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G = D3012D4order 480 = 25·3·5

4th semidirect product of D30 and D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D3012D4, Dic1515D4, D6⋊C412D5, C6.64(D4×D5), C52(Dic3⋊D4), C10.66(S3×D4), C32(D10⋊D4), C1513(C4⋊D4), (C2×C20).203D6, C30.155(C2×D4), D10⋊C412S3, C30.93(C4○D4), C6.41(C4○D20), (C2×C12).201D10, Dic155C426C2, (C2×Dic5).47D6, (C22×D5).21D6, C10.44(C4○D12), C2.22(C20⋊D6), (C2×C60).169C22, (C2×C30).151C23, (C2×Dic3).47D10, (C22×S3).19D10, C2.17(D10⋊D6), (C6×Dic5).91C22, C2.30(D6.D10), (C10×Dic3).91C22, (C2×Dic15).213C22, (C22×D15).108C22, (C2×C4×D15)⋊11C2, (C5×D6⋊C4)⋊12C2, (C2×C3⋊D20)⋊7C2, (C2×C15⋊D4)⋊7C2, (C2×C5⋊D12)⋊7C2, (C2×C4).182(S3×D5), (D5×C2×C6).35C22, C22.203(C2×S3×D5), (S3×C2×C10).35C22, (C3×D10⋊C4)⋊12C2, (C2×C6).163(C22×D5), (C2×C10).163(C22×S3), SmallGroup(480,537)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D3012D4
C1C5C15C30C2×C30D5×C2×C6C2×C15⋊D4 — D3012D4
C15C2×C30 — D3012D4
C1C22C2×C4

Generators and relations for D3012D4
 G = < a,b,c,d | a30=b2=c4=d2=1, bab=cac-1=a-1, dad=a19, cbc-1=a28b, dbd=a3b, dcd=c-1 >

Subgroups: 1196 in 188 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×10], C5, S3 [×3], C6 [×3], C6, C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], D5 [×3], C10 [×3], C10, Dic3 [×3], C12 [×2], D6 [×7], C2×C6, C2×C6 [×3], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×3], C20 [×2], D10 [×7], C2×C10, C2×C10 [×3], C4×S3 [×2], D12 [×2], C2×Dic3, C2×Dic3, C3⋊D4 [×4], C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, D15 [×2], C30 [×3], C4⋊D4, C4×D5 [×2], D20 [×2], C2×Dic5, C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4 [×2], C5×Dic3, C3×Dic5, Dic15 [×2], C60, C6×D5 [×3], S3×C10 [×3], D30 [×2], D30 [×2], C2×C30, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4 [×2], Dic3⋊D4, C15⋊D4 [×2], C3⋊D20 [×2], C5⋊D12 [×2], C6×Dic5, C10×Dic3, C4×D15 [×2], C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, C22×D15, D10⋊D4, Dic155C4, C3×D10⋊C4, C5×D6⋊C4, C2×C15⋊D4, C2×C3⋊D20, C2×C5⋊D12, C2×C4×D15, D3012D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], C22×S3, C4⋊D4, C22×D5, C4○D12, S3×D4 [×2], S3×D5, C4○D20, D4×D5 [×2], Dic3⋊D4, C2×S3×D5, D10⋊D4, D6.D10, C20⋊D6, D10⋊D6, D3012D4

