)(169 174)(170 173)(171 172)(181 185)(182 184)(186 210)(187 209)(188 208)(189 207)(190 206)(191 205)(192 204)(193 203)(194 202)(195 201)(196 200)(197 199)(211 237)(212 236)(213 235)(214 234)(215 233)(216 232)(217 231)(218 230)(219 229)(220 228)(221 227)(222 226)(223 225)(238 240)
(1 217 157 206)(2 218 158 207)(3 219 159 208)(4 220 160 209)(5 221 161 210)(6 222 162 181)(7 223 163 182)(8 224 164 183)(9 225 165 184)(10 226 166 185)(11 227 167 186)(12 228 168 187)(13 229 169 188)(14 230 170 189)(15 231 171 190)(16 232 172 191)(17 233 173 192)(18 234 174 193)(19 235 175 194)(20 236 176 195)(21 237 177 196)(22 238 178 197)(23 239 179 198)(24 240 180 199)(25 211 151 200)(26 212 152 201)(27 213 153 202)(28 214 154 203)(29 215 155 204)(30 216 156 205)(31 101 80 129)(32 102 81 130)(33 103 82 131)(34 104 83 132)(35 105 84 133)(36 106 85 134)(37 107 86 135)(38 108 87 136)(39 109 88 137)(40 110 89 138)(41 111 90 139)(42 112 61 140)(43 113 62 141)(44 114 63 142)(45 115 64 143)(46 116 65 144)(47 117 66 145)(48 118 67 146)(49 119 68 147)(50 120 69 148)(51 91 70 149)(52 92 71 150)(53 93 72 121)(54 94 73 122)(55 95 74 123)(56 96 75 124)(57 97 76 125)(58 98 77 126)(59 99 78 127)(60 100 79 128)
(1 32)(2 43)(3 54)(4 35)(5 46)(6 57)(7 38)(8 49)(9 60)(10 41)(11 52)(12 33)(13 44)(14 55)(15 36)(16 47)(17 58)(18 39)(19 50)(20 31)(21 42)(22 53)(23 34)(24 45)(25 56)(26 37)(27 48)(28 59)(29 40)(30 51)(61 177)(62 158)(63 169)(64 180)(65 161)(66 172)(67 153)(68 164)(69 175)(70 156)(71 167)(72 178)(73 159)(74 170)(75 151)(76 162)(77 173)(78 154)(79 165)(80 176)(81 157)(82 168)(83 179)(84 160)(85 171)(86 152)(87 163)(88 174)(89 155)(90 166)(91 190)(92 201)(93 182)(94 193)(95 204)(96 185)(97 196)(98 207)(99 188)(100 199)(101 210)(102 191)(103 202)(104 183)(105 194)(106 205)(107 186)(108 197)(109 208)(110 189)(111 200)(112 181)(113 192)(114 203)(115 184)(116 195)(117 206)(118 187)(119 198)(120 209)(121 223)(122 234)(123 215)(124 226)(125 237)(126 218)(127 229)(128 240)(129 221)(130 232)(131 213)(132 224)(133 235)(134 216)(135 227)(136 238)(137 219)(138 230)(139 211)(140 222)(141 233)(142 214)(143 225)(144 236)(145 217)(146 228)(147 239)(148 220)(149 231)(150 212)
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(86,90)(87,89)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)(113,120)(114,119)(115,118)(116,117)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)(139,150)(140,149)(141,148)(142,147)(143,146)(144,145)(151,162)(152,161)(153,160)(154,159)(155,158)(156,157)(163,180)(164,179)(165,178)(166,177)(167,176)(168,175)(169,174)(170,173)(171,172)(181,185)(182,184)(186,210)(187,209)(188,208)(189,207)(190,206)(191,205)(192,204)(193,203)(194,202)(195,201)(196,200)(197,199)(211,237)(212,236)(213,235)(214,234)(215,233)(216,232)(217,231)(218,230)(219,229)(220,228)(221,227)(222,226)(223,225)(238,240), (1,217,157,206)(2,218,158,207)(3,219,159,208)(4,220,160,209)(5,221,161,210)(6,222,162,181)(7,223,163,182)(8,224,164,183)(9,225,165,184)(10,226,166,185)(11,227,167,186)(12,228,168,187)(13,229,169,188)(14,230,170,189)(15,231