metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.C2⋊1C4, Dic3⋊C4⋊20D5, C5⋊3(C42⋊2S3), (C4×Dic5)⋊14S3, D30.22(C2×C4), (C2×C20).194D6, (C12×Dic5)⋊24C2, D30⋊4C4.9C2, (C2×C12).264D10, C30.53(C22×C4), (C2×C30).92C23, C6.Dic10⋊14C2, Dic3.12(C4×D5), Dic5.36(C4×S3), C15⋊16(C42⋊C2), (Dic3×Dic5)⋊16C2, D30⋊3C4.13C2, C30.114(C4○D4), C10.10(C4○D12), C2.3(C12.28D10), C6.43(D4⋊2D5), (C2×C60).387C22, C6.12(Q8⋊2D5), (C2×Dic5).173D6, (C2×Dic3).94D10, C2.3(Dic3.D10), (C10×Dic3).53C22, (C6×Dic5).198C22, (C2×Dic15).74C22, (C22×D15).28C22, C6.21(C2×C4×D5), C2.23(C4×S3×D5), C10.54(S3×C2×C4), C3⋊1(C4⋊C4⋊7D5), C22.47(C2×S3×D5), (C2×C4).127(S3×D5), (C5×Dic3⋊C4)⋊27C2, (C2×D30.C2).1C2, (C5×Dic3).28(C2×C4), (C3×Dic5).41(C2×C4), (C2×C6).104(C22×D5), (C2×C10).104(C22×S3), SmallGroup(480,478)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.C2⋊C4
G = < a,b,c,d | a30=b2=d4=1, c2=a15, bab=a-1, cac-1=a19, ad=da, cbc-1=a18b, dbd-1=a15b, cd=dc >
Subgroups: 748 in 152 conjugacy classes, 58 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, Dic5, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C42⋊C2, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, Dic3⋊C4, Dic3⋊C4, D6⋊C4, C4×C12, S3×C2×C4, C5×Dic3, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, D30, D30, C2×C30, C4×Dic5, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C42⋊2S3, D30.C2, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C22×D15, C4⋊C4⋊7D5, Dic3×Dic5, D30⋊4C4, C6.Dic10, C12×Dic5, C5×Dic3⋊C4, D30⋊3C4, C2×D30.C2, D30.C2⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, C4○D4, D10, C4×S3, C22×S3, C42⋊C2, C4×D5, C22×D5, S3×C2×C4, C4○D12, S3×D5, C2×C4×D5, D4⋊2D5, Q8⋊2D5, C42⋊2S3, C2×S3×D5, C4⋊C4⋊7D5, C12.28D10, C4×S3×D5, Dic3.D10, D30.C2⋊C4
►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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 60)(9 59)(10 58)(11 57)(12 56)(13 55)(14 54)(15 53)(16 52)(17 51)(18 50)(19 49)(20 48)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 40)(29 39)(30 38)(61 98)(62 97)(63 96)(64 95)(65 94)(66 93)(67 92)(68 91)(69 120)(70 119)(71 118)(72 117)(73 116)(74 115)(75 114)(76 113)(77 112)(78 111)(79 110)(80 109)(81 108)(82 107)(83 106)(84 105)(85 104)(86 103)(87 102)(88 101)(89 100)(90 99)(121 170)(122 169)(123 168)(124 167)(125 166)(126 165)(127 164)(128 163)(129 162)(130 161)(131 160)(132 159)(133 158)(134 157)(135 156)(136 155)(137 154)(138 153)(139 152)(140 151)(141 180)(142 179)(143 178)(144 177)(145 176)(146 175)(147 174)(148 173)(149 172)(150 171)(181 215)(182 214)(183 213)(184 212)(185 211)(186 240)(187 239)(188 238)(189 237)(190 236)(191 235)(192 234)(193 233)(194 232)(195 231)(196 230)(197 229)(198 228)(199 227)(200 226)(201 225)(202 224)(203 223)(204 222)(205 221)(206 220)(207 219)(208 218)(209 217)(210 216)
