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## G = D30.3D4order 480 = 25·3·5

### 3rd non-split extension by D30 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D30.3D4
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — D20⋊S3 — D30.3D4
 Lower central C15 — C30 — C60 — D30.3D4
 Upper central C1 — C2 — C4 — C8

Generators and relations for D30.3D4
G = < a,b,c,d | a20=b2=1, c6=a15, d2=a10, bab=a-1, ac=ca, dad-1=a11, cbc-1=dbd-1=a15b, dcd-1=a5c5 >

Subgroups: 684 in 120 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×2], C5, S3, C6, C6, C8, C8, C2×C4 [×3], D4 [×2], Q8 [×4], D5 [×2], C10, Dic3 [×3], C12, C12, D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5 [×2], C20, C20 [×2], D10 [×2], C3⋊C8, C24, Dic6 [×2], Dic6, C4×S3 [×2], C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C3×D5, D15, C30, C8.C22, C52C8, C40, Dic10, Dic10, C4×D5 [×3], D20, D20, C5×Q8 [×2], C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D42S3, S3×Q8, C5×Dic3 [×2], C3×Dic5, Dic15, C60, C6×D5, D30, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q82D5, D4.D6, C153C8, C120, D5×Dic3, D30.C2, C3⋊D20, C15⋊Q8, C3×Dic10, C3×D20, C5×Dic6 [×2], C4×D15, Q16⋊D5, C30.D4, C15⋊Q16, C3×C40⋊C2, C5×Dic12, C40⋊S3, D20⋊S3, D15⋊Q8, D30.3D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8.C22, C22×D5, S3×D4, S3×D5, D4×D5, D4.D6, C2×S3×D5, Q16⋊D5, C20⋊D6, D30.3D4

