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## G = D30⋊9Q8order 480 = 25·3·5

### 5th semidirect product of D30 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — D30⋊9Q8
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C30.Q8 — D30⋊9Q8
 Lower central C15 — C2×C30 — D30⋊9Q8
 Upper central C1 — C22 — C2×C4

Generators and relations for D309Q8
G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a25b, dcd-1=c-1 >

Subgroups: 780 in 148 conjugacy classes, 54 normal (32 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×7], Q8 [×2], C23, D5 [×2], C10 [×3], Dic3 [×3], C12 [×2], C12 [×2], D6 [×4], C2×C6, C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×4], C2×C10, Dic6 [×2], C4×S3 [×2], C2×Dic3 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, D15 [×2], C30 [×3], C22⋊Q8, C4×D5 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C4⋊Dic3 [×2], D6⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15, C60 [×2], D30 [×2], D30 [×2], C2×C30, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C2×C4×D5, Q8×C10, C4.D12, C6×Dic5 [×2], C5×Dic6 [×2], C10×Dic3 [×2], C4×D15 [×2], C2×Dic15, C2×C60, C22×D15, D103Q8, D304C4 [×2], C30.Q8 [×2], C3×C4⋊Dic5, C10×Dic6, C2×C4×D15, D309Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], D12 [×2], C22×S3, C22⋊Q8, C5⋊D4 [×2], C22×D5, C2×D12, D42S3, S3×Q8, S3×D5, Q8×D5, Q82D5, C2×C5⋊D4, C4.D12, C5⋊D12 [×2], C2×S3×D5, D103Q8, D20⋊S3, D15⋊Q8, C2×C5⋊D12, D309Q8

Smallest permutation representation of D309Q8
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)
(1 225)(2 224)(3 223)(4 222)(5 221)(6 220)(7 219)(8 218)(9 217)(10 216)(11 215)(12 214)(13 213)(14 212)(15 211)(16 240)(17 239)(18 238)(19 237)(20 236)(21 235)(22 234)(23 233)(24 232)(25 231)(26 230)(27 229)(28 228)(29 227)(30 226)(31 169)(32 168)(33 167)(34 166)(35 165)(36 164)(37 163)(38 162)(39 161)(40 160)(41 159)(42 158)(43 157)(44 156)(45 155)(46 154)(47 153)(48 152)(49 151)(50 180)(51 179)(52 178)(53 177)(54 176)(55 175)(56 174)(57 173)(58 172)(59 171)(60 170)(61 111)(62 110)(63 109)(64 108)(65 107)(66 106)(67 105)(68 104)(69 103)(70 102)(71 101)(72 100)(73 99)(74 98)(75 97)(76 96)(77 95)(78 94)(79 93)(80 92)(81 91)(82 120)(83 119)(84 118)(85 117)(86 116)(87 115)(88 114)(89 113)(90 112)(121 205)(122 204)(123 203)(124 202)(125 201)(126 200)(127 199)(128 198)(129 197)(130 196)(131 195)(132 194)(133 193)(134 192)(135 191)(136 190)(137 189)(138 188)(139 187)(140 186)(141 185)(142 184)(143 183)(144 182)(145 181)(146 210)(147 209)(148 208)(149 207)(150 206)
(1 144 226 183)(2 145 227 184)(3 146 228 185)(4 147 229 186)(5 148 230 187)(6 149 231 188)(7 150 232 189)(8 121 233 190)(9 122 234 191)(10 123 235 192)(11 124 236 193)(12 125 237 194)(13 126 238 195)(14 127 239 196)(15 128 240 197)(16 129 211 198)(17 130 212 199)(18 131 213 200)(19 132 214 201)(20 133 215 202)(21 134 216 203)(22 135 217 204)(23 136 218 205)(24 137 219 206)(25 138 220 207)(26 139 221 208)(27 140 222 209)(28 141 223 210)(29 142 224 181)(30 143 225 182)(31 103 163 63)(32 104 164 64)(33 105 165 65)(34 106 166 66)(35 107 167 67)(36 108 168 68)(37 109 169 69)(38 110 170 70)(39 111 171 71)(40 112 172 72)(41 113 173 73)(42 114 174 74)(43 115 175 75)(44 116 176 76)(45 117 177 77)(46 118 178 78)(47 119 179 79)(48 120 180 80)(49 91 151 81)(50 92 152 82)(51 93 153 83)(52 94 154 84)(53 95 155 85)(54 96 156 86)(55 97 157 87)(56 98 158 88)(57 99 159 89)(58 100 160 90)(59 101 161 61)(60 102 162 62)
(1 159 226 57)(2 170 227 38)(3 151 228 49)(4 162 229 60)(5 173 230 41)(6 154 231 52)(7 165 232 33)(8 176 233 44)(9 157 234 55)(10 168 235 36)(11 179 236 47)(12 160 237 58)(13 171 238 39)(14 152 239 50)(15 163 240 31)(16 174 211 42)(17 155 212 53)(18 166 213 34)(19 177 214 45)(20 158 215 56)(21 169 216 37)(22 180 217 48)(23 161 218 59)(24 172 219 40)(25 153 220 51)(26 164 221 32)(27 175 222 43)(28 156 223 54)(29 167 224 35)(30 178 225 46)(61 136 101 205)(62 147 102 186)(63 128 103 197)(64 139 104 208)(65 150 105 189)(66 131 106 200)(67 142 107 181)(68 123 108 192)(69 134 109 203)(70 145 110 184)(71 126 111 195)(72 137 112 206)(73 148 113 187)(74 129 114 198)(75 140 115 209)(76 121 116 190)(77 132 117 201)(78 143 118 182)(79 124 119 193)(80 135 120 204)(81 146 91 185)(82 127 92 196)(83 138 93 207)(84 149 94 188)(85 130 95 199)(86 141 96 210)(87 122 97 191)(88 133 98 202)(89 144 99 183)(90 125 100 194)```