,171,190)(16,232,172,191)(17,233,173,192)(18,234,174,193)(19,235,175,194)(20,236,176,195)(21,237,177,196)(22,238,178,197)(23,239,179,198)(24,240,180,199)(25,211,151,200)(26,212,152,201)(27,213,153,202)(28,214,154,203)(29,215,155,204)(30,216,156,205)(31,101,80,129)(32,102,81,130)(33,103,82,131)(34,104,83,132)(35,105,84,133)(36,106,85,134)(37,107,86,135)(38,108,87,136)(39,109,88,137)(40,110,89,138)(41,111,90,139)(42,112,61,140)(43,113,62,141)(44,114,63,142)(45,115,64,143)(46,116,65,144)(47,117,66,145)(48,118,67,146)(49,119,68,147)(50,120,69,148)(51,91,70,149)(52,92,71,150)(53,93,72,121)(54,94,73,122)(55,95,74,123)(56,96,75,124)(57,97,76,125)(58,98,77,126)(59,99,78,127)(60,100,79,128), (1,32)(2,43)(3,54)(4,35)(5,46)(6,57)(7,38)(8,49)(9,60)(10,41)(11,52)(12,33)(13,44)(14,55)(15,36)(16,47)(17,58)(18,39)(19,50)(20,31)(21,42)(22,53)(23,34)(24,45)(25,56)(26,37)(27,48)(28,59)(29,40)(30,51)(61,177)(62,158)(63,169)(64,180)(65,161)(66,172)(67,153)(68,164)(69,175)(70,156)(71,167)(72,178)(73,159)(74,170)(75,151)(76,162)(77,173)(78,154)(79,165)(80,176)(81,157)(82,168)(83,179)(84,160)(85,171)(86,152)(87,163)(88,174)(89,155)(90,166)(91,190)(92,201)(93,182)(94,193)(95,204)(96,185)(97,196)(98,207)(99,188)(100,199)(101,210)(102,191)(103,202)(104,183)(105,194)(106,205)(107,186)(108,197)(109,208)(110,189)(111,200)(112,181)(113,192)(114,203)(115,184)(116,195)(117,206)(118,187)(119,198)(120,209)(121,223)(122,234)(123,215)(124,226)(125,237)(126,218)(127,229)(128,240)(129,221)(130,232)(131,213)(132,224)(133,235)(134,216)(135,227)(136,238)(137,219)(138,230)(139,211)(140,222)(141,233)(142,214)(143,225)(144,236)(145,217)(146,228)(147,239)(148,220)(149,231)(150,212)>;
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(86,90)(87,89)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)(113,120)(114,119)(115,118)(116,117)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)(139,150)(140,149)(141,148)(142,147)(143,146)(144,145)(151,162)(152,161)(153,160)(154,159)(155,158)(156,157)(163,180)(164,179)(165,178)(166,177)(167,176)(168,175)(169,174)(170,173)(171,172)(181,185)(182,184)(186,210)(187,209)(188,208)(189,207)(190,206)(191,205)(192,204)(193,203)(194,202)(195,201)(196,200)(197,199)(211,237)(212,236)(213,235)(214,234)(215,233)(216,232)(217,231)(218,230)(219,229)(220,228)(221,227)(222,226)(223,225)(238,240), (1,217,157,206)(2,218,158,207)(3,219,159,208)(4,220,160,209)(5,221,161,210)(6,222,162,181)(7,223,163,182)(8,224,164,183)(9,225,165,184)(10,226,166,185)(11,227,167,186)(12,228,168,187)(13,229,169,188)(14,230,170,189)(15,231,171,190)(16,232,172,191)(17,233,173,192)(18,234,174,193)(19,235,175,194)(20,236,176,195)(21,237,177,196)(22,238,178,197)(23,239,179,198)(24,240,180,199)(25,211,151,200)(26,212,152,201)(27,213,153,202)(28,214,154,203)(29,215,155,204)(30,216,156,205)(31,101,80,129)(32,102,81,130)(33,103,82,131)(34,104,83,132)(35,105,84,133)(36,106,85,134)(37,107,86,135)(38,108,87,136)(39,109,88,137)(40,110,89,138)(41,111,90,139)(42,112,61,140)(43,113,62,141)(44,114,63,142)(45,115,64,143)(46,116,65,144)(47,117,