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, C15⋊7Q16⋊13C2, (C2×C30).149D4, (C2×C20).152D6, C30.386(C2×D4), (C3×Q8).32D10, C60.7C4⋊19C2, Q8⋊2D15⋊13C2, (C2×C12).151D10, C4.17(C15⋊7D4), C3⋊5(C20.C23), C5⋊5(Q8.11D6), C20.44(C3⋊D4), C12.46(C5⋊D4), C15⋊34(C8.C22), (C2×C60).78C22, C4.15(C22×D15), C20.116(C22×S3), C15⋊3C8.21C22, C12.116(C22×D5), D60⋊11C2.11C2, (Q8×C15).37C22, C22.11(C15⋊7D4), C6.113(C2×C5⋊D4), C2.18(C2×C15⋊7D4), (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
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, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8, C3×Q8, D15, C30, C30, C8.C22, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×Q8, C5×Q8, C4.Dic3, Q8⋊2S3, C3⋊Q16, C4○D12, C6×Q8, Dic15, C60, C60, D30, C2×C30, C4.Dic5, Q8⋊D5, C5⋊Q16, C4○D20, Q8×C10, Q8.11D6, C15⋊3C8, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, C2×C60, Q8×C15, Q8×C15, C20.C23, C60.7C4, Q8⋊2D15, C15⋊7Q16, D60⋊11C2, Q8×C30, Q8.11D30
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, D15, C8.C22, C5⋊D4, C22×D5, C2×C3⋊D4, D30, C2×C5⋊D4, Q8.11D6, C15⋊7D4, C22×D15, C20.C23, C2×C15⋊7D4, 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 61 31 91)(2 92 32 62)(3 63 33 93)(4 94 34 64)(5 65 35 95)(6 96 36 66)(7 67 37 97)(8 98 38 68)(9 69 39 99)(10 100 40 70)(11 71 41 101)(12 102 42 72)(13 73 43 103)(14 104 44 74)(15 75 45 105)(16 106 46 76)(17 77 47 107)(18 108 48 78)(19 79 49 109)(20 110 50 80)(21 81 51 111)(22 112 52 82)(23 83 53 113)(24 114 54 84)(25 85 55 115)(26 116 56 86)(27 87 57 117)(28 118 58 88)(29 89 59 119)(30 120 60 90)(121 190 151 220)(122 221 152 191)(123 192 153 222)(124 223 154 193)(125 194 155 224)(126 225 156 195)(127 196 157 226)(128 227 158 197)(129 198 159 228)(130 229 160 199)(131 200 161 230)(132 231 162 201)(133 202 163 232)(134 233 164 203)(135 204 165 234)(136 235 166 205)(137 206 167 236)(138 237 168 207)(139 208 169 238)(140 239 170 209)(141 210 171 240)(142 181 172 211)(143 212 173 182)(144 183 174 213)(145 214 175 184)(146 185 176 215)(147 216 177 186)(148 187 178 217)(149 218 179 188)(150 189 180 219)
(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)
(1 140 31 170)(2 169 32 139)(3 138 33 168)(4 167 34 137)(5 136 35 166)(6 165 36 135)(7 134 37 164)(8 163 38 133)(9 132 39 162)(10 161 40 131)(11 130 41 160)(12 159 42 129)(13 128 43 158)(14 157 44 127)(15 126 45 156)(16 155 46 125)(17 124 47 154)(18 153 48 123)(19 122 49 152)(20 151 50 121)(21 180 51 150)(22 149 52 179)(23 178 53 148)(24 147 54 177)(25 176 55 146)(26 145 56 175)(27 174 57 144)(28 143 58 173)(29 172 59 142)(30 141 60 171)(61 194 91 224)(62 223 92 193)(63 192 93 222)(64 221 94 191)(65 190 95 220)(66 219 96 189)(67 188 97 218)(68 217 98 187)(69 186 99 216)(70 215 100 185)(71 184 101 214)(72 213 102 183)(73 182 103 212)(74 211 104 181)(75 240 105 210)(76 209 106 239)(77 238 107 208)(78 207 108 237)(79 236 109 206)(80 205 110 235)(81 234 111 204)(82 203 112 233)(83 232 113 202)(84 201 114 231)(85 230 115 200)(86 199 116 229)(87 228 117 198)(88 197 118 227)(89 226 119 196)(90 195 120 225)
