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G = D30.C2⋊C4order 480 = 25·3·5

1st semidirect product of D30.C2 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D30.C21C4, Dic3⋊C420D5, C53(C422S3), (C4×Dic5)⋊14S3, D30.22(C2×C4), (C2×C20).194D6, (C12×Dic5)⋊24C2, D304C4.9C2, (C2×C12).264D10, C30.53(C22×C4), (C2×C30).92C23, C6.Dic1014C2, Dic3.12(C4×D5), Dic5.36(C4×S3), C1516(C42⋊C2), (Dic3×Dic5)⋊16C2, D303C4.13C2, C30.114(C4○D4), C10.10(C4○D12), C2.3(C12.28D10), C6.43(D42D5), (C2×C60).387C22, C6.12(Q82D5), (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), C31(C4⋊C47D5), 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

C1C30 — D30.C2⋊C4
C1C5C15C30C2×C30C6×Dic5C2×D30.C2 — D30.C2⋊C4
C15C30 — D30.C2⋊C4
C1C22C2×C4

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 [×3], C2 [×2], C3, C4 [×8], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×9], C23, D5 [×2], C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×4], D6 [×4], C2×C6, C15, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×2], Dic5 [×2], C20 [×4], D10 [×4], C2×C10, C4×S3 [×4], C2×Dic3 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, D15 [×2], C30 [×3], C42⋊C2, C4×D5 [×4], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C4×Dic3, Dic3⋊C4, Dic3⋊C4, D6⋊C4 [×2], C4×C12, S3×C2×C4, C5×Dic3 [×2], C5×Dic3, C3×Dic5 [×2], C3×Dic5, Dic15, C60, D30 [×2], D30 [×2], C2×C30, C4×Dic5, C4×Dic5, C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C422S3, D30.C2 [×4], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15, C2×C60, C22×D15, C4⋊C47D5, Dic3×Dic5, D304C4, C6.Dic10, C12×Dic5, C5×Dic3⋊C4, D303C4, C2×D30.C2, D30.C2⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, C4○D4 [×2], D10 [×3], C4×S3 [×2], C22×S3, C42⋊C2, C4×D5 [×2], C22×D5, S3×C2×C4, C4○D12 [×2], S3×D5, C2×C4×D5, D42D5, Q82D5, C422S3, C2×S3×D5, C4⋊C47D5, C12.28D10, C4×S3×D5, Dic3.D10, D30.C2⋊C4

Smallest permutation representation of 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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 60)(20 59)(21 58)(22 57)(23 56)(24 55)(25 54)(26 53)(27 52)(28 51)(29 50)(30 49)(61 93)(62 92)(63 91)(64 120)(65 119)(66 118)(67 117)(68 116)(69 115)(70 114)(71 113)(72 112)(73 111)(74 110)(75 109)(76 108)(77 107)(78 106)(79 105)(80 104)(81 103)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(121 163)(122 162)(123 161)(124 160)(125 159)(126 158)(127 157)(128 156)(129 155)(130 154)(131 153)(132 152)(133 151)(134 180)(135 179)(136 178)(137 177)(138 176)(139 175)(140 174)(141 173)(142 172)(143 171)(144 170)(145 169)(146 168)(147 167)(148 166)(149 165)(150 164)(181 222)(182 221)(183 220)(184 219)(185 218)(186 217)(187 216)(188 215)(189 214)(190 213)(191 212)(192 211)(193 240)(194 239)(195 238)(196 237)(197 236)(198 235)(199 234)(200 233)(201 232)(202 231)(203 230)(204 229)(205 228)(206 227)(207 226)(208 225)(209 224)(210 223)
(1 172 16 157)(2 161 17 176)(3 180 18 165)(4 169 19 154)(5 158 20 173)(6 177 21 162)(7 166 22 151)(8 155 23 170)(9 174 24 159)(10 163 25 178)(11 152 26 167)(12 171 27 156)(13 160 28 175)(14 179 29 164)(15 168 30 153)(31 131 46 146)(32 150 47 135)(33 139 48 124)(34 128 49 143)(35 147 50 132)(36 136 51 121)(37 125 52 140)(38 144 53 129)(39 133 54 148)(40 122 55 137)(41 141 56 126)(42 130 57 145)(43 149 58 134)(44 138 59 123)(45 127 60 142)(61 234 76 219)(62 223 77 238)(63 212 78 227)(64 231 79 216)(65 220 80 235)(66 239 81 224)(67 228 82 213)(68 217 83 232)(69 236 84 221)(70 225 85 240)(71 214 86 229)(72 233 87 218)(73 222 88 237)(74 211 89 226)(75 230 90 215)(91 203 106 188)(92 192 107 207)(93 181 108 196)(94 200 109 185)(95 189 110 204)(96 208 111 193)(97 197 112 182)(98 186 113 201)(99 205 114 190)(100 194 115 209)(101 183 116 198)(102 202 117 187)(103 191 118 206)(104 210 119 195)(105 199 120 184)
(1 102 49 68)(2 103 50 69)(3 104 51 70)(4 105 52 71)(5 106 53 72)(6 107 54 73)(7 108 55 74)(8 109 56 75)(9 110 57 76)(10 111 58 77)(11 112 59 78)(12 113 60 79)(13 114 31 80)(14 115 32 81)(15 116 33 82)(16 117 34 83)(17 118 35 84)(18 119 36 85)(19 120 37 86)(20 91 38 87)(21 92 39 88)(22 93 40 89)(23 94 41 90)(24 95 42 61)(25 96 43 62)(26 97 44 63)(27 98 45 64)(28 99 46 65)(29 100 47 66)(30 101 48 67)(121 225 180 210)(122 226 151 181)(123 227 152 182)(124 228 153 183)(125 229 154 184)(126 230 155 185)(127 231 156 186)(128 232 157 187)(129 233 158 188)(130 234 159 189)(131 235 160 190)(132 236 161 191)(133 237 162 192)(134 238 163 193)(135 239 164 194)(136 240 165 195)(137 211 166 196)(138 212 167 197)(139 213 168 198)(140 214 169 199)(141 215 170 200)(142 216 171 201)(143 217 172 202)(144 218 173 203)(145 219 174 204)(146 220 175 205)(147 221 176 206)(148 222 177 207)(149 223 178 208)(150 224 179 209)