66,145)(48,118,67,146)(49,119,68,147)(50,120,69,148)(51,91,70,149)(52,92,71,150)(53,93,72,121)(54,94,73,122)(55,95,74,123)(56,96,75,124)(57,97,76,125)(58,98,77,126)(59,99,78,127)(60,100,79,128), (1,32)(2,43)(3,54)(4,35)(5,46)(6,57)(7,38)(8,49)(9,60)(10,41)(11,52)(12,33)(13,44)(14,55)(15,36)(16,47)(17,58)(18,39)(19,50)(20,31)(21,42)(22,53)(23,34)(24,45)(25,56)(26,37)(27,48)(28,59)(29,40)(30,51)(61,177)(62,158)(63,169)(64,180)(65,161)(66,172)(67,153)(68,164)(69,175)(70,156)(71,167)(72,178)(73,159)(74,170)(75,151)(76,162)(77,173)(78,154)(79,165)(80,176)(81,157)(82,168)(83,179)(84,160)(85,171)(86,152)(87,163)(88,174)(89,155)(90,166)(91,190)(92,201)(93,182)(94,193)(95,204)(96,185)(97,196)(98,207)(99,188)(100,199)(101,210)(102,191)(103,202)(104,183)(105,194)(106,205)(107,186)(108,197)(109,208)(110,189)(111,200)(112,181)(113,192)(114,203)(115,184)(116,195)(117,206)(118,187)(119,198)(120,209)(121,223)(122,234)(123,215)(124,226)(125,237)(126,218)(127,229)(128,240)(129,221)(130,232)(131,213)(132,224)(133,235)(134,216)(135,227)(136,238)(137,219)(138,230)(139,211)(140,222)(141,233)(142,214)(143,225)(144,236)(145,217)(146,228)(147,239)(148,220)(149,231)(150,212) );
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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55),(61,85),(62,84),(63,83),(64,82),(65,81),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(86,90),(87,89),(91,112),(92,111),(93,110),(94,109),(95,108),(96,107),(97,106),(98,105),(99,104),(100,103),(101,102),(113,120),(114,119),(115,118),(116,117),(121,138),(122,137),(123,136),(124,135),(125,134),(126,133),(127,132),(128,131),(129,130),(139,150),(140,149),(141,148),(142,147),(143,146),(144,145),(151,162),(152,161),(153,160),(154,159),(155,158),(156,157),(163,180),(164,179),(165,178),(166,177),(167,176),(168,175),(169,174),(170,173),(171,172),(181,185),(182,184),(186,210),(187,209),(188,208),(189,207),(190,206),(191,205),(192,204),(193,203),(194,202),(195,201),(196,200),(197,199),(211,237),(212,236),(213,235),(214,234),(215,233),(216,232),(217,231),(218,230),(219,229),(220,228),(221,227),(222,226),(223,225),(238,240)], [(1,217,157,206),(2,218,158,207),(3,219,159,208),(4,220,160,209),(5,221,161,210),(6,222,162,181),(7,223,163,182),(8,224,164,183),(9,225,165,184),(10,226,166,185),(11,227,167,186),(12,228,168,187),(13,229,169,188),(14,230,170,189),(15,231,171,190),(16,232,172,191),(17,233,173,192),(18,234,174,193),(19,235,175,194),(20,236,176,195),(21,237,177,196),(22,238,178,197),(23,239,179,198),(24,240,180,199),(25,211,151,200),(26,212,152,201),(27,213,153,202),(28,214,154,203),(29,215,155,204),(30,216,156,205),(31,101,80,129),(32,102,81,130),(33,103,82,131),(34,104,83,132),(35,105,84,133),(36,106,85,134),(37,107,86,135),(38,108,87,136),(39,109,88,137),(40,110,89,138),(41,111,90,139),(42,112,61,140),(43,113,62,141),(44,114,63,142),(45,115,64,143),(46,116,65,144),(47,117,66,145),(48,118,67,146),(49,119,68,147),(50,120,69,148),(51,91,70,149),(52,92,71,150),(53,93,72,121),(54,94,73,122),(55,95,74,123),(56,96,75,124),(57,97,76,125),(