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G = D30.16D4order 480 = 25·3·5

16th non-split extension by D30 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D30.16D4, C6.90(D4×D5), C10.91(S3×D4), C30.242(C2×D4), C54(C23.9D6), C23.D512S3, D304C433C2, C23.23(S3×D5), C6.86(C4○D20), C6.D412D5, C6.Dic1037C2, C30.Q836C2, (C2×Dic5).64D6, (C22×C6).41D10, (C22×C10).56D6, C10.86(C4○D12), C30.155(C4○D4), C6.59(D42D5), C34(D10.12D4), (C2×C30).204C23, C2.41(D10⋊D6), C10.58(D42S3), (C2×Dic3).126D10, C1524(C22.D4), (C22×C30).66C22, C2.29(Dic3.D10), C2.29(Dic5.D6), (C6×Dic5).119C22, (C22×D15).67C22, (C2×Dic15).140C22, (C10×Dic3).119C22, C22.233(C2×S3×D5), (C2×D30.C2)⋊18C2, (C2×C157D4).13C2, (C3×C23.D5)⋊14C2, (C5×C6.D4)⋊14C2, (C2×C6).216(C22×D5), (C2×C10).216(C22×S3), SmallGroup(480,638)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D30.16D4
C1C5C15C30C2×C30C6×Dic5C2×D30.C2 — D30.16D4
C15C2×C30 — D30.16D4
C1C22C23

Generators and relations for D30.16D4
 G = < a,b,c,d | a30=b2=c4=1, d2=a15, bab=a-1, cac-1=dad-1=a19, cbc-1=a3b, dbd-1=a18b, dcd-1=a15c-1 >

Subgroups: 876 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 [×7], D4 [×2], C23, C23, D5 [×2], C10 [×3], C10, Dic3 [×3], C12 [×2], D6 [×4], C2×C6, C2×C6 [×3], 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], C2×Dic3 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C22×S3, C22×C6, D15 [×2], C30 [×3], C30, C22.D4, C4×D5 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×C10, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15, D30 [×2], D30 [×2], C2×C30, C2×C30 [×3], C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, C23.9D6, D30.C2 [×2], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15, C157D4 [×2], C22×D15, C22×C30, D10.12D4, D304C4, C30.Q8, C6.Dic10, C3×C23.D5, C5×C6.D4, C2×D30.C2, C2×C157D4, D30.16D4
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, D42D5, C23.9D6, C2×S3×D5, D10.12D4, Dic5.D6, Dic3.D10, D10⋊D6, D30.16D4

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

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B20A···20H30A···30N
order122222234444444556666610···101010101012121212151520···2030···30
size111143030266101012206022222442···24444202020204412···124···4

60 irreducible representations

dim111111112222222222444444444
type++++++++++++++++-++-++
imageC1C2C2C2C2C2C2C2S3D4D5D6D6C4○D4D10D10C4○D12C4○D20S3×D4D42S3S3×D5D4×D5D42D5C2×S3×D5Dic5.D6Dic3.D10D10⋊D6
kernelD30.16D4D304C4C30.Q8C6.Dic10C3×C23.D5C5×C6.D4C2×D30.C2C2×C157D4C23.D5D30C6.D4C2×Dic5C22×C10C30C2×Dic3C22×C6C10C6C10C10C23C6C6C22C2C2C2
# reps111111111222144248112222444

Matrix representation of D30.16D4 in GL4(𝔽61) generated by

0100
601800
00160
0010
,
43100
431800
00600
00601
,
272700
253400
003846
001523
,
154600
114600
00110
00011
G:=sub<GL(4,GF(61))| [0,60,0,0,1,18,0,0,0,0,1,1,0,0,60,0],[43,43,0,0,1,18,0,0,0,0,60,60,0,0,0,1],[27,25,0,0,27,34,0,0,0,0,38,15,0,0,46,23],[15,11,0,0,46,46,0,0,0,0,11,0,0,0,0,11] >;

D30.16D4 in GAP, Magma, Sage, TeX

D_{30}._{16}D_4
% in TeX

G:=Group("D30.16D4");
// GroupNames label

G:=SmallGroup(480,638);
// by ID

G=gap.SmallGroup(480,638);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,303,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=a^15,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^19,c*b*c^-1=a^3*b,d*b*d^-1=a^18*b,d*c*d^-1=a^15*c^-1>;
// generators/relations

׿
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