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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(61,93)(62,92)(63,91)(64,120)(65,119)(66,118)(67,117)(68,116)(69,115)(70,114)(71,113)(72,112)(73,111)(74,110)(75,109)(76,108)(77,107)(78,106)(79,105)(80,104)(81,103)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(121,163)(122,162)(123,161)(124,160)(125,159)(126,158)(127,157)(128,156)(129,155)(130,154)(131,153)(132,152)(133,151)(134,180)(135,179)(136,178)(137,177)(138,176)(139,175)(140,174)(141,173)(142,172)(143,171)(144,170)(145,169)(146,168)(147,167)(148,166)(149,165)(150,164)(181,222)(182,221)(183,220)(184,219)(185,218)(186,217)(187,216)(188,215)(189,214)(190,213)(191,212)(192,211)(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,234)(200,233)(201,232)(202,231)(203,230)(204,229)(205,228)(206,227)(207,226)(208,225)(209,224)(210,223), (1,172,16,157)(2,161,17,176)(3,180,18,165)(4,169,19,154)(5,158,20,173)(6,177,21,162)(7,166,22,151)(8,155,23,170)(9,174,24,159)(10,163,25,178)(11,152,26,167)(12,171,27,156)(13,160,28,175)(14,179,29,164)(15,168,30,153)(31,131,46,146)(32,150,47,135)(33,139,48,124)(34,128,49,143)(35,147,50,132)(36,136,51,121)(37,125,52,140)(38,144,53,129)(39,133,54,148)(40,122,55,137)(41,141,56,126)(42,130,57,145)(43,149,58,134)(44,138,59,123)(45,127,60,142)(61,234,76,219)(62,223,77,238)(63,212,78,227)(64,231,79,216)(65,220,80,235)(66,239,81,224)(67,228,82,213)(68,217,83,232)(69,236,84,221)(70,225,85,240)(71,214,86,229)(72,233,87,218)(73,222,88,237)(74,211,89,226)(75,230,90,215)(91,203,106,188)(92,192,107,207)(93,181,108,196)(94,200,109,185)(95,189,110,204)(96,208,111,193)(97,197,112,182)(98,186,113,201)(99,205,114,190)(100,194,115,209)(101,183,116,198)(102,202,117,187)(103,191,118,206)(104,210,119,195)(105,199,120,184), (1,102,49,68)(2,103,50,69)(3,104,51,70)(4,105,52,71)(5,106,53,72)(6,107,54,73)(7,108,55,74)(8,109,56,75)(9,110,57,76)(10,111,58,77)(11,112,59,78)(12,113,60,79)(13,114,31,80)(14,115,32,81)(15,116,33,82)(16,117,34,83)(17,118,35,84)(18,119,36,85)(19,120,37,86)(20,91,38,87)(21,92,39,88)(22,93,40,89)(23,94,41,90)(24,95,42,61)(25,96,43,62)(26,97,44,63)(27,98,45,64)(28,99,46,65)(29,100,47,66)(30,101,48,67)(121,225,180,210)(122,226,151,181)(123,227,152,182)(124,228,153,183)(125,229,154,184)(126,230,155,185)(127,231,156,186)(128,232,157,187)(129,233,158,188)(130,234,159,189)(131,235,160,190)(132,236,161,191)(133,237,162,192)(134,238,163,193)(135,239,164,194)(136,240,165,195)(137,211,166,196)(138,212,167,197)(139,213,168,198)(140,214,169,199)(141,215,170,200)(142,216,171,201)(143,217,172,202)(144,218,173,203)(145,219,174,204)(146,220,175,205)(147,221,176,206)(148,222,177,207)(149,223,178,208)(150,224,179,209)>;

