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G = Q8.11D30order 480 = 25·3·5

1st non-split extension by Q8 of D30 acting via D30/C30=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.19D4, Q8.11D30, C60.78C23, D60.39C22, Dic30.40C22, (C6×Q8)⋊2D5, (Q8×C30)⋊2C2, (Q8×C10)⋊6S3, (C2×Q8)⋊4D15, (C2×C4).19D30, (C5×Q8).49D6, C157Q1613C2, (C2×C30).149D4, (C2×C20).152D6, C30.386(C2×D4), (C3×Q8).32D10, C60.7C419C2, Q82D1513C2, (C2×C12).151D10, C4.17(C157D4), C35(C20.C23), C55(Q8.11D6), C20.44(C3⋊D4), C12.46(C5⋊D4), C1534(C8.C22), (C2×C60).78C22, C4.15(C22×D15), C20.116(C22×S3), C153C8.21C22, C12.116(C22×D5), D6011C2.11C2, (Q8×C15).37C22, C22.11(C157D4), C6.113(C2×C5⋊D4), C2.18(C2×C157D4), (C2×C6).81(C5⋊D4), C10.113(C2×C3⋊D4), (C2×C10).81(C3⋊D4), SmallGroup(480,907)

Series: Derived Chief Lower central Upper central

C1C60 — Q8.11D30
C1C5C15C30C60D60D6011C2 — Q8.11D30
C15C30C60 — Q8.11D30
C1C2C2×C4C2×Q8

Generators and relations for Q8.11D30
 G = < a,b,c,d | a4=1, b2=c30=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=ab, dcd-1=c29 >

Subgroups: 628 in 120 conjugacy classes, 47 normal (33 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, S3, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×2], Q8 [×2], D5, C10, C10, Dic3, C12 [×2], C12 [×2], D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5, C20 [×2], C20 [×2], D10, C2×C10, C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8 [×2], C3×Q8, D15, C30, C30, C8.C22, C52C8 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×Q8 [×2], C5×Q8, C4.Dic3, Q82S3 [×2], C3⋊Q16 [×2], C4○D12, C6×Q8, Dic15, C60 [×2], C60 [×2], D30, C2×C30, C4.Dic5, Q8⋊D5 [×2], C5⋊Q16 [×2], C4○D20, Q8×C10, Q8.11D6, C153C8 [×2], Dic30, C4×D15, D60, C157D4, C2×C60, C2×C60, Q8×C15 [×2], Q8×C15, C20.C23, C60.7C4, Q82D15 [×2], C157Q16 [×2], D6011C2, Q8×C30, Q8.11D30
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, D15, C8.C22, C5⋊D4 [×2], C22×D5, C2×C3⋊D4, D30 [×3], C2×C5⋊D4, Q8.11D6, C157D4 [×2], C22×D15, C20.C23, C2×C157D4, Q8.11D30

