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G = D306D4order 480 = 25·3·5

6th semidirect product of D30 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D306D4, C6.84(D4×D5), C52(D63D4), (C3×Dic5)⋊3D4, C10.85(S3×D4), C36(D10⋊D4), C1516(C4⋊D4), C6.D44D5, C30.213(C2×D4), C23.15(S3×D5), C6.78(C4○D20), Dic52(C3⋊D4), C30.Q830C2, (C22×D5).26D6, (C22×C6).23D10, (C22×C10).38D6, C30.135(C4○D4), D10⋊Dic325C2, (C2×C30).175C23, (C2×Dic3).54D10, (C2×Dic5).126D6, C10.50(D42S3), C2.37(D10⋊D6), (C22×C30).37C22, C2.23(Dic5.D6), (C6×Dic5).103C22, (C22×D15).59C22, (C10×Dic3).103C22, (C2×Dic15).124C22, (C6×C5⋊D4)⋊1C2, (C2×C5⋊D4)⋊1S3, C2.36(D5×C3⋊D4), (C2×C3⋊D20)⋊11C2, (C2×C157D4)⋊12C2, C10.57(C2×C3⋊D4), (D5×C2×C6).44C22, C22.219(C2×S3×D5), (C2×D30.C2)⋊13C2, (C5×C6.D4)⋊4C2, (C2×C6).187(C22×D5), (C2×C10).187(C22×S3), SmallGroup(480,609)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D306D4
C1C5C15C30C2×C30D5×C2×C6D10⋊Dic3 — D306D4
C15C2×C30 — D306D4
C1C22C23

Generators and relations for D306D4
 G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=dad=a19, cbc-1=a18b, dbd=a3b, dcd=c-1 >

Subgroups: 1116 in 188 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×10], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4 [×6], D4 [×6], C23, C23 [×2], D5 [×3], C10 [×3], C10, Dic3 [×3], C12 [×2], D6 [×4], C2×C6, C2×C6 [×6], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×2], Dic5, C20 [×2], D10 [×7], C2×C10, C2×C10 [×3], C4×S3 [×2], C2×Dic3 [×2], C2×Dic3, C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6, C22×C6, C3×D5, D15 [×2], C30 [×3], C30, C4⋊D4, C4×D5 [×2], D20 [×2], C2×Dic5, C2×Dic5, C5⋊D4 [×4], C2×C20 [×2], C22×D5, C22×D5, C22×C10, C4⋊Dic3, C6.D4, C6.D4, S3×C2×C4, C2×C3⋊D4 [×2], C6×D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15, C6×D5 [×3], D30 [×2], D30 [×2], C2×C30, C2×C30 [×3], C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C2×C5⋊D4, D63D4, D30.C2 [×2], C3⋊D20 [×2], C6×Dic5, C3×C5⋊D4 [×2], C10×Dic3 [×2], C2×Dic15, C157D4 [×2], D5×C2×C6, C22×D15, C22×C30, D10⋊D4, D10⋊Dic3, C30.Q8, C5×C6.D4, C2×D30.C2, C2×C3⋊D20, C6×C5⋊D4, C2×C157D4, D306D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], C3⋊D4 [×2], C22×S3, C4⋊D4, C22×D5, S3×D4, D42S3, C2×C3⋊D4, S3×D5, C4○D20, D4×D5 [×2], D63D4, C2×S3×D5, D10⋊D4, Dic5.D6, D5×C3⋊D4, D10⋊D6, D306D4