Smallest permutation representation of D3012D4
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)
(1 221)(2 220)(3 219)(4 218)(5 217)(6 216)(7 215)(8 214)(9 213)(10 212)(11 211)(12 240)(13 239)(14 238)(15 237)(16 236)(17 235)(18 234)(19 233)(20 232)(21 231)(22 230)(23 229)(24 228)(25 227)(26 226)(27 225)(28 224)(29 223)(30 222)(31 200)(32 199)(33 198)(34 197)(35 196)(36 195)(37 194)(38 193)(39 192)(40 191)(41 190)(42 189)(43 188)(44 187)(45 186)(46 185)(47 184)(48 183)(49 182)(50 181)(51 210)(52 209)(53 208)(54 207)(55 206)(56 205)(57 204)(58 203)(59 202)(60 201)(61 149)(62 148)(63 147)(64 146)(65 145)(66 144)(67 143)(68 142)(69 141)(70 140)(71 139)(72 138)(73 137)(74 136)(75 135)(76 134)(77 133)(78 132)(79 131)(80 130)(81 129)(82 128)(83 127)(84 126)(85 125)(86 124)(87 123)(88 122)(89 121)(90 150)(91 170)(92 169)(93 168)(94 167)(95 166)(96 165)(97 164)(98 163)(99 162)(100 161)(101 160)(102 159)(103 158)(104 157)(105 156)(106 155)(107 154)(108 153)(109 152)(110 151)(111 180)(112 179)(113 178)(114 177)(115 176)(116 175)(117 174)(118 173)(119 172)(120 171)
(1 125 237 71)(2 124 238 70)(3 123 239 69)(4 122 240 68)(5 121 211 67)(6 150 212 66)(7 149 213 65)(8 148 214 64)(9 147 215 63)(10 146 216 62)(11 145 217 61)(12 144 218 90)(13 143 219 89)(14 142 220 88)(15 141 221 87)(16 140 222 86)(17 139 223 85)(18 138 224 84)(19 137 225 83)(20 136 226 82)(21 135 227 81)(22 134 228 80)(23 133 229 79)(24 132 230 78)(25 131 231 77)(26 130 232 76)(27 129 233 75)(28 128 234 74)(29 127 235 73)(30 126 236 72)(31 173 181 109)(32 172 182 108)(33 171 183 107)(34 170 184 106)(35 169 185 105)(36 168 186 104)(37 167 187 103)(38 166 188 102)(39 165 189 101)(40 164 190 100)(41 163 191 99)(42 162 192 98)(43 161 193 97)(44 160 194 96)(45 159 195 95)(46 158 196 94)(47 157 197 93)(48 156 198 92)(49 155 199 91)(50 154 200 120)(51 153 201 119)(52 152 202 118)(53 151 203 117)(54 180 204 116)(55 179 205 115)(56 178 206 114)(57 177 207 113)(58 176 208 112)(59 175 209 111)(60 174 210 110)
(1 206)(2 195)(3 184)(4 203)(5 192)(6 181)(7 200)(8 189)(9 208)(10 197)(11 186)(12 205)(13 194)(14 183)(15 202)(16 191)(17 210)(18 199)(19 188)(20 207)(21 196)(22 185)(23 204)(24 193)(25 182)(26 201)(27 190)(28 209)(29 198)(30 187)(31 212)(32 231)(33 220)(34 239)(35 228)(36 217)(37 236)(38 225)(39 214)(40 233)(41 222)(42 211)(43 230)(44 219)(45 238)(46 227)(47 216)(48 235)(49 224)(50 213)(51 232)(52 221)(53 240)(54 229)(55 218)(56 237)(57 226)(58 215)(59 234)(60 223)(61 104)(62 93)(63 112)(64 101)(65 120)(66 109)(67 98)(68 117)(69 106)(70 95)(71 114)(72 103)(73 92)(74 111)(75 100)(76 119)(77 108)(78 97)(79 116)(80 105)(81 94)(82 113)(83 102)(84 91)(85 110)(86 99)(87 118)(88 107)(89 96)(90 115)(121 162)(122 151)(123 170)(124 159)(125 178)(126 167)(127 156)(128 175)(129 164)(130 153)(131 172)(132 161)(133 180)(134 169)(135 158)(136 177)(137 166)(138 155)(139 174)(140 163)(141 152)(142 171)(143 160)(144 179)(145 168)(146 157)(147 176)(148 165)(149 154)(150 173)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order122222223444444556666610···1010101010121212121515202020202020202030···3060···60
size111112203030222122030302222220202···212121212442020444444121212124···44···4

60 irreducible representations

dim1111111122222222222224444444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6C4○D4D10D10D10C4○D12C4○D20S3×D4S3×D5D4×D5C2×S3×D5D6.D10C20⋊D6D10⋊D6
kernelD3012D4Dic155C4C3×D10⋊C4C5×D6⋊C4C2×C15⋊D4C2×C3⋊D20C2×C5⋊D12C2×C4×D15D10⋊C4Dic15D30D6⋊C4C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12C22×S3C10C6C10C2×C4C6C22C2C2C2
# reps1111111112221112222482242444

Matrix representation of D3012D4 in GL6(𝔽61)

0600000
1430000
0016000
001000
0000600
0000060
,
18600000
18430000
0006000
0060000
000010
00003060
,
43180000
60180000
00501100
0001100
000010
000001
,
43180000
60180000
00521800
0043900
00002955
00001832

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,43,0,0,0,0,0,0,1,1,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[18,18,0,0,0,0,60,43,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,1,30,0,0,0,0,0,60],[43,60,0,0,0,0,18,18,0,0,0,0,0,0,50,0,0,0,0,0,11,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[43,60,0,0,0,0,18,18,0,0,0,0,0,0,52,43,0,0,0,0,18,9,0,0,0,0,0,0,29,18,0,0,0,0,55,32] >;

D3012D4 in GAP, Magma, Sage, TeX

D_{30}\rtimes_{12}D_4
% in TeX

G:=Group("D30:12D4");
// GroupNames label

G:=SmallGroup(480,537);
// by ID

G=gap.SmallGroup(480,537);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,422,219,58,1356,18822]);
// 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^28*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