Smallest permutation representation of D30.3D4
On 240 points
Generators in S240
```(1 147 128 234 99 19 165 122 228 117 13 159 140 222 111 7 153 134 240 105)(2 148 129 235 100 20 166 123 229 118 14 160 141 223 112 8 154 135 217 106)(3 149 130 236 101 21 167 124 230 119 15 161 142 224 113 9 155 136 218 107)(4 150 131 237 102 22 168 125 231 120 16 162 143 225 114 10 156 137 219 108)(5 151 132 238 103 23 145 126 232 97 17 163 144 226 115 11 157 138 220 109)(6 152 133 239 104 24 146 127 233 98 18 164 121 227 116 12 158 139 221 110)(25 58 178 94 201 43 52 172 88 195 37 70 190 82 213 31 64 184 76 207)(26 59 179 95 202 44 53 173 89 196 38 71 191 83 214 32 65 185 77 208)(27 60 180 96 203 45 54 174 90 197 39 72 192 84 215 33 66 186 78 209)(28 61 181 73 204 46 55 175 91 198 40 49 169 85 216 34 67 187 79 210)(29 62 182 74 205 47 56 176 92 199 41 50 170 86 193 35 68 188 80 211)(30 63 183 75 206 48 57 177 93 200 42 51 171 87 194 36 69 189 81 212)
(1 202)(2 197)(3 216)(4 211)(5 206)(6 201)(7 196)(8 215)(9 210)(10 205)(11 200)(12 195)(13 214)(14 209)(15 204)(16 199)(17 194)(18 213)(19 208)(20 203)(21 198)(22 193)(23 212)(24 207)(25 104)(26 99)(27 118)(28 113)(29 108)(30 103)(31 98)(32 117)(33 112)(34 107)(35 102)(36 97)(37 116)(38 111)(39 106)(40 101)(41 120)(42 115)(43 110)(44 105)(45 100)(46 119)(47 114)(48 109)(49 236)(50 231)(51 226)(52 221)(53 240)(54 235)(55 230)(56 225)(57 220)(58 239)(59 234)(60 229)(61 224)(62 219)(63 238)(64 233)(65 228)(66 223)(67 218)(68 237)(69 232)(70 227)(71 222)(72 217)(73 161)(74 156)(75 151)(76 146)(77 165)(78 160)(79 155)(80 150)(81 145)(82 164)(83 159)(84 154)(85 149)(86 168)(87 163)(88 158)(89 153)(90 148)(91 167)(92 162)(93 157)(94 152)(95 147)(96 166)(121 190)(122 185)(123 180)(124 175)(125 170)(126 189)(127 184)(128 179)(129 174)(130 169)(131 188)(132 183)(133 178)(134 173)(135 192)(136 187)(137 182)(138 177)(139 172)(140 191)(141 186)(142 181)(143 176)(144 171)
(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 31 13 43)(2 30 14 42)(3 29 15 41)(4 28 16 40)(5 27 17 39)(6 26 18 38)(7 25 19 37)(8 48 20 36)(9 47 21 35)(10 46 22 34)(11 45 23 33)(12 44 24 32)(49 162 61 150)(50 161 62 149)(51 160 63 148)(52 159 64 147)(53 158 65 146)(54 157 66 145)(55 156 67 168)(56 155 68 167)(57 154 69 166)(58 153 70 165)(59 152 71 164)(60 151 72 163)(73 237 85 225)(74 236 86 224)(75 235 87 223)(76 234 88 222)(77 233 89 221)(78 232 90 220)(79 231 91 219)(80 230 92 218)(81 229 93 217)(82 228 94 240)(83 227 95 239)(84 226 96 238)(97 209 109 197)(98 208 110 196)(99 207 111 195)(100 206 112 194)(101 205 113 193)(102 204 114 216)(103 203 115 215)(104 202 116 214)(105 201 117 213)(106 200 118 212)(107 199 119 211)(108 198 120 210)(121 191 133 179)(122 190 134 178)(123 189 135 177)(124 188 136 176)(125 187 137 175)(126 186 138 174)(127 185 139 173)(128 184 140 172)(129 183 141 171)(130 182 142 170)(131 181 143 169)(132 180 144 192)```

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 8A 8B 10A 10B 12A 12B 15A 15B 20A 20B 20C 20D 20E 20F 24A 24B 30A 30B 40A 40B 40C 40D 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 3 4 4 4 4 4 5 5 6 6 8 8 10 10 12 12 15 15 20 20 20 20 20 20 24 24 30 30 40 40 40 40 60 60 60 60 120 ··· 120 size 1 1 20 30 2 2 12 12 20 30 2 2 2 40 4 60 2 2 4 40 4 4 4 4 24 24 24 24 4 4 4 4 4 4 4 4 4 4 4 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 C8.C22 S3×D4 S3×D5 D4×D5 D4.D6 C2×S3×D5 Q16⋊D5 C20⋊D6 D30.3D4 kernel D30.3D4 C30.D4 C15⋊Q16 C3×C40⋊C2 C5×Dic12 C40⋊S3 D20⋊S3 D15⋊Q8 C40⋊C2 Dic15 D30 Dic12 C40 Dic10 D20 C24 Dic6 C15 C10 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 4 1 1 2 2 2 2 4 4 8

Matrix representation of D30.3D4 in GL6(𝔽241)

 0 1 0 0 0 0 240 51 0 0 0 0 0 0 144 47 133 0 0 0 194 97 0 133 0 0 118 0 97 194 0 0 0 118 47 144
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 240 130 104 208 0 0 111 110 33 137 0 0 96 27 1 111 0 0 214 69 130 131
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 132 141 237 4 0 0 100 232 237 233 0 0 49 192 100 109 0 0 49 98 132 232
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 156 179 142 142 0 0 23 85 0 99 0 0 173 173 62 85 0 0 0 68 23 179

`G:=sub<GL(6,GF(241))| [0,240,0,0,0,0,1,51,0,0,0,0,0,0,144,194,118,0,0,0,47,97,0,118,0,0,133,0,97,47,0,0,0,133,194,144],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,240,111,96,214,0,0,130,110,27,69,0,0,104,33,1,130,0,0,208,137,111,131],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,132,100,49,49,0,0,141,232,192,98,0,0,237,237,100,132,0,0,4,233,109,232],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,156,23,173,0,0,0,179,85,173,68,0,0,142,0,62,23,0,0,142,99,85,179] >;`

D30.3D4 in GAP, Magma, Sage, TeX

`D_{30}._3D_4`
`% in TeX`

`G:=Group("D30.3D4");`
`// GroupNames label`

`G:=SmallGroup(480,354);`
`// by ID`

`G=gap.SmallGroup(480,354);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,303,58,675,346,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=1,c^6=a^15,d^2=a^10,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,c*b*c^-1=d*b*d^-1=a^15*b,d*c*d^-1=a^5*c^5>;`
`// generators/relations`

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