Smallest permutation representation of D306D4
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 69)(62 68)(63 67)(64 66)(70 90)(71 89)(72 88)(73 87)(74 86)(75 85)(76 84)(77 83)(78 82)(79 81)(91 119)(92 118)(93 117)(94 116)(95 115)(96 114)(97 113)(98 112)(99 111)(100 110)(101 109)(102 108)(103 107)(104 106)(121 124)(122 123)(125 150)(126 149)(127 148)(128 147)(129 146)(130 145)(131 144)(132 143)(133 142)(134 141)(135 140)(136 139)(137 138)(151 172)(152 171)(153 170)(154 169)(155 168)(156 167)(157 166)(158 165)(159 164)(160 163)(161 162)(173 180)(174 179)(175 178)(176 177)(181 183)(184 210)(185 209)(186 208)(187 207)(188 206)(189 205)(190 204)(191 203)(192 202)(193 201)(194 200)(195 199)(196 198)(211 230)(212 229)(213 228)(214 227)(215 226)(216 225)(217 224)(218 223)(219 222)(220 221)(231 240)(232 239)(233 238)(234 237)(235 236)
(1 138 177 221)(2 127 178 240)(3 146 179 229)(4 135 180 218)(5 124 151 237)(6 143 152 226)(7 132 153 215)(8 121 154 234)(9 140 155 223)(10 129 156 212)(11 148 157 231)(12 137 158 220)(13 126 159 239)(14 145 160 228)(15 134 161 217)(16 123 162 236)(17 142 163 225)(18 131 164 214)(19 150 165 233)(20 139 166 222)(21 128 167 211)(22 147 168 230)(23 136 169 219)(24 125 170 238)(25 144 171 227)(26 133 172 216)(27 122 173 235)(28 141 174 224)(29 130 175 213)(30 149 176 232)(31 92 64 184)(32 111 65 203)(33 100 66 192)(34 119 67 181)(35 108 68 200)(36 97 69 189)(37 116 70 208)(38 105 71 197)(39 94 72 186)(40 113 73 205)(41 102 74 194)(42 91 75 183)(43 110 76 202)(44 99 77 191)(45 118 78 210)(46 107 79 199)(47 96 80 188)(48 115 81 207)(49 104 82 196)(50 93 83 185)(51 112 84 204)(52 101 85 193)(53 120 86 182)(54 109 87 201)(55 98 88 190)(56 117 89 209)(57 106 90 198)(58 95 61 187)(59 114 62 206)(60 103 63 195)
(1 190)(2 209)(3 198)(4 187)(5 206)(6 195)(7 184)(8 203)(9 192)(10 181)(11 200)(12 189)(13 208)(14 197)(15 186)(16 205)(17 194)(18 183)(19 202)(20 191)(21 210)(22 199)(23 188)(24 207)(25 196)(26 185)(27 204)(28 193)(29 182)(30 201)(31 215)(32 234)(33 223)(34 212)(35 231)(36 220)(37 239)(38 228)(39 217)(40 236)(41 225)(42 214)(43 233)(44 222)(45 211)(46 230)(47 219)(48 238)(49 227)(50 216)(51 235)(52 224)(53 213)(54 232)(55 221)(56 240)(57 229)(58 218)(59 237)(60 226)(61 135)(62 124)(63 143)(64 132)(65 121)(66 140)(67 129)(68 148)(69 137)(70 126)(71 145)(72 134)(73 123)(74 142)(75 131)(76 150)(77 139)(78 128)(79 147)(80 136)(81 125)(82 144)(83 133)(84 122)(85 141)(86 130)(87 149)(88 138)(89 127)(90 146)(91 164)(92 153)(93 172)(94 161)(95 180)(96 169)(97 158)(98 177)(99 166)(100 155)(101 174)(102 163)(103 152)(104 171)(105 160)(106 179)(107 168)(108 157)(109 176)(110 165)(111 154)(112 173)(113 162)(114 151)(115 170)(116 159)(117 178)(118 167)(119 156)(120 175)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C6D6E6F6G10A···10F10G10H10I10J12A12B15A15B20A···20H30A···30N
order12222222344444455666666610···10101010101212151520···2030···30
size1111420303026610101260222224420202···2444420204412···124···4

60 irreducible representations

dim1111111122222222222244444444
type++++++++++++++++++-++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6C4○D4D10D10C3⋊D4C4○D20S3×D4D42S3S3×D5D4×D5C2×S3×D5Dic5.D6D5×C3⋊D4D10⋊D6
kernelD306D4D10⋊Dic3C30.Q8C5×C6.D4C2×D30.C2C2×C3⋊D20C6×C5⋊D4C2×C157D4C2×C5⋊D4C3×Dic5D30C6.D4C2×Dic5C22×D5C22×C10C30C2×Dic3C22×C6Dic5C6C10C10C23C6C22C2C2C2
# reps1111111112221112424811242444

Matrix representation of D306D4 in GL4(𝔽61) generated by

14500
601700
00215
001260
,
01700
18000
0010
001260
,
4400
115700
00600
00060
,
59500
36200
003435
002827
G:=sub<GL(4,GF(61))| [1,60,0,0,45,17,0,0,0,0,2,12,0,0,15,60],[0,18,0,0,17,0,0,0,0,0,1,12,0,0,0,60],[4,11,0,0,4,57,0,0,0,0,60,0,0,0,0,60],[59,36,0,0,5,2,0,0,0,0,34,28,0,0,35,27] >;

D306D4 in GAP, Magma, Sage, TeX

D_{30}\rtimes_6D_4
% in TeX

G:=Group("D30:6D4");
// GroupNames label

G:=SmallGroup(480,609);
// by ID

G=gap.SmallGroup(480,609);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,64,254,219,1356,18822]);
// 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=d*a*d=a^19,c*b*c^-1=a^18*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