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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(61,93)(62,92)(63,91)(64,120)(65,119)(66,118)(67,117)(68,116)(69,115)(70,114)(71,113)(72,112)(73,111)(74,110)(75,109)(76,108)(77,107)(78,106)(79,105)(80,104)(81,103)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(121,163)(122,162)(123,161)(124,160)(125,159)(126,158)(127,157)(128,156)(129,155)(130,154)(131,153)(132,152)(133,151)(134,180)(135,179)(136,178)(137,177)(138,176)(139,175)(140,174)(141,173)(142,172)(143,171)(144,170)(145,169)(146,168)(147,167)(148,166)(149,165)(150,164)(181,222)(182,221)(183,220)(184,219)(185,218)(186,217)(187,216)(188,215)(189,214)(190,213)(191,212)(192,211)(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,234)(200,233)(201,232)(202,231)(203,230)(204,229)(205,228)(206,227)(207,226)(208,225)(209,224)(210,223), (1,172,16,157)(2,161,17,176)(3,180,18,165)(4,169,19,154)(5,158,20,173)(6,177,21,162)(7,166,22,151)(8,155,23,170)(9,174,24,159)(10,163,25,178)(11,152,26,167)(12,171,27,156)(13,160,28,175)(14,179,29,164)(15,168,30,153)(31,131,46,146)(32,150,47,135)(33,139,48,124)(34,128,49,143)(35,147,50,132)(36,136,51,121)(37,125,52,140)(38,144,53,129)(39,133,54,148)(40,122,55,137)(41,141,56,126)(42,130,57,145)(43,149,58,134)(44,138,59,123)(45,127,60,142)(61,234,76,219)(62,223,77,238)(63,212,78,227)(64,231,79,216)(65,220,80,235)(66,239,81,224)(67,228,82,213)(68,217,83,232)(69,236,84,221)(70,225,85,240)(71,214,86,229)(72,233,87,218)(73,222,88,237)(74,211,89,226)(75,230,90,215)(91,203,106,188)(92,192,107,207)(93,181,108,196)(94,200,109,185)(95,189,110,204)(96,208,111,193)(97,197,112,182)(98,186,113,201)(99,205,114,190)(100,194,115,209)(101,183,116,198)(102,202,117,187)(103,191,118,206)(104,210,119,195)(105,199,120,184), (1,102,49,68)(2,103,50,69)(3,104,51,70)(4,105,52,71)(5,106,53,72)(6,107,54,73)(7,108,55,74)(8,109,56,75)(9,110,57,76)(10,111,58,77)(11,112,59,78)(12,113,60,79)(13,114,31,80)(14,115,32,81)(15,116,33,82)(16,117,34,83)(17,118,35,84)(18,119,36,85)(19,120,37,86)(20,91,38,87)(21,92,39,88)(22,93,40,89)(23,94,41,90)(24,95,42,61)(25,96,43,62)(26,97,44,63)(27,98,45,64)(28,99,46,65)(29,100,47,66)(30,101,48,67)(121,225,180,210)(122,226,151,181)(123,227,152,182)(124,228,153,183)(125,229,154,184)(126,230,155,185)(127,231,156,186)(128,232,157,187)(129,233,158,188)(130,234,159,189)(131,235,160,190)(132,236,161,191)(133,237,162,192)(134,238,163,193)(135,239,164,194)(136,240,165,195)(137,211,166,196)(138,212,167,197)(139,213,168,198)(140,214,169,199)(141,215,170,200)(142,216,171,201)(143,217,172,202)(144,218,173,203)(145,219,174,204)(146,220,175,205)(147,221,176,206)(148,222,177,207)(149,223,178,208)(150,224,179,209) );