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 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 3 4 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 2 2 2 12 12 20 20 30 30 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 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + - + + + + + + - - + - + + + image C1 C2 C2 C2 C2 C2 S3 D4 Q8 D5 D6 D6 C4○D4 D10 D10 D12 C5⋊D4 D4⋊2S3 S3×Q8 S3×D5 Q8×D5 Q8⋊2D5 C5⋊D12 C2×S3×D5 D20⋊S3 D15⋊Q8 kernel D30⋊9Q8 D30⋊4C4 C30.Q8 C3×C4⋊Dic5 C10×Dic6 C2×C4×D15 C4⋊Dic5 C60 D30 C2×Dic6 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 C20 C12 C10 C10 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 2 1 1 1 1 2 2 2 2 1 2 4 2 4 8 1 1 2 2 2 4 2 4 4

Matrix representation of D309Q8 in GL6(𝔽61)

 0 60 0 0 0 0 1 43 0 0 0 0 0 0 60 15 0 0 0 0 12 2 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 18 60 0 0 0 0 18 43 0 0 0 0 0 0 60 15 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 52 60
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 50 0 0 0 0 0 38 11
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 53 19 0 0 0 0 32 8 0 0 0 0 0 0 11 16 0 0 0 0 0 50

`G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,43,0,0,0,0,0,0,60,12,0,0,0,0,15,2,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[18,18,0,0,0,0,60,43,0,0,0,0,0,0,60,0,0,0,0,0,15,1,0,0,0,0,0,0,1,52,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,38,0,0,0,0,0,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,53,32,0,0,0,0,19,8,0,0,0,0,0,0,11,0,0,0,0,0,16,50] >;`

D309Q8 in GAP, Magma, Sage, TeX

`D_{30}\rtimes_9Q_8`
`% in TeX`

`G:=Group("D30:9Q8");`
`// GroupNames label`

`G:=SmallGroup(480,459);`
`// by ID`

`G=gap.SmallGroup(480,459);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,120,422,219,100,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^25*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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