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## G = D30⋊5Q8order 480 = 25·3·5

### 1st semidirect product of D30 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — D30⋊5Q8
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×D15 — C2×C4×D15 — D30⋊5Q8
 Lower central C15 — C2×C30 — D30⋊5Q8
 Upper central C1 — C22 — C4⋊C4

Generators and relations for D305Q8
G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=cac-1=dad-1=a-1, cbc-1=a13b, dbd-1=a28b, dcd-1=c-1 >

Subgroups: 900 in 148 conjugacy classes, 51 normal (47 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4 [×3], C2×C4 [×5], Q8 [×2], C23, D5 [×2], C10 [×3], Dic3 [×4], C12 [×3], D6 [×4], C2×C6, C15, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, Dic5 [×4], C20 [×3], D10 [×4], C2×C10, Dic6 [×2], C4×S3 [×2], C2×Dic3 [×3], C2×C12 [×3], C22×S3, D15 [×2], C30 [×3], C22⋊Q8, Dic10 [×2], C4×D5 [×2], C2×Dic5 [×3], C2×C20 [×3], C22×D5, Dic3⋊C4 [×2], D6⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, Dic15 [×2], Dic15 [×2], C60 [×3], D30 [×2], D30 [×2], C2×C30, C10.D4 [×2], D10⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, D6⋊Q8, Dic30 [×2], C4×D15 [×2], C2×Dic15 [×3], C2×C60 [×3], C22×D15, D10⋊Q8, C30.4Q8 [×2], D303C4 [×2], C15×C4⋊C4, C2×Dic30, C2×C4×D15, D305Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], C22×S3, D15, C22⋊Q8, C22×D5, C4○D12, S3×D4, S3×Q8, D30 [×3], C4○D20, D4×D5, Q8×D5, D6⋊Q8, C22×D15, D10⋊Q8, D6011C2, D4×D15, Q8×D15, D305Q8

Smallest permutation representation of D305Q8
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)
(1 100)(2 99)(3 98)(4 97)(5 96)(6 95)(7 94)(8 93)(9 92)(10 91)(11 120)(12 119)(13 118)(14 117)(15 116)(16 115)(17 114)(18 113)(19 112)(20 111)(21 110)(22 109)(23 108)(24 107)(25 106)(26 105)(27 104)(28 103)(29 102)(30 101)(31 188)(32 187)(33 186)(34 185)(35 184)(36 183)(37 182)(38 181)(39 210)(40 209)(41 208)(42 207)(43 206)(44 205)(45 204)(46 203)(47 202)(48 201)(49 200)(50 199)(51 198)(52 197)(53 196)(54 195)(55 194)(56 193)(57 192)(58 191)(59 190)(60 189)(61 137)(62 136)(63 135)(64 134)(65 133)(66 132)(67 131)(68 130)(69 129)(70 128)(71 127)(72 126)(73 125)(74 124)(75 123)(76 122)(77 121)(78 150)(79 149)(80 148)(81 147)(82 146)(83 145)(84 144)(85 143)(86 142)(87 141)(88 140)(89 139)(90 138)(151 236)(152 235)(153 234)(154 233)(155 232)(156 231)(157 230)(158 229)(159 228)(160 227)(161 226)(162 225)(163 224)(164 223)(165 222)(166 221)(167 220)(168 219)(169 218)(170 217)(171 216)(172 215)(173 214)(174 213)(175 212)(176 211)(177 240)(178 239)(179 238)(180 237)
(1 207 116 43)(2 206 117 42)(3 205 118 41)(4 204 119 40)(5 203 120 39)(6 202 91 38)(7 201 92 37)(8 200 93 36)(9 199 94 35)(10 198 95 34)(11 197 96 33)(12 196 97 32)(13 195 98 31)(14 194 99 60)(15 193 100 59)(16 192 101 58)(17 191 102 57)(18 190 103 56)(19 189 104 55)(20 188 105 54)(21 187 106 53)(22 186 107 52)(23 185 108 51)(24 184 109 50)(25 183 110 49)(26 182 111 48)(27 181 112 47)(28 210 113 46)(29 209 114 45)(30 208 115 44)(61 213 128 170)(62 212 129 169)(63 211 130 168)(64 240 131 167)(65 239 132 166)(66 238 133 165)(67 237 134 164)(68 236 135 163)(69 235 136 162)(70 234 137 161)(71 233 138 160)(72 232 139 159)(73 231 140 158)(74 230 141 157)(75 229 142 156)(76 228 143 155)(77 227 144 154)(78 226 145 153)(79 225 146 152)(80 224 147 151)(81 223 148 180)(82 222 149 179)(83 221 150 178)(84 220 121 177)(85 219 122 176)(86 218 123 175)(87 217 124 174)(88 216 125 173)(89 215 126 172)(90 214 127 171)
(1 223 116 180)(2 222 117 179)(3 221 118 178)(4 220 119 177)(5 219 120 176)(6 218 91 175)(7 217 92 174)(8 216 93 173)(9 215 94 172)(10 214 95 171)(11 213 96 170)(12 212 97 169)(13 211 98 168)(14 240 99 167)(15 239 100 166)(16 238 101 165)(17 237 102 164)(18 236 103 163)(19 235 104 162)(20 234 105 161)(21 233 106 160)(22 232 107 159)(23 231 108 158)(24 230 109 157)(25 229 110 156)(26 228 111 155)(27 227 112 154)(28 226 113 153)(29 225 114 152)(30 224 115 151)(31 130 195 63)(32 129 196 62)(33 128 197 61)(34 127 198 90)(35 126 199 89)(36 125 200 88)(37 124 201 87)(38 123 202 86)(39 122 203 85)(40 121 204 84)(41 150 205 83)(42 149 206 82)(43 148 207 81)(44 147 208 80)(45 146 209 79)(46 145 210 78)(47 144 181 77)(48 143 182 76)(49 142 183 75)(50 141 184 74)(51 140 185 73)(52 139 186 72)(53 138 187 71)(54 137 188 70)(55 136 189 69)(56 135 190 68)(57 134 191 67)(58 133 192 66)(59 132 193 65)(60 131 194 64)```

