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G = D30.34D4order 480 = 25·3·5

7th non-split extension by D30 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D30.34D4, C6.53(D4×D5), Dic3⋊C49D5, C10.55(S3×D4), D304C46C2, D10⋊C49S3, (C2×C20).180D6, C30.106(C2×D4), C51(C23.9D6), C30.Q84C2, (C22×D5).4D6, C30.21(C4○D4), C6.21(C4○D20), D10⋊Dic33C2, (C2×C12).178D10, (C2×C30).44C23, (C2×Dic3).8D10, (C2×Dic5).10D6, C10.24(C4○D12), C2.8(D20⋊S3), C10.5(D42S3), C2.9(D10⋊D6), C32(D10.13D4), (C2×C60).157C22, C6.24(Q82D5), C153(C22.D4), (C6×Dic5).26C22, C2.14(D6.D10), (C10×Dic3).25C22, (C22×D15).95C22, (C2×Dic15).183C22, (C2×C4×D15)⋊7C2, (D5×C2×C6).2C22, (C5×Dic3⋊C4)⋊9C2, (C2×C4).170(S3×D5), (C2×C3⋊D20).3C2, (C3×D10⋊C4)⋊9C2, C22.133(C2×S3×D5), (C2×C6).56(C22×D5), (C2×C10).56(C22×S3), SmallGroup(480,430)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D30.34D4
C1C5C15C30C2×C30D5×C2×C6D10⋊Dic3 — D30.34D4
C15C2×C30 — D30.34D4
C1C22C2×C4

Generators and relations for D30.34D4
 G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=a11, dad=a19, cbc-1=a25b, 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, C3⋊D20 [×2], C6×Dic5, C10×Dic3 [×2], C4×D15 [×2], C2×Dic15, C2×C60, D5×C2×C6, C22×D15, D10.13D4, D10⋊Dic3, D304C4, C30.Q8, C3×D10⋊C4, C5×Dic3⋊C4, C2×C3⋊D20, C2×C4×D15, D30.34D4
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, D20⋊S3, D6.D10, D10⋊D6, D30.34D4

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

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C6D6E10A···10F12A12B12C12D15A15B20A20B20C20D20E···20L30A···30F60A···60H
order122222234444444556666610···101212121215152020202020···2030···3060···60
size111120303022212122030302222220202···244202044444412···124···44···4

60 irreducible representations

dim1111111122222222222444444444
type+++++++++++++++++-+++++
imageC1C2C2C2C2C2C2C2S3D4D5D6D6D6C4○D4D10D10C4○D12C4○D20S3×D4D42S3S3×D5D4×D5Q82D5C2×S3×D5D20⋊S3D6.D10D10⋊D6
kernelD30.34D4D10⋊Dic3D304C4C30.Q8C3×D10⋊C4C5×Dic3⋊C4C2×C3⋊D20C2×C4×D15D10⋊C4D30Dic3⋊C4C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12C10C6C10C10C2×C4C6C6C22C2C2C2
# reps1111111112211144248112222444

Matrix representation of D30.34D4 in GL6(𝔽61)

010000
60600000
0014500
00601700
0000600
0000060
,
010000
100000
0001700
0018000
0000600
0000381
,
6000000
110000
00142800
00174700
00004260
00005719
,
6000000
0600000
0059500
0036200
00003550
00001726

G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,60,0,0,0,0,45,17,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,18,0,0,0,0,17,0,0,0,0,0,0,0,60,38,0,0,0,0,0,1],[60,1,0,0,0,0,0,1,0,0,0,0,0,0,14,17,0,0,0,0,28,47,0,0,0,0,0,0,42,57,0,0,0,0,60,19],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,59,36,0,0,0,0,5,2,0,0,0,0,0,0,35,17,0,0,0,0,50,26] >;

D30.34D4 in GAP, Magma, Sage, TeX

D_{30}._{34}D_4
% in TeX

G:=Group("D30.34D4");
// GroupNames label

G:=SmallGroup(480,430);
// by ID

G=gap.SmallGroup(480,430);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,120,422,219,142,1356,18822]);
// 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^11,d*a*d=a^19,c*b*c^-1=a^25*b,d*b*d=a^3*b,d*c*d=a^15*c^-1>;
// generators/relations

׿
×
𝔽