(1 158 16 173)(2 177 17 162)(3 166 18 151)(4 155 19 170)(5 174 20 159)(6 163 21 178)(7 152 22 167)(8 171 23 156)(9 160 24 175)(10 179 25 164)(11 168 26 153)(12 157 27 172)(13 176 28 161)(14 165 29 180)(15 154 30 169)(31 121 46 136)(32 140 47 125)(33 129 48 144)(34 148 49 133)(35 137 50 122)(36 126 51 141)(37 145 52 130)(38 134 53 149)(39 123 54 138)(40 142 55 127)(41 131 56 146)(42 150 57 135)(43 139 58 124)(44 128 59 143)(45 147 60 132)(61 224 76 239)(62 213 77 228)(63 232 78 217)(64 221 79 236)(65 240 80 225)(66 229 81 214)(67 218 82 233)(68 237 83 222)(69 226 84 211)(70 215 85 230)(71 234 86 219)(72 223 87 238)(73 212 88 227)(74 231 89 216)(75 220 90 235)(91 201 106 186)(92 190 107 205)(93 209 108 194)(94 198 109 183)(95 187 110 202)(96 206 111 191)(97 195 112 210)(98 184 113 199)(99 203 114 188)(100 192 115 207)(101 181 116 196)(102 200 117 185)(103 189 118 204)(104 208 119 193)(105 197 120 182)
(1 114 38 61)(2 115 39 62)(3 116 40 63)(4 117 41 64)(5 118 42 65)(6 119 43 66)(7 120 44 67)(8 91 45 68)(9 92 46 69)(10 93 47 70)(11 94 48 71)(12 95 49 72)(13 96 50 73)(14 97 51 74)(15 98 52 75)(16 99 53 76)(17 100 54 77)(18 101 55 78)(19 102 56 79)(20 103 57 80)(21 104 58 81)(22 105 59 82)(23 106 60 83)(24 107 31 84)(25 108 32 85)(26 109 33 86)(27 110 34 87)(28 111 35 88)(29 112 36 89)(30 113 37 90)(121 211 175 205)(122 212 176 206)(123 213 177 207)(124 214 178 208)(125 215 179 209)(126 216 180 210)(127 217 151 181)(128 218 152 182)(129 219 153 183)(130 220 154 184)(131 221 155 185)(132 222 156 186)(133 223 157 187)(134 224 158 188)(135 225 159 189)(136 226 160 190)(137 227 161 191)(138 228 162 192)(139 229 163 193)(140 230 164 194)(141 231 165 195)(142 232 166 196)(143 233 167 197)(144 234 168 198)(145 235 169 199)(146 236 170 200)(147 237 171 201)(148 238 172 202)(149 239 173 203)(150 240 174 204)
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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,60)(9,59)(10,58)(11,57)(12,56)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,105)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(121,170)(122,169)(123,168)(124,167)(125,166)(126,165)(127,164)(128,163)(129,162)(130,161)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151)(141,180)(142,179)(143,178)(144,177)(145,176)(146,175)(147,174)(148,173)(149,172)(150,171)(181,215)(182,214)(183,213)(184,212)(185,211)(186,240)(187,239)(188,238)(189,237)(190,236)(191,235)(192,234)(193,233)(194,232)(195,231)(196,230)(197,229)(198,228)(199,227)(200,226)(201,225)(202,224)(203,223)(204,222)(205,221)(206,220)(207,219)(208,218)(209,217)(210,216), (1,158,16,173)(2,177,17,162)(3,166,18,151)(4,155,19,170)(5,174,20,159)(6,163,21,178)(7,152,22,167)(8,171,23,156)(9,160,24,175)(10,179,25,164)(11,168,26,153)(12,157,27,172)(13,176,28,161)(14,165,29,180)(15,154,30,169)(31,121,46,136)(32,140,47,125)(33,129,48,144)(34,148,49,133)(35,137,50,122)(36,126,51,141)(37,145,52,130)(38,134,53,