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

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

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

84 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 ··· 12F 15A 15B 15C 15D 20A ··· 20L 30A ··· 30L 60A ··· 60X 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 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 30 30 2 2 2 4 4 30 30 60 60 2 2 2 2 2 2 ··· 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

84 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 4 4 type + + + + + + + + - + + + + + + - + - + - image C1 C2 C2 C2 C2 C2 S3 D4 Q8 D5 D6 C4○D4 D10 D15 C4○D12 D30 C4○D20 D60⋊11C2 S3×D4 S3×Q8 D4×D5 Q8×D5 D4×D15 Q8×D15 kernel D30⋊5Q8 C30.4Q8 D30⋊3C4 C15×C4⋊C4 C2×Dic30 C2×C4×D15 C5×C4⋊C4 Dic15 D30 C3×C4⋊C4 C2×C20 C30 C2×C12 C4⋊C4 C10 C2×C4 C6 C2 C10 C10 C6 C6 C2 C2 # reps 1 2 2 1 1 1 1 2 2 2 3 2 6 4 4 12 8 16 1 1 2 2 4 4

Matrix representation of D305Q8 in GL6(𝔽61)

 4 15 0 0 0 0 52 58 0 0 0 0 0 0 0 60 0 0 0 0 1 17 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 4 15 0 0 0 0 60 57 0 0 0 0 0 0 14 24 0 0 0 0 30 47 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 2 26 0 0 0 0 35 59 0 0 0 0 0 0 31 14 0 0 0 0 36 30 0 0 0 0 0 0 1 55 0 0 0 0 41 60
,
 50 0 0 0 0 0 58 11 0 0 0 0 0 0 30 47 0 0 0 0 25 31 0 0 0 0 0 0 52 58 0 0 0 0 7 9

`G:=sub<GL(6,GF(61))| [4,52,0,0,0,0,15,58,0,0,0,0,0,0,0,1,0,0,0,0,60,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,60,0,0,0,0,15,57,0,0,0,0,0,0,14,30,0,0,0,0,24,47,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[2,35,0,0,0,0,26,59,0,0,0,0,0,0,31,36,0,0,0,0,14,30,0,0,0,0,0,0,1,41,0,0,0,0,55,60],[50,58,0,0,0,0,0,11,0,0,0,0,0,0,30,25,0,0,0,0,47,31,0,0,0,0,0,0,52,7,0,0,0,0,58,9] >;`

D305Q8 in GAP, Magma, Sage, TeX

`D_{30}\rtimes_5Q_8`
`% in TeX`

`G:=Group("D30:5Q8");`
`// GroupNames label`

`G:=SmallGroup(480,861);`
`// by ID`

`G=gap.SmallGroup(480,861);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,590,219,100,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^13*b,d*b*d^-1=a^28*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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