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G = D30.35D4order 480 = 25·3·5

8th non-split extension by D30 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D30.35D4, D6⋊C49D5, C6.54(D4×D5), D6⋊Dic54C2, C10.56(S3×D4), D304C47C2, C30.107(C2×D4), C10.D49S3, (C2×C20).181D6, C52(D6.D4), C6.Dic103C2, C30.22(C4○D4), C6.22(C4○D20), (C2×C12).179D10, C6.6(D42D5), (C2×C30).45C23, (C2×Dic3).9D10, (C2×Dic5).11D6, (C22×S3).5D10, C10.25(C4○D12), C31(D10.12D4), (C2×C60).158C22, C154(C22.D4), C2.10(D10⋊D6), C2.10(D12⋊D5), C10.25(Q83S3), (C6×Dic5).27C22, C2.15(D6.D10), (C10×Dic3).26C22, (C22×D15).96C22, (C2×Dic15).184C22, (C2×C4×D15)⋊8C2, (C5×D6⋊C4)⋊9C2, (C2×C4).171(S3×D5), (C2×C5⋊D12).4C2, (S3×C2×C10).5C22, C22.134(C2×S3×D5), (C3×C10.D4)⋊9C2, (C2×C6).57(C22×D5), (C2×C10).57(C22×S3), SmallGroup(480,431)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D30.35D4
C1C5C15C30C2×C30C6×Dic5D6⋊Dic5 — D30.35D4
C15C2×C30 — D30.35D4
C1C22C2×C4

Generators and relations for D30.35D4
 G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=a19, dad=a11, cbc-1=a3b, dbd=a25b, dcd=a15c-1 >

Subgroups: 908 in 156 conjugacy classes, 46 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×5], C22, C22 [×7], C5, S3 [×3], C6 [×3], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], D5 [×2], C10 [×3], C10, Dic3 [×2], C12 [×3], D6 [×7], C2×C6, C15, C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×4], C2×C10, C2×C10 [×3], C4×S3 [×2], D12 [×2], C2×Dic3, C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, C22×S3, C5×S3, D15 [×2], C30 [×3], C22.D4, C4×D5 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20, C2×C20, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C2×D12, C5×Dic3, C3×Dic5 [×2], Dic15, C60, S3×C10 [×3], D30 [×2], D30 [×2], C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, D6.D4, C5⋊D12 [×2], C6×Dic5 [×2], C10×Dic3, C4×D15 [×2], C2×Dic15, C2×C60, S3×C2×C10, C22×D15, D10.12D4, D6⋊Dic5, D304C4, C6.Dic10, C3×C10.D4, C5×D6⋊C4, C2×C5⋊D12, C2×C4×D15, D30.35D4
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, Q83S3, S3×D5, C4○D20, D4×D5, D42D5, D6.D4, C2×S3×D5, D10.12D4, D12⋊D5, D6.D10, D10⋊D6, D30.35D4

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

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C10A···10F10G10H10I10J12A12B12C12D12E12F15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order1222222344444445566610···10101010101212121212121515202020202020202030···3060···60
size11111230302221220203030222222···2121212124420202020444444121212124···44···4

60 irreducible representations

dim1111111122222222222444444444
type++++++++++++++++++++-++
imageC1C2C2C2C2C2C2C2S3D4D5D6D6C4○D4D10D10D10C4○D12C4○D20S3×D4Q83S3S3×D5D4×D5D42D5C2×S3×D5D12⋊D5D6.D10D10⋊D6
kernelD30.35D4D6⋊Dic5D304C4C6.Dic10C3×C10.D4C5×D6⋊C4C2×C5⋊D12C2×C4×D15C10.D4D30D6⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3C10C6C10C10C2×C4C6C6C22C2C2C2
# reps1111111112221422248112222444

Matrix representation of D30.35D4 in GL8(𝔽61)

600000000
060000000
00010000
0060600000
000006000
000011700
000000600
000000060
,
600000000
161000000
00010000
00100000
0000446000
0000441700
000000600
00000031
,
2711000000
1734000000
006000000
000600000
0000174400
0000604400
000000417
0000005620
,
81000000
5953000000
006000000
00110000
00001000
00000100
0000003745
0000005524

G:=sub<GL(8,GF(61))| [60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,17,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[60,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,44,44,0,0,0,0,0,0,60,17,0,0,0,0,0,0,0,0,60,3,0,0,0,0,0,0,0,1],[27,17,0,0,0,0,0,0,11,34,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,17,60,0,0,0,0,0,0,44,44,0,0,0,0,0,0,0,0,41,56,0,0,0,0,0,0,7,20],[8,59,0,0,0,0,0,0,1,53,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,37,55,0,0,0,0,0,0,45,24] >;

D30.35D4 in GAP, Magma, Sage, TeX

D_{30}._{35}D_4
% in TeX

G:=Group("D30.35D4");
// GroupNames label

G:=SmallGroup(480,431);
// by ID

G=gap.SmallGroup(480,431);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,422,219,142,1356,18822]);
// 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^3*b,d*b*d=a^25*b,d*c*d=a^15*c^-1>;
// generators/relations

׿
×
𝔽