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## G = D30⋊2D4order 480 = 25·3·5

### 2nd semidirect product of D30 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — D30⋊2D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×C3⋊D20 — D30⋊2D4
 Lower central C15 — C2×C30 — D30⋊2D4
 Upper central C1 — C22 — C2×C4

Generators and relations for D302D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=a11, dad=a19, cbc-1=a10b, dbd=a3b, dcd=c-1 >

Subgroups: 1324 in 188 conjugacy classes, 50 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 [×4], C10 [×3], Dic3 [×2], Dic3, C12 [×2], D6 [×7], C2×C6, C2×C6 [×3], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5, C20 [×4], D10 [×10], C2×C10, C4×S3 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C2×C12, C22×S3 [×2], C22×C6, C3×D5, D15 [×3], C30 [×3], C4⋊D4, C4×D5 [×2], D20 [×6], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5 [×2], Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4 [×2], C5×Dic3 [×2], C5×Dic3, C3×Dic5, C60, C6×D5 [×3], D30 [×2], D30 [×5], C2×C30, D10⋊C4, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20 [×3], Dic3⋊D4, D30.C2 [×2], C3⋊D20 [×4], C6×Dic5, C10×Dic3 [×2], D60 [×2], C2×C60, D5×C2×C6, C22×D15 [×2], C4⋊D20, D304C4, C3×D10⋊C4, C5×Dic3⋊C4, C2×D30.C2, C2×C3⋊D20 [×2], C2×D60, D302D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], C22×S3, C4⋊D4, D20 [×2], C22×D5, C4○D12, S3×D4 [×2], S3×D5, C2×D20, D4×D5, Q82D5, Dic3⋊D4, C2×S3×D5, C4⋊D20, C12.28D10, S3×D20, D10⋊D6, D302D4

