Copied to
clipboard

## G = D30.6D4order 480 = 25·3·5

### 6th non-split extension by D30 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — D30.6D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C2×D30.C2 — D30.6D4
 Lower central C15 — C2×C30 — D30.6D4
 Upper central C1 — C22 — C2×C4

Generators and relations for D30.6D4
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=c-1 >

Subgroups: 1036 in 156 conjugacy classes, 46 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, C12, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C22.D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C5×Dic3, C3×Dic5, C60, D30, D30, C2×C30, C10.D4, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, D6.D4, D30.C2, C6×Dic5, C10×Dic3, D60, C2×C60, C22×D15, D10.13D4, D304C4, C3×C10.D4, C5×Dic3⋊C4, C2×D30.C2, C2×D60, D30.6D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C22.D4, C22×D5, C4○D12, S3×D4, Q83S3, S3×D5, C4○D20, D4×D5, Q82D5, D6.D4, C2×S3×D5, D10.13D4, D60⋊C2, C12.28D10, D10⋊D6, D30.6D4

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

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 2 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 S3 D4 D5 D6 D6 C4○D4 D10 D10 C4○D12 C4○D20 S3×D4 Q8⋊3S3 S3×D5 D4×D5 Q8⋊2D5 C2×S3×D5 D60⋊C2 C12.28D10 D10⋊D6 kernel D30.6D4 D30⋊4C4 C3×C10.D4 C5×Dic3⋊C4 C2×D30.C2 C2×D60 C10.D4 D30 Dic3⋊C4 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 C10 C6 C10 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 3 1 1 1 1 1 2 2 2 1 4 4 2 4 8 1 1 2 2 2 2 4 4 4

Matrix representation of D30.6D4 in GL4(𝔽61) generated by

 1 45 0 0 60 17 0 0 0 0 0 60 0 0 1 1
,
 0 17 0 0 18 0 0 0 0 0 0 60 0 0 60 0
,
 22 6 0 0 31 39 0 0 0 0 52 43 0 0 18 9
,
 57 57 0 0 50 4 0 0 0 0 11 0 0 0 0 11
`G:=sub<GL(4,GF(61))| [1,60,0,0,45,17,0,0,0,0,0,1,0,0,60,1],[0,18,0,0,17,0,0,0,0,0,0,60,0,0,60,0],[22,31,0,0,6,39,0,0,0,0,52,18,0,0,43,9],[57,50,0,0,57,4,0,0,0,0,11,0,0,0,0,11] >;`

D30.6D4 in GAP, Magma, Sage, TeX

`D_{30}._6D_4`
`% in TeX`

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

`G:=SmallGroup(480,509);`
`// by ID`

`G=gap.SmallGroup(480,509);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,590,219,58,1356,18822]);`
`// 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=c^-1>;`
`// generators/relations`

׿
×
𝔽