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G = D30.D4order 480 = 25·3·5

5th non-split extension by D30 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D30.5D4, C4⋊Dic33D5, C6.38(D4×D5), C30.33(C2×D4), C10.38(S3×D4), (C2×C12).9D10, (C2×C20).221D6, C52(C23.9D6), Dic155C44C2, D303C418C2, D10⋊C415S3, (C22×D5).5D6, C6.64(C4○D20), C10.4(C4○D12), C30.23(C4○D4), D10⋊Dic34C2, (C2×C30).46C23, C6.5(Q82D5), (C2×Dic5).94D6, C2.9(C12.28D10), C33(D10.13D4), C2.16(C20⋊D6), (C2×C60).314C22, (C2×Dic3).87D10, C155(C22.D4), C10.40(D42S3), (C6×Dic5).28C22, C2.12(Dic5.D6), (C2×Dic15).49C22, (C10×Dic3).27C22, (C22×D15).22C22, (C2×C4).33(S3×D5), (D5×C2×C6).3C22, (C2×D30.C2)⋊1C2, (C5×C4⋊Dic3)⋊15C2, (C2×C3⋊D20).4C2, C22.135(C2×S3×D5), (C3×D10⋊C4)⋊20C2, (C2×C6).58(C22×D5), (C2×C10).58(C22×S3), SmallGroup(480,432)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D30.D4
C1C5C15C30C2×C30D5×C2×C6D10⋊Dic3 — 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=cac-1=a-1, dad=a19, cbc-1=a13b, dbd=a3b, dcd=a15c-1 >

Subgroups: 940 in 156 conjugacy classes, 46 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×5], C22, C22 [×7], C5, S3 [×2], C6 [×3], C6, C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], D5 [×3], C10 [×3], Dic3 [×3], C12 [×2], D6 [×4], C2×C6, C2×C6 [×3], C15, C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×2], C20 [×3], D10 [×7], C2×C10, C4×S3 [×2], C2×Dic3 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12, C2×C12, C22×S3, C22×C6, C3×D5, D15 [×2], C30 [×3], C22.D4, C4×D5 [×2], D20 [×2], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4, C5×Dic3 [×2], C3×Dic5, Dic15, C60, C6×D5 [×3], D30 [×2], D30 [×2], C2×C30, C10.D4, D10⋊C4, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×D20, C23.9D6, D30.C2 [×2], C3⋊D20 [×2], C6×Dic5, C10×Dic3 [×2], C2×Dic15, C2×C60, D5×C2×C6, C22×D15, D10.13D4, D10⋊Dic3, Dic155C4, C3×D10⋊C4, C5×C4⋊Dic3, D303C4, C2×D30.C2, C2×C3⋊D20, D30.D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C22×S3, C22.D4, C22×D5, C4○D12, S3×D4, D42S3, S3×D5, C4○D20, D4×D5, Q82D5, C23.9D6, C2×S3×D5, D10.13D4, C12.28D10, C20⋊D6, Dic5.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 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(78 90)(79 89)(80 88)(81 87)(82 86)(83 85)(91 120)(92 119)(93 118)(94 117)(95 116)(96 115)(97 114)(98 113)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)(121 150)(122 149)(123 148)(124 147)(125 146)(126 145)(127 144)(128 143)(129 142)(130 141)(131 140)(132 139)(133 138)(134 137)(135 136)(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)(181 207)(182 206)(183 205)(184 204)(185 203)(186 202)(187 201)(188 200)(189 199)(190 198)(191 197)(192 196)(193 195)(208 210)(211 215)(212 214)(216 240)(217 239)(218 238)(219 237)(220 236)(221 235)(222 234)(223 233)(224 232)(225 231)(226 230)(227 229)
(1 31 106 221)(2 60 107 220)(3 59 108 219)(4 58 109 218)(5 57 110 217)(6 56 111 216)(7 55 112 215)(8 54 113 214)(9 53 114 213)(10 52 115 212)(11 51 116 211)(12 50 117 240)(13 49 118 239)(14 48 119 238)(15 47 120 237)(16 46 91 236)(17 45 92 235)(18 44 93 234)(19 43 94 233)(20 42 95 232)(21 41 96 231)(22 40 97 230)(23 39 98 229)(24 38 99 228)(25 37 100 227)(26 36 101 226)(27 35 102 225)(28 34 103 224)(29 33 104 223)(30 32 105 222)(61 122 186 168)(62 121 187 167)(63 150 188 166)(64 149 189 165)(65 148 190 164)(66 147 191 163)(67 146 192 162)(68 145 193 161)(69 144 194 160)(70 143 195 159)(71 142 196 158)(72 141 197 157)(73 140 198 156)(74 139 199 155)(75 138 200 154)(76 137 201 153)(77 136 202 152)(78 135 203 151)(79 134 204 180)(80 133 205 179)(81 132 206 178)(82 131 207 177)(83 130 208 176)(84 129 209 175)(85 128 210 174)(86 127 181 173)(87 126 182 172)(88 125 183 171)(89 124 184 170)(90 123 185 169)
(1 62)(2 81)(3 70)(4 89)(5 78)(6 67)(7 86)(8 75)(9 64)(10 83)(11 72)(12 61)(13 80)(14 69)(15 88)(16 77)(17 66)(18 85)(19 74)(20 63)(21 82)(22 71)(23 90)(24 79)(25 68)(26 87)(27 76)(28 65)(29 84)(30 73)(31 152)(32 171)(33 160)(34 179)(35 168)(36 157)(37 176)(38 165)(39 154)(40 173)(41 162)(42 151)(43 170)(44 159)(45 178)(46 167)(47 156)(48 175)(49 164)(50 153)(51 172)(52 161)(53 180)(54 169)(55 158)(56 177)(57 166)(58 155)(59 174)(60 163)(91 202)(92 191)(93 210)(94 199)(95 188)(96 207)(97 196)(98 185)(99 204)(100 193)(101 182)(102 201)(103 190)(104 209)(105 198)(106 187)(107 206)(108 195)(109 184)(110 203)(111 192)(112 181)(113 200)(114 189)(115 208)(116 197)(117 186)(118 205)(119 194)(120 183)(121 236)(122 225)(123 214)(124 233)(125 222)(126 211)(127 230)(128 219)(129 238)(130 227)(131 216)(132 235)(133 224)(134 213)(135 232)(136 221)(137 240)(138 229)(139 218)(140 237)(141 226)(142 215)(143 234)(144 223)(145 212)(146 231)(147 220)(148 239)(149 228)(150 217)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C6D6E10A···10F12A12B12C12D15A15B20A20B20C20D20E···20L30A···30F60A···60H
order122222234444444556666610···101212121215152020202020···2030···3060···60
size11112030302466101012602222220202···244202044444412···124···44···4