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)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,60),(20,59),(21,58),(22,57),(23,56),(24,55),(25,54),(26,53),(27,52),(28,51),(29,50),(30,49),(61,93),(62,92),(63,91),(64,120),(65,119),(66,118),(67,117),(68,116),(69,115),(70,114),(71,113),(72,112),(73,111),(74,110),(75,109),(76,108),(77,107),(78,106),(79,105),(80,104),(81,103),(82,102),(83,101),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(121,163),(122,162),(123,161),(124,160),(125,159),(126,158),(127,157),(128,156),(129,155),(130,154),(131,153),(132,152),(133,151),(134,180),(135,179),(136,178),(137,177),(138,176),(139,175),(140,174),(141,173),(142,172),(143,171),(144,170),(145,169),(146,168),(147,167),(148,166),(149,165),(150,164),(181,222),(182,221),(183,220),(184,219),(185,218),(186,217),(187,216),(188,215),(189,214),(190,213),(191,212),(192,211),(193,240),(194,239),(195,238),(196,237),(197,236),(198,235),(199,234),(200,233),(201,232),(202,231),(203,230),(204,229),(205,228),(206,227),(207,226),(208,225),(209,224),(210,223)], [(1,172,16,157),(2,161,17,176),(3,180,18,165),(4,169,19,154),(5,158,20,173),(6,177,21,162),(7,166,22,151),(8,155,23,170),(9,174,24,159),(10,163,25,178),(11,152,26,167),(12,171,27,156),(13,160,28,175),(14,179,29,164),(15,168,30,153),(31,131,46,146),(32,150,47,135),(33,139,48,124),(34,128,49,143),(35,147,50,132),(36,136,51,121),(37,125,52,140),(38,144,53,129),(39,133,54,148),(40,122,55,137),(41,141,56,126),(42,130,57,145),(43,149,58,134),(44,138,59,123),(45,127,60,142),(61,234,76,219),(62,223,77,238),(63,212,78,227),(64,231,79,216),(65,220,80,235),(66,239,81,224),(67,228,82,213),(68,217,83,232),(69,236,84,221),(70,225,85,240),(71,214,86,229),(72,233,87,218),(73,222,88,237),(74,211,89,226),(75,230,90,215),(91,203,106,188),(92,192,107,207),(93,181,108,196),(94,200,109,185),(95,189,110,204),(96,208,111,193),(97,197,112,182),(98,186,113,201),(99,205,114,190),(100,194,115,209),(101,183,116,198),(102,202,117,187),(103,191,118,206),(104,210,119,195),(105,199,120,184)], [(1,102,49,68),(2,103,50,69),(3,104,51,70),(4,105,52,71),(5,106,53,72),(6,107,54,73),(7,108,55,74),(8,109,56,75),(9,110,57,76),(10,111,58,77),(11,112,59,78),(12,113,60,79),(13,114,31,80),(14,115,32,81),(15,116,33,82),(16,117,34,83),(17,118,35,84),(18,119,36,85),(19,120,37,86),(20,91,38,87),(21,92,39,88),(22,93,40,89),(23,94,41,90),(24,95,42,61),(25,96,43,62),(26,97,44,63),(27,98,45,64),(28,99,46,65),(29,100,47,66),(30,101,48,67),(121,225,180,210),(122,226,151,181),(123,227,152,182),(124,228,153,183),(125,229,154,184),(126,230,155,185),(127,231,156,186),(128,232,157,187),(129,233,158,188),(130,234,159,189),(131,235,160,190),(132,236,161,191),(133,237,162,192),(134,238,163,193),(135,239,164,194),(136,240,165,195),(137,211,166,196),(138,212,167,197),(139,213,168,198),(140,214,169,199),(141,215,170,200),(142,216,171,201),(143,217,172,202),(144,218,173,203),(145,219,174,204),(146,220,175,205),(147,221,176,206),(148,222,177,207),(149,223,178,208),(150,224,179,209)])

72 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B6A6B6C10A···10F12A12B12C12D12E···12L15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222223444444444444445566610···101212121212···1215152020202020···2030···3060···60
size111130302225555666610103030222222···2222210···1044444412···124···44···4

72 irreducible representations

dim11111111122222222224444444
type+++++++++++++++-+++
imageC1C2C2C2C2C2C2C2C4S3D5D6D6C4○D4D10D10C4×S3C4×D5C4○D12S3×D5D42D5Q82D5C2×S3×D5C12.28D10C4×S3×D5Dic3.D10
kernelD30.C2⋊C4Dic3×Dic5D304C4C6.Dic10C12×Dic5C5×Dic3⋊C4D303C4C2×D30.C2D30.C2C4×Dic5Dic3⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12Dic5Dic3C10C2×C4C6C6C22C2C2C2
# reps11111111812214424882222444

Matrix representation of D30.C2⋊C4 in GL4(𝔽61) generated by

0100
601700
0011
00600
,
06000
60000
0011
00060
,
1000
176000
00500
00050
,
50000
05000
00918
004352
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

׿
×
𝔽