Smallest permutation representation of Q8.11D30
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 76 91 106)(62 77 92 107)(63 78 93 108)(64 79 94 109)(65 80 95 110)(66 81 96 111)(67 82 97 112)(68 83 98 113)(69 84 99 114)(70 85 100 115)(71 86 101 116)(72 87 102 117)(73 88 103 118)(74 89 104 119)(75 90 105 120)(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)(181 226 211 196)(182 227 212 197)(183 228 213 198)(184 229 214 199)(185 230 215 200)(186 231 216 201)(187 232 217 202)(188 233 218 203)(189 234 219 204)(190 235 220 205)(191 236 221 206)(192 237 222 207)(193 238 223 208)(194 239 224 209)(195 240 225 210)
(1 81 31 111)(2 112 32 82)(3 83 33 113)(4 114 34 84)(5 85 35 115)(6 116 36 86)(7 87 37 117)(8 118 38 88)(9 89 39 119)(10 120 40 90)(11 91 41 61)(12 62 42 92)(13 93 43 63)(14 64 44 94)(15 95 45 65)(16 66 46 96)(17 97 47 67)(18 68 48 98)(19 99 49 69)(20 70 50 100)(21 101 51 71)(22 72 52 102)(23 103 53 73)(24 74 54 104)(25 105 55 75)(26 76 56 106)(27 107 57 77)(28 78 58 108)(29 109 59 79)(30 80 60 110)(121 192 151 222)(122 223 152 193)(123 194 153 224)(124 225 154 195)(125 196 155 226)(126 227 156 197)(127 198 157 228)(128 229 158 199)(129 200 159 230)(130 231 160 201)(131 202 161 232)(132 233 162 203)(133 204 163 234)(134 235 164 205)(135 206 165 236)(136 237 166 207)(137 208 167 238)(138 239 168 209)(139 210 169 240)(140 181 170 211)(141 212 171 182)(142 183 172 213)(143 214 173 184)(144 185 174 215)(145 216 175 186)(146 187 176 217)(147 218 177 188)(148 189 178 219)(149 220 179 190)(150 191 180 221)
(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)
(1 173 31 143)(2 142 32 172)(3 171 33 141)(4 140 34 170)(5 169 35 139)(6 138 36 168)(7 167 37 137)(8 136 38 166)(9 165 39 135)(10 134 40 164)(11 163 41 133)(12 132 42 162)(13 161 43 131)(14 130 44 160)(15 159 45 129)(16 128 46 158)(17 157 47 127)(18 126 48 156)(19 155 49 125)(20 124 50 154)(21 153 51 123)(22 122 52 152)(23 151 53 121)(24 180 54 150)(25 149 55 179)(26 178 56 148)(27 147 57 177)(28 176 58 146)(29 145 59 175)(30 174 60 144)(61 219 91 189)(62 188 92 218)(63 217 93 187)(64 186 94 216)(65 215 95 185)(66 184 96 214)(67 213 97 183)(68 182 98 212)(69 211 99 181)(70 240 100 210)(71 209 101 239)(72 238 102 208)(73 207 103 237)(74 236 104 206)(75 205 105 235)(76 234 106 204)(77 203 107 233)(78 232 108 202)(79 201 109 231)(80 230 110 200)(81 199 111 229)(82 228 112 198)(83 197 113 227)(84 226 114 196)(85 195 115 225)(86 224 116 194)(87 193 117 223)(88 222 118 192)(89 191 119 221)(90 220 120 190)

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

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

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

81 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E5A5B6A6B6C8A8B10A···10F12A···12F15A15B15C15D20A···20L30A···30L60A···60X
order1222344444556668810···1012···121515151520···2030···3060···60
size1126022244602222260602···24···422224···42···24···4

81 irreducible representations

dim111111222222222222222224444
type+++++++++++++++++-
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10C3⋊D4C3⋊D4D15C5⋊D4C5⋊D4D30D30C157D4C157D4C8.C22Q8.11D6C20.C23Q8.11D30
kernelQ8.11D30C60.7C4Q82D15C157Q16D6011C2Q8×C30Q8×C10C60C2×C30C6×Q8C2×C20C5×Q8C2×C12C3×Q8C20C2×C10C2×Q8C12C2×C6C2×C4Q8C4C22C15C5C3C1
# reps112211111212242244448881248

Matrix representation of Q8.11D30 in GL4(𝔽241) generated by

124000
224000
002401
002391
,
13713600
6410400
007332
00210168
,
8116000
16216000
00122119
003119
,
00122119
003119
8116000
16216000
G:=sub<GL(4,GF(241))| [1,2,0,0,240,240,0,0,0,0,240,239,0,0,1,1],[137,64,0,0,136,104,0,0,0,0,73,210,0,0,32,168],[81,162,0,0,160,160,0,0,0,0,122,3,0,0,119,119],[0,0,81,162,0,0,160,160,122,3,0,0,119,119,0,0] >;

Q8.11D30 in GAP, Magma, Sage, TeX

Q_8._{11}D_{30}
% in TeX

G:=Group("Q8.11D30");
// GroupNames label

G:=SmallGroup(480,907);
// by ID

G=gap.SmallGroup(480,907);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,100,675,185,80,2693,18822]);
// Polycyclic

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

׿
×
𝔽