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G = D30⋊D4order 480 = 25·3·5

1st semidirect product of D30 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D301D4, Dic53D12, D6⋊C45D5, (C2×D60)⋊1C2, C6.59(D4×D5), C51(C12⋊D4), C154(C4⋊D4), (C3×Dic5)⋊2D4, (C2×C20).19D6, C2.22(D5×D12), C10.61(S3×D4), C31(D10⋊D4), C6.9(C4○D20), C10.D46S3, C10.21(C2×D12), C30.133(C2×D4), (C2×C12).17D10, (C2×C60).8C22, D304C414C2, C30.63(C4○D4), (C2×Dic5).33D6, (C2×C30).110C23, (C22×S3).12D10, C2.12(D10⋊D6), C2.16(D60⋊C2), C10.14(Q83S3), (C2×Dic3).103D10, (C6×Dic5).64C22, (C10×Dic3).68C22, (C22×D15).37C22, (C5×D6⋊C4)⋊5C2, (C2×C5⋊D12)⋊2C2, (C2×C4).46(S3×D5), (C2×D30.C2)⋊5C2, C22.176(C2×S3×D5), (C3×C10.D4)⋊6C2, (S3×C2×C10).20C22, (C2×C6).122(C22×D5), (C2×C10).122(C22×S3), SmallGroup(480,496)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D30⋊D4
C1C5C15C30C2×C30C6×Dic5C2×D30.C2 — D30⋊D4
C15C2×C30 — D30⋊D4
C1C22C2×C4

Generators and relations for D30⋊D4
 G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=a19, dad=a11, cbc-1=a18b, dbd=a25b, dcd=c-1 >

Subgroups: 1292 in 188 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×10], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], D5 [×3], C10 [×3], C10, Dic3, C12 [×4], D6 [×10], C2×C6, C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×2], Dic5, C20 [×2], D10 [×7], C2×C10, C2×C10 [×3], C4×S3 [×2], D12 [×6], C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, C22×S3 [×2], C5×S3, D15 [×3], C30 [×3], C4⋊D4, C4×D5 [×2], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C2×C20, C22×D5 [×2], C22×C10, D6⋊C4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12 [×3], C5×Dic3, C3×Dic5 [×2], C3×Dic5, C60, S3×C10 [×3], D30 [×2], D30 [×5], C2×C30, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4 [×2], C12⋊D4, D30.C2 [×2], C5⋊D12 [×4], C6×Dic5 [×2], C10×Dic3, D60 [×2], C2×C60, S3×C2×C10, C22×D15 [×2], D10⋊D4, D304C4, C3×C10.D4, C5×D6⋊C4, C2×D30.C2, C2×C5⋊D12 [×2], C2×D60, D30⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], D12 [×2], C22×S3, C4⋊D4, C22×D5, C2×D12, S3×D4, Q83S3, S3×D5, C4○D20, D4×D5 [×2], C12⋊D4, C2×S3×D5, D10⋊D4, D60⋊C2, D5×D12, D10⋊D6, D30⋊D4