149)(39,123,54,138)(40,142,55,127)(41,131,56,146)(42,150,57,135)(43,139,58,124)(44,128,59,143)(45,147,60,132)(61,224,76,239)(62,213,77,228)(63,232,78,217)(64,221,79,236)(65,240,80,225)(66,229,81,214)(67,218,82,233)(68,237,83,222)(69,226,84,211)(70,215,85,230)(71,234,86,219)(72,223,87,238)(73,212,88,227)(74,231,89,216)(75,220,90,235)(91,201,106,186)(92,190,107,205)(93,209,108,194)(94,198,109,183)(95,187,110,202)(96,206,111,191)(97,195,112,210)(98,184,113,199)(99,203,114,188)(100,192,115,207)(101,181,116,196)(102,200,117,185)(103,189,118,204)(104,208,119,193)(105,197,120,182), (1,114,38,61)(2,115,39,62)(3,116,40,63)(4,117,41,64)(5,118,42,65)(6,119,43,66)(7,120,44,67)(8,91,45,68)(9,92,46,69)(10,93,47,70)(11,94,48,71)(12,95,49,72)(13,96,50,73)(14,97,51,74)(15,98,52,75)(16,99,53,76)(17,100,54,77)(18,101,55,78)(19,102,56,79)(20,103,57,80)(21,104,58,81)(22,105,59,82)(23,106,60,83)(24,107,31,84)(25,108,32,85)(26,109,33,86)(27,110,34,87)(28,111,35,88)(29,112,36,89)(30,113,37,90)(121,211,175,205)(122,212,176,206)(123,213,177,207)(124,214,178,208)(125,215,179,209)(126,216,180,210)(127,217,151,181)(128,218,152,182)(129,219,153,183)(130,220,154,184)(131,221,155,185)(132,222,156,186)(133,223,157,187)(134,224,158,188)(135,225,159,189)(136,226,160,190)(137,227,161,191)(138,228,162,192)(139,229,163,193)(140,230,164,194)(141,231,165,195)(142,232,166,196)(143,233,167,197)(144,234,168,198)(145,235,169,199)(146,236,170,200)(147,237,171,201)(148,238,172,202)(149,239,173,203)(150,240,174,204)>;
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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,60)(9,59)(10,58)(11,57)(12,56)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,105)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(121,170)(122,169)(123,168)(124,167)(125,166)(126,165)(127,164)(128,163)(129,162)(130,161)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151)(141,180)(142,179)(143,178)(144,177)(145,176)(146,175)(147,174)(148,173)(149,172)(150,171)(181,215)(182,214)(183,213)(184,212)(185,211)(186,240)(187,239)(188,238)(189,237)(190,236)(191,235)(192,234)(193,233)(194,232)(195,231)(196,230)(197,229)(198,228)(199,227)(200,226)(201,225)(202,224)(203,223)(204,222)(205,221)(206,220)(207,219)(208,218)(209,217)(210,216), (1,158,16,173)(2,177,17,162)(3,166,18,151)(4,155,19,170)(5,174,20,159)(6,163,21,178)(7,152,22,167)(8,171,23,156)(9,160,24,175)(10,179,25,164)(11,168,26,153)(12,157,27,172)(13,176,28,161)(14,165,29,180)(15,154,30,169)(31,121,46,136)(32,140,47,125)(33,129,48,144)(34,148,49,133)(35,137,50,122)(36,126,51,141)(37,145,52,130)(38,134,53,149)(39,123,54,138)(40,142,55,127)(41,131,56,146)(42,150,57,135)(43,139,58,124)(44,128,59,143)(45,147,60,132)(61,224,76,239)(62,213,77,228)(63,232,78,217)(64,221,79,236)(65,240,80,225)(66,229,81,214)(67,218,82,233)(68,237,83,222)(69,226,84,211)(70,215,85,230