60 irreducible representations

dim1111111122222222222444444444
type+++++++++++++++++-+++++
imageC1C2C2C2C2C2C2C2S3D4D5D6D6D6C4○D4D10D10C4○D12C4○D20S3×D4D42S3S3×D5D4×D5Q82D5C2×S3×D5C12.28D10C20⋊D6Dic5.D6
kernelD30.D4D10⋊Dic3Dic155C4C3×D10⋊C4C5×C4⋊Dic3D303C4C2×D30.C2C2×C3⋊D20D10⋊C4D30C4⋊Dic3C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12C10C6C10C10C2×C4C6C6C22C2C2C2
# reps1111111112211144248112222444

Matrix representation of D30.D4 in GL6(𝔽61)

010000
60600000
0060000
0006000
0000171
0000161
,
010000
100000
00603600
000100
0000043
0000440
,
6000000
110000
00184300
0014300
00003930
00005522
,
100000
010000
00152400
00114600
0000225
00005659

G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,17,16,0,0,0,0,1,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,36,1,0,0,0,0,0,0,0,44,0,0,0,0,43,0],[60,1,0,0,0,0,0,1,0,0,0,0,0,0,18,1,0,0,0,0,43,43,0,0,0,0,0,0,39,55,0,0,0,0,30,22],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,11,0,0,0,0,24,46,0,0,0,0,0,0,2,56,0,0,0,0,25,59] >;

D30.D4 in GAP, Magma, Sage, TeX

D_{30}.D_4
% in TeX

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

G:=SmallGroup(480,432);
// by ID

G=gap.SmallGroup(480,432);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,64,590,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,d*a*d=a^19,c*b*c^-1=a^13*b,d*b*d=a^3*b,d*c*d=a^15*c^-1>;
// generators/relations

׿
×
𝔽