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G = D4.10D30order 480 = 25·3·5

The non-split extension by D4 of D30 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D4.10D30
 Chief series C1 — C5 — C15 — C30 — D30 — C4×D15 — Q8×D15 — D4.10D30
 Lower central C15 — C30 — D4.10D30
 Upper central C1 — C2 — C4○D4

Generators and relations for D4.10D30
G = < a,b,c,d | a4=b2=1, c30=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c29 >

Subgroups: 1412 in 292 conjugacy classes, 119 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, D15, C30, C30, 2- 1+4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C2×Dic6, C4○D12, D42S3, S3×Q8, C3×C4○D4, Dic15, C60, C60, D30, C2×C30, C2×Dic10, C4○D20, D42D5, Q8×D5, C5×C4○D4, Q8○D12, Dic30, C4×D15, D60, C2×Dic15, C157D4, C2×C60, D4×C15, Q8×C15, D4.10D10, C2×Dic30, D6011C2, D42D15, Q8×D15, C15×C4○D4, D4.10D30
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, D15, 2- 1+4, C22×D5, S3×C23, D30, C23×D5, Q8○D12, C22×D15, D4.10D10, C23×D15, D4.10D30

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

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

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

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

87 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E ··· 4J 5A 5B 6A 6B 6C 6D 10A 10B 10C ··· 10H 12A 12B 12C 12D 12E 15A 15B 15C 15D 20A 20B 20C 20D 20E ··· 20J 30A 30B 30C 30D 30E ··· 30P 60A ··· 60H 60I ··· 60T order 1 2 2 2 2 2 2 3 4 4 4 4 4 ··· 4 5 5 6 6 6 6 10 10 10 ··· 10 12 12 12 12 12 15 15 15 15 20 20 20 20 20 ··· 20 30 30 30 30 30 ··· 30 60 ··· 60 60 ··· 60 size 1 1 2 2 2 30 30 2 2 2 2 2 30 ··· 30 2 2 2 4 4 4 2 2 4 ··· 4 2 2 4 4 4 2 2 2 2 2 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

87 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + - - - - image C1 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 D10 D10 D10 D15 D30 D30 D30 2- 1+4 Q8○D12 D4.10D10 D4.10D30 kernel D4.10D30 C2×Dic30 D60⋊11C2 D4⋊2D15 Q8×D15 C15×C4○D4 C5×C4○D4 C3×C4○D4 C2×C20 C5×D4 C5×Q8 C2×C12 C3×D4 C3×Q8 C4○D4 C2×C4 D4 Q8 C15 C5 C3 C1 # reps 1 3 3 6 2 1 1 2 3 3 1 6 6 2 4 12 12 4 1 2 4 8

Matrix representation of D4.10D30 in GL4(𝔽61) generated by

 60 0 14 14 0 60 47 0 0 35 1 0 26 26 0 1
,
 38 35 2 5 26 23 2 58 43 31 3 26 49 12 55 58
,
 25 53 1 10 8 11 11 52 17 31 58 8 29 47 31 28
,
 59 57 13 48 23 2 45 58 26 50 21 23 24 50 38 40
`G:=sub<GL(4,GF(61))| [60,0,0,26,0,60,35,26,14,47,1,0,14,0,0,1],[38,26,43,49,35,23,31,12,2,2,3,55,5,58,26,58],[25,8,17,29,53,11,31,47,1,11,58,31,10,52,8,28],[59,23,26,24,57,2,50,50,13,45,21,38,48,58,23,40] >;`

D4.10D30 in GAP, Magma, Sage, TeX

`D_4._{10}D_{30}`
`% in TeX`

`G:=Group("D4.10D30");`
`// GroupNames label`

`G:=SmallGroup(480,1177);`
`// by ID`

`G=gap.SmallGroup(480,1177);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,219,100,675,80,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^30=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^29>;`
`// generators/relations`

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