Smallest permutation representation of D302D4
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 67)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 90)(9 89)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 82)(17 81)(18 80)(19 79)(20 78)(21 77)(22 76)(23 75)(24 74)(25 73)(26 72)(27 71)(28 70)(29 69)(30 68)(31 115)(32 114)(33 113)(34 112)(35 111)(36 110)(37 109)(38 108)(39 107)(40 106)(41 105)(42 104)(43 103)(44 102)(45 101)(46 100)(47 99)(48 98)(49 97)(50 96)(51 95)(52 94)(53 93)(54 92)(55 91)(56 120)(57 119)(58 118)(59 117)(60 116)(121 191)(122 190)(123 189)(124 188)(125 187)(126 186)(127 185)(128 184)(129 183)(130 182)(131 181)(132 210)(133 209)(134 208)(135 207)(136 206)(137 205)(138 204)(139 203)(140 202)(141 201)(142 200)(143 199)(144 198)(145 197)(146 196)(147 195)(148 194)(149 193)(150 192)(151 223)(152 222)(153 221)(154 220)(155 219)(156 218)(157 217)(158 216)(159 215)(160 214)(161 213)(162 212)(163 211)(164 240)(165 239)(166 238)(167 237)(168 236)(169 235)(170 234)(171 233)(172 232)(173 231)(174 230)(175 229)(176 228)(177 227)(178 226)(179 225)(180 224)
(1 194 68 149)(2 205 69 130)(3 186 70 141)(4 197 71 122)(5 208 72 133)(6 189 73 144)(7 200 74 125)(8 181 75 136)(9 192 76 147)(10 203 77 128)(11 184 78 139)(12 195 79 150)(13 206 80 131)(14 187 81 142)(15 198 82 123)(16 209 83 134)(17 190 84 145)(18 201 85 126)(19 182 86 137)(20 193 87 148)(21 204 88 129)(22 185 89 140)(23 196 90 121)(24 207 61 132)(25 188 62 143)(26 199 63 124)(27 210 64 135)(28 191 65 146)(29 202 66 127)(30 183 67 138)(31 176 111 234)(32 157 112 215)(33 168 113 226)(34 179 114 237)(35 160 115 218)(36 171 116 229)(37 152 117 240)(38 163 118 221)(39 174 119 232)(40 155 120 213)(41 166 91 224)(42 177 92 235)(43 158 93 216)(44 169 94 227)(45 180 95 238)(46 161 96 219)(47 172 97 230)(48 153 98 211)(49 164 99 222)(50 175 100 233)(51 156 101 214)(52 167 102 225)(53 178 103 236)(54 159 104 217)(55 170 105 228)(56 151 106 239)(57 162 107 220)(58 173 108 231)(59 154 109 212)(60 165 110 223)
(1 56)(2 45)(3 34)(4 53)(5 42)(6 31)(7 50)(8 39)(9 58)(10 47)(11 36)(12 55)(13 44)(14 33)(15 52)(16 41)(17 60)(18 49)(19 38)(20 57)(21 46)(22 35)(23 54)(24 43)(25 32)(26 51)(27 40)(28 59)(29 48)(30 37)(61 93)(62 112)(63 101)(64 120)(65 109)(66 98)(67 117)(68 106)(69 95)(70 114)(71 103)(72 92)(73 111)(74 100)(75 119)(76 108)(77 97)(78 116)(79 105)(80 94)(81 113)(82 102)(83 91)(84 110)(85 99)(86 118)(87 107)(88 96)(89 115)(90 104)(121 159)(122 178)(123 167)(124 156)(125 175)(126 164)(127 153)(128 172)(129 161)(130 180)(131 169)(132 158)(133 177)(134 166)(135 155)(136 174)(137 163)(138 152)(139 171)(140 160)(141 179)(142 168)(143 157)(144 176)(145 165)(146 154)(147 173)(148 162)(149 151)(150 170)(181 232)(182 221)(183 240)(184 229)(185 218)(186 237)(187 226)(188 215)(189 234)(190 223)(191 212)(192 231)(193 220)(194 239)(195 228)(196 217)(197 236)(198 225)(199 214)(200 233)(201 222)(202 211)(203 230)(204 219)(205 238)(206 227)(207 216)(208 235)(209 224)(210 213)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 6D 6E 10A ··· 10F 12A 12B 12C 12D 15A 15B 20A 20B 20C 20D 20E ··· 20L 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 6 6 10 ··· 10 12 12 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 20 30 30 60 2 4 6 6 10 10 12 2 2 2 2 2 20 20 2 ··· 2 4 4 20 20 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 C4○D4 D10 D10 D20 C4○D12 S3×D4 S3×D5 D4×D5 Q8⋊2D5 C2×S3×D5 C12.28D10 S3×D20 D10⋊D6 kernel D30⋊2D4 D30⋊4C4 C3×D10⋊C4 C5×Dic3⋊C4 C2×D30.C2 C2×C3⋊D20 C2×D60 D10⋊C4 C5×Dic3 D30 Dic3⋊C4 C2×Dic5 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 Dic3 C10 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 2 1 1 2 2 2 1 1 1 2 4 2 8 4 2 2 2 2 2 4 4 4

Matrix representation of D302D4 in GL4(𝔽61) generated by

 0 60 0 0 1 1 0 0 0 0 0 1 0 0 60 44
,
 0 60 0 0 60 0 0 0 0 0 17 1 0 0 17 44
,
 11 0 0 0 50 50 0 0 0 0 1 0 0 0 0 1
,
 9 18 0 0 43 52 0 0 0 0 2 2 0 0 29 59
G:=sub<GL(4,GF(61))| [0,1,0,0,60,1,0,0,0,0,0,60,0,0,1,44],[0,60,0,0,60,0,0,0,0,0,17,17,0,0,1,44],[11,50,0,0,0,50,0,0,0,0,1,0,0,0,0,1],[9,43,0,0,18,52,0,0,0,0,2,29,0,0,2,59] >;

D302D4 in GAP, Magma, Sage, TeX

D_{30}\rtimes_2D_4
% in TeX

G:=Group("D30:2D4");
// GroupNames label

G:=SmallGroup(480,535);
// by ID

G=gap.SmallGroup(480,535);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,254,219,142,1356,18822]);
// 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^10*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