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G = D4.8D30order 480 = 25·3·5

3rd non-split extension by D4 of D30 acting via D30/C30=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D4.8D30
 Chief series C1 — C5 — C15 — C30 — C60 — D60 — D60⋊11C2 — D4.8D30
 Lower central C15 — C30 — C60 — D4.8D30
 Upper central C1 — C4 — C2×C4 — C4○D4

Generators and relations for D4.8D30
G = < a,b,c,d | a4=b2=1, c30=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c29 >

Subgroups: 644 in 124 conjugacy classes, 47 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, S3, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8, D5, C10, C10 [×2], Dic3, C12 [×2], C12, D6, C2×C6, C2×C6, C15, C2×C8, D8, SD16 [×2], Q16, C4○D4, C4○D4, Dic5, C20 [×2], C20, D10, C2×C10, C2×C10, C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, D15, C30, C30 [×2], C4○D8, C52C8 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C4○D12, C3×C4○D4, Dic15, C60 [×2], C60, D30, C2×C30, C2×C30, C2×C52C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, Q8.13D6, C153C8 [×2], Dic30, C4×D15, D60, C157D4, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, D4.8D10, C2×C153C8, D4⋊D15, D4.D15, Q82D15, C157Q16, D6011C2, C15×C4○D4, D4.8D30
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, D15, C4○D8, C5⋊D4 [×2], C22×D5, C2×C3⋊D4, D30 [×3], C2×C5⋊D4, Q8.13D6, C157D4 [×2], C22×D15, D4.8D10, C2×C157D4, D4.8D30

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

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 6D 8A 8B 8C 8D 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 3 4 4 4 4 4 5 5 6 6 6 6 8 8 8 8 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 4 60 2 1 1 2 4 60 2 2 2 4 4 4 30 30 30 30 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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 D10 C3⋊D4 C3⋊D4 D15 C4○D8 C5⋊D4 C5⋊D4 D30 D30 D30 C15⋊7D4 C15⋊7D4 Q8.13D6 D4.8D10 D4.8D30 kernel D4.8D30 C2×C15⋊3C8 D4⋊D15 D4.D15 Q8⋊2D15 C15⋊7Q16 D60⋊11C2 C15×C4○D4 C5×C4○D4 C60 C2×C30 C3×C4○D4 C2×C20 C5×D4 C5×Q8 C2×C12 C3×D4 C3×Q8 C20 C2×C10 C4○D4 C15 C12 C2×C6 C2×C4 D4 Q8 C4 C22 C5 C3 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 2 4 4 4 4 4 4 4 8 8 2 4 8

Matrix representation of D4.8D30 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 177 0 0 0 64 64
,
 240 0 0 0 0 240 0 0 0 0 177 113 0 0 64 64
,
 225 64 0 0 177 30 0 0 0 0 64 0 0 0 0 64
,
 78 72 0 0 113 163 0 0 0 0 233 225 0 0 230 8
`G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,177,64,0,0,0,64],[240,0,0,0,0,240,0,0,0,0,177,64,0,0,113,64],[225,177,0,0,64,30,0,0,0,0,64,0,0,0,0,64],[78,113,0,0,72,163,0,0,0,0,233,230,0,0,225,8] >;`

D4.8D30 in GAP, Magma, Sage, TeX

`D_4._8D_{30}`
`% in TeX`

`G:=Group("D4.8D30");`
`// GroupNames label`

`G:=SmallGroup(480,915);`
`// by ID`

`G=gap.SmallGroup(480,915);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,675,185,80,2693,18822]);`
`// Polycyclic`

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

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