58,98,77,126),(59,99,78,127),(60,100,79,128)], [(1,32),(2,43),(3,54),(4,35),(5,46),(6,57),(7,38),(8,49),(9,60),(10,41),(11,52),(12,33),(13,44),(14,55),(15,36),(16,47),(17,58),(18,39),(19,50),(20,31),(21,42),(22,53),(23,34),(24,45),(25,56),(26,37),(27,48),(28,59),(29,40),(30,51),(61,177),(62,158),(63,169),(64,180),(65,161),(66,172),(67,153),(68,164),(69,175),(70,156),(71,167),(72,178),(73,159),(74,170),(75,151),(76,162),(77,173),(78,154),(79,165),(80,176),(81,157),(82,168),(83,179),(84,160),(85,171),(86,152),(87,163),(88,174),(89,155),(90,166),(91,190),(92,201),(93,182),(94,193),(95,204),(96,185),(97,196),(98,207),(99,188),(100,199),(101,210),(102,191),(103,202),(104,183),(105,194),(106,205),(107,186),(108,197),(109,208),(110,189),(111,200),(112,181),(113,192),(114,203),(115,184),(116,195),(117,206),(118,187),(119,198),(120,209),(121,223),(122,234),(123,215),(124,226),(125,237),(126,218),(127,229),(128,240),(129,221),(130,232),(131,213),(132,224),(133,235),(134,216),(135,227),(136,238),(137,219),(138,230),(139,211),(140,222),(141,233),(142,214),(143,225),(144,236),(145,217),(146,228),(147,239),(148,220),(149,231),(150,212)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 30 | 30 | 2 | 4 | 6 | 6 | 10 | 10 | 20 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C4○D12 | C4○D20 | S3×D4 | Q8⋊3S3 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | D60⋊C2 | C20⋊D6 | Dic3.D10 |
| kernel | D30.7D4 | D6⋊Dic5 | Dic15⋊5C4 | C3×C4⋊Dic5 | C5×D6⋊C4 | D30⋊3C4 | C2×D30.C2 | C2×C5⋊D12 | C4⋊Dic5 | D30 | D6⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | C10 | C6 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D30.7D4 ►in GL6(𝔽61)
| 1 | 19 | 0 | 0 | 0 | 0 |
| 60 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 59 | 20 |
| 0 | 0 | 0 | 0 | 9 | 1 |
| 0 | 43 | 0 | 0 | 0 | 0 |
| 44 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 15 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 9 | 1 |
| 25 | 50 | 0 | 0 | 0 | 0 |
| 7 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 21 | 0 | 0 |
| 0 | 0 | 25 | 56 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 30 | 60 | 0 | 0 | 0 | 0 |
| 45 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 13 | 0 | 0 |
| 0 | 0 | 2 | 55 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 20 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(61))| [1,60,0,0,0,0,19,43,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,59,9,0,0,0,0,20,1],[0,44,0,0,0,0,43,0,0,0,0,0,0,0,60,15,0,0,0,0,0,1,0,0,0,0,0,0,60,9,0,0,0,0,0,1],[25,7,0,0,0,0,50,36,0,0,0,0,0,0,5,25,0,0,0,0,21,56,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,45,0,0,0,0,60,31,0,0,0,0,0,0,6,2,0,0,0,0,13,55,0,0,0,0,0,0,60,0,0,0,0,0,20,1] >;
D30.7D4 in GAP, Magma, Sage, TeX
D_{30}._7D_4 % in TeX
G:=Group("D30.7D4"); // GroupNames label
G:=SmallGroup(480,514);
// by ID
G=gap.SmallGroup(480,514);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,590,219,142,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^11,c*b*c^-1=a^15*b,d*b*d=a^25*b,d*c*d=a^15*c^-1>;
// generators/relations