)(71,234,86,219)(72,223,87,238)(73,212,88,227)(74,231,89,216)(75,220,90,235)(91,201,106,186)(92,190,107,205)(93,209,108,194)(94,198,109,183)(95,187,110,202)(96,206,111,191)(97,195,112,210)(98,184,113,199)(99,203,114,188)(100,192,115,207)(101,181,116,196)(102,200,117,185)(103,189,118,204)(104,208,119,193)(105,197,120,182), (1,114,38,61)(2,115,39,62)(3,116,40,63)(4,117,41,64)(5,118,42,65)(6,119,43,66)(7,120,44,67)(8,91,45,68)(9,92,46,69)(10,93,47,70)(11,94,48,71)(12,95,49,72)(13,96,50,73)(14,97,51,74)(15,98,52,75)(16,99,53,76)(17,100,54,77)(18,101,55,78)(19,102,56,79)(20,103,57,80)(21,104,58,81)(22,105,59,82)(23,106,60,83)(24,107,31,84)(25,108,32,85)(26,109,33,86)(27,110,34,87)(28,111,35,88)(29,112,36,89)(30,113,37,90)(121,211,175,205)(122,212,176,206)(123,213,177,207)(124,214,178,208)(125,215,179,209)(126,216,180,210)(127,217,151,181)(128,218,152,182)(129,219,153,183)(130,220,154,184)(131,221,155,185)(132,222,156,186)(133,223,157,187)(134,224,158,188)(135,225,159,189)(136,226,160,190)(137,227,161,191)(138,228,162,192)(139,229,163,193)(140,230,164,194)(141,231,165,195)(142,232,166,196)(143,233,167,197)(144,234,168,198)(145,235,169,199)(146,236,170,200)(147,237,171,201)(148,238,172,202)(149,239,173,203)(150,240,174,204) );
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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,60),(9,59),(10,58),(11,57),(12,56),(13,55),(14,54),(15,53),(16,52),(17,51),(18,50),(19,49),(20,48),(21,47),(22,46),(23,45),(24,44),(25,43),(26,42),(27,41),(28,40),(29,39),(30,38),(61,98),(62,97),(63,96),(64,95),(65,94),(66,93),(67,92),(68,91),(69,120),(70,119),(71,118),(72,117),(73,116),(74,115),(75,114),(76,113),(77,112),(78,111),(79,110),(80,109),(81,108),(82,107),(83,106),(84,105),(85,104),(86,103),(87,102),(88,101),(89,100),(90,99),(121,170),(122,169),(123,168),(124,167),(125,166),(126,165),(127,164),(128,163),(129,162),(130,161),(131,160),(132,159),(133,158),(134,157),(135,156),(136,155),(137,154),(138,153),(139,152),(140,151),(141,180),(142,179),(143,178),(144,177),(145,176),(146,175),(147,174),(148,173),(149,172),(150,171),(181,215),(182,214),(183,213),(184,212),(185,211),(186,240),(187,239),(188,238),(189,237),(190,236),(191,235),(192,234),(193,233),(194,232),(195,231),(196,230),(197,229),(198,228),(199,227),(200,226),(201,225),(202,224),(203,223),(204,222),(205,221),(206,220),(207,219),(208,218),(209,217),(210,216)], [(1,158,16,173),(2,177,17,162),(3,166,18,151),(4,155,19,170),(5,174,20,159),(6,163,21,178),(7,152,22,167),(8,171,23,156),(9,160,24,175),(10,179,25,164),(11,168,26,153),(12,157,27,172),(13,176,28,161),(14,165,29,180),(15,154,30,169),(31,121,46,136),(32,140,47,125),(33,129,48,144),(34,148,49,133),(35,137,50,122),(36,126,51,141),(37,145,52,130),(38,134,53,149),(39,123,54,138),(40,142,55,127),(41,131,56,146),(42,150,57,135),(43,139,58,124),(44,128,59,143