Smallest permutation representation of D30⋊D4
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 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(56 60)(57 59)(61 71)(62 70)(63 69)(64 68)(65 67)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(91 94)(92 93)(95 120)(96 119)(97 118)(98 117)(99 116)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(121 128)(122 127)(123 126)(124 125)(129 150)(130 149)(131 148)(132 147)(133 146)(134 145)(135 144)(136 143)(137 142)(138 141)(139 140)(151 152)(153 180)(154 179)(155 178)(156 177)(157 176)(158 175)(159 174)(160 173)(161 172)(162 171)(163 170)(164 169)(165 168)(166 167)(181 197)(182 196)(183 195)(184 194)(185 193)(186 192)(187 191)(188 190)(198 210)(199 209)(200 208)(201 207)(202 206)(203 205)(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 93 152 125)(2 112 153 144)(3 101 154 133)(4 120 155 122)(5 109 156 141)(6 98 157 130)(7 117 158 149)(8 106 159 138)(9 95 160 127)(10 114 161 146)(11 103 162 135)(12 92 163 124)(13 111 164 143)(14 100 165 132)(15 119 166 121)(16 108 167 140)(17 97 168 129)(18 116 169 148)(19 105 170 137)(20 94 171 126)(21 113 172 145)(22 102 173 134)(23 91 174 123)(24 110 175 142)(25 99 176 131)(26 118 177 150)(27 107 178 139)(28 96 179 128)(29 115 180 147)(30 104 151 136)(31 192 69 227)(32 181 70 216)(33 200 71 235)(34 189 72 224)(35 208 73 213)(36 197 74 232)(37 186 75 221)(38 205 76 240)(39 194 77 229)(40 183 78 218)(41 202 79 237)(42 191 80 226)(43 210 81 215)(44 199 82 234)(45 188 83 223)(46 207 84 212)(47 196 85 231)(48 185 86 220)(49 204 87 239)(50 193 88 228)(51 182 89 217)(52 201 90 236)(53 190 61 225)(54 209 62 214)(55 198 63 233)(56 187 64 222)(57 206 65 211)(58 195 66 230)(59 184 67 219)(60 203 68 238)
(1 74)(2 85)(3 66)(4 77)(5 88)(6 69)(7 80)(8 61)(9 72)(10 83)(11 64)(12 75)(13 86)(14 67)(15 78)(16 89)(17 70)(18 81)(19 62)(20 73)(21 84)(22 65)(23 76)(24 87)(25 68)(26 79)(27 90)(28 71)(29 82)(30 63)(31 157)(32 168)(33 179)(34 160)(35 171)(36 152)(37 163)(38 174)(39 155)(40 166)(41 177)(42 158)(43 169)(44 180)(45 161)(46 172)(47 153)(48 164)(49 175)(50 156)(51 167)(52 178)(53 159)(54 170)(55 151)(56 162)(57 173)(58 154)(59 165)(60 176)(91 205)(92 186)(93 197)(94 208)(95 189)(96 200)(97 181)(98 192)(99 203)(100 184)(101 195)(102 206)(103 187)(104 198)(105 209)(106 190)(107 201)(108 182)(109 193)(110 204)(111 185)(112 196)(113 207)(114 188)(115 199)(116 210)(117 191)(118 202)(119 183)(120 194)(121 218)(122 229)(123 240)(124 221)(125 232)(126 213)(127 224)(128 235)(129 216)(130 227)(131 238)(132 219)(133 230)(134 211)(135 222)(136 233)(137 214)(138 225)(139 236)(140 217)(141 228)(142 239)(143 220)(144 231)(145 212)(146 223)(147 234)(148 215)(149 226)(150 237)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C10A···10F10G10H10I10J12A12B12C12D12E12F15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order1222222234444445566610···10101010101212121212121515202020202020202030···3060···60
size1111123030602466101020222222···2121212124420202020444444121212124···44···4

60 irreducible representations

dim111111122222222222244444444
type+++++++++++++++++++++++++
imageC1C2C2C2C2C2C2S3D4D4D5D6D6C4○D4D10D10D10D12C4○D20S3×D4Q83S3S3×D5D4×D5C2×S3×D5D60⋊C2D5×D12D10⋊D6
kernelD30⋊D4D304C4C3×C10.D4C5×D6⋊C4C2×D30.C2C2×C5⋊D12C2×D60C10.D4C3×Dic5D30D6⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3Dic5C6C10C10C2×C4C6C22C2C2C2
# reps111112112222122224811242444

Matrix representation of D30⋊D4 in GL4(𝔽61) generated by

06000
1100
00181
00600
,
06000
60000
00181
004343
,
1000
0100
00500
001511
,
381500
382300
003117
004430
G:=sub<GL(4,GF(61))| [0,1,0,0,60,1,0,0,0,0,18,60,0,0,1,0],[0,60,0,0,60,0,0,0,0,0,18,43,0,0,1,43],[1,0,0,0,0,1,0,0,0,0,50,15,0,0,0,11],[38,38,0,0,15,23,0,0,0,0,31,44,0,0,17,30] >;

D30⋊D4 in GAP, Magma, Sage, TeX

D_{30}\rtimes D_4
% in TeX

G:=Group("D30:D4");
// GroupNames label

G:=SmallGroup(480,496);
// by ID

G=gap.SmallGroup(480,496);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,254,219,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=a^19,d*a*d=a^11,c*b*c^-1=a^18*b,d*b*d=a^25*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