),(45,147,60,132),(61,224,76,239),(62,213,77,228),(63,232,78,217),(64,221,79,236),(65,240,80,225),(66,229,81,214),(67,218,82,233),(68,237,83,222),(69,226,84,211),(70,215,85,230),(71,234,86,219),(72,223,87,238),(73,212,88,227),(74,231,89,216),(75,220,90,235),(91,201,106,186),(92,190,107,205),(93,209,108,194),(94,198,109,183),(95,187,110,202),(96,206,111,191),(97,195,112,210),(98,184,113,199),(99,203,114,188),(100,192,115,207),(101,181,116,196),(102,200,117,185),(103,189,118,204),(104,208,119,193),(105,197,120,182)], [(1,114,38,61),(2,115,39,62),(3,116,40,63),(4,117,41,64),(5,118,42,65),(6,119,43,66),(7,120,44,67),(8,91,45,68),(9,92,46,69),(10,93,47,70),(11,94,48,71),(12,95,49,72),(13,96,50,73),(14,97,51,74),(15,98,52,75),(16,99,53,76),(17,100,54,77),(18,101,55,78),(19,102,56,79),(20,103,57,80),(21,104,58,81),(22,105,59,82),(23,106,60,83),(24,107,31,84),(25,108,32,85),(26,109,33,86),(27,110,34,87),(28,111,35,88),(29,112,36,89),(30,113,37,90),(121,211,175,205),(122,212,176,206),(123,213,177,207),(124,214,178,208),(125,215,179,209),(126,216,180,210),(127,217,151,181),(128,218,152,182),(129,219,153,183),(130,220,154,184),(131,221,155,185),(132,222,156,186),(133,223,157,187),(134,224,158,188),(135,225,159,189),(136,226,160,190),(137,227,161,191),(138,228,162,192),(139,229,163,193),(140,230,164,194),(141,231,165,195),(142,232,166,196),(143,233,167,197),(144,234,168,198),(145,235,169,199),(146,236,170,200),(147,237,171,201),(148,238,172,202),(149,239,173,203),(150,240,174,204)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 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 | 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 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C4×S3 | C4×D5 | C4○D12 | S3×D5 | D4⋊2D5 | Q8⋊2D5 | C2×S3×D5 | C12.28D10 | C4×S3×D5 | Dic3.D10 |
| kernel | D30.C2⋊C4 | Dic3×Dic5 | D30⋊4C4 | C6.Dic10 | C12×Dic5 | C5×Dic3⋊C4 | D30⋊3C4 | C2×D30.C2 | D30.C2 | C4×Dic5 | Dic3⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | Dic5 | Dic3 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D30.C2⋊C4 ►in GL4(𝔽61) generated by
| 0 | 1 | 0 | 0 |
| 60 | 17 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 60 | 0 |
| 0 | 60 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 17 | 60 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 9 | 18 |
| 0 | 0 | 43 | 52 |
G:=sub<GL(4,GF(61))| [0,60,0,0,1,17,0,0,0,0,1,60,0,0,1,0],[0,60,0,0,60,0,0,0,0,0,1,0,0,0,1,60],[1,17,0,0,0,60,0,0,0,0,50,0,0,0,0,50],[50,0,0,0,0,50,0,0,0,0,9,43,0,0,18,52] >;
D30.C2⋊C4 in GAP, Magma, Sage, TeX
D_{30}.C_2\rtimes C_4 % in TeX
G:=Group("D30.C2:C4"); // GroupNames label
G:=SmallGroup(480,478);
// by ID
G=gap.SmallGroup(480,478);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=d^4=1,c^2=a^15,b*a*b=a^-1,c*a*c^-1=a^19,a*d=d*a,c*b*c^-1=a^18*b,d*b*d^-1=a^15*b,c*d=d*c>;
// generators/relations