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), (1,61,31,91)(2,92,32,62)(3,63,33,93)(4,94,34,64)(5,65,35,95)(6,96,36,66)(7,67,37,97)(8,98,38,68)(9,69,39,99)(10,100,40,70)(11,71,41,101)(12,102,42,72)(13,73,43,103)(14,104,44,74)(15,75,45,105)(16,106,46,76)(17,77,47,107)(18,108,48,78)(19,79,49,109)(20,110,50,80)(21,81,51,111)(22,112,52,82)(23,83,53,113)(24,114,54,84)(25,85,55,115)(26,116,56,86)(27,87,57,117)(28,118,58,88)(29,89,59,119)(30,120,60,90)(121,190,151,220)(122,221,152,191)(123,192,153,222)(124,223,154,193)(125,194,155,224)(126,225,156,195)(127,196,157,226)(128,227,158,197)(129,198,159,228)(130,229,160,199)(131,200,161,230)(132,231,162,201)(133,202,163,232)(134,233,164,203)(135,204,165,234)(136,235,166,205)(137,206,167,236)(138,237,168,207)(139,208,169,238)(140,239,170,209)(141,210,171,240)(142,181,172,211)(143,212,173,182)(144,183,174,213)(145,214,175,184)(146,185,176,215)(147,216,177,186)(148,187,178,217)(149,218,179,188)(150,189,180,219), (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), (1,140,31,170)(2,169,32,139)(3,138,33,168)(4,167,34,137)(5,136,35,166)(6,165,36,135)(7,134,37,164)(8,163,38,133)(9,132,39,162)(10,161,40,131)(11,130,41,160)(12,159,42,129)(13,128,43,158)(14,157,44,127)(15,126,45,156)(16,155,46,125)(17,124,47,154)(18,153,48,123)(19,122,49,152)(20,151,50,121)(21,180,51,150)(22,149,52,179)(23,178,53,148)(24,147,54,177)(25,176,55,146)(26,145,56,175)(27,174,57,144)(28,143,58,173)(29,172,59,142)(30,141,60,171)(61,194,91,224)(62,223,92,193)(63,192,93,222)(64,221,94,191)(65,190,95,220)(66,219,96,189)(67,188,97,218)(68,217,98,187)(69,186,99,216)(70,215,100,185)(71,184,101,214)(72,213,102,183)(73,182,103,212)(74,211,104,181)(75,240,105,210)(76,209,106,239)(77,238,107,208)(78,207,108,237)(79,236,109,206)(80,205,110,235)(81,234,111,204)(82,203,112,233)(83,232,113,202)(84,201,114,231)(85,230,115,200)(86,199,116,229)(87,228,117,198)(88,197,118,227)(89,226,119,196)(90,195,120,225)>;
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), (1,61,31,91)(2,92,32,62)(3,63,33,93)(4,94,34,64)(5,65,35,95)(6,96,36,66)(7,67,37,97)(8,98,38,68)(9,69,39,99)(10,100,40,70)(11,71,41,101)(12,102,42,72)(13,73,43,103)(14,104,44,74)(15,75,45,105)(16,106,46,76)(17,77,47,107)(18,108,48,78)(19,79,49,109)(20,110,50,80)(21,81,51,111)(22,112,52,82)(23,83,53,113)(24,114,54,84)(25,85,55,115)(26,116,56,86)(27,87,57,117)(28,118,58,88)(29,89,59,119)(30,120,60,90)(121,190,151,220)(122,221,152,191)(123,192,153,222)(124,223,154,193)(125,194,155,224)(126,225,156,195)(127,196,157,226)(128,227,158,197)(129,198,159,228)(130,229,160,199)(131,200,161,230)(132,231,162,201)(133,202,163,232)(134,233,164,203)(135,204,165,234)(136,235,166,205)(137,206,167,236)(138,237,168,207)(139,208,169,238)(140,239,170,209)(141,210,171,240)(142,181,172,211)(143,212,173,182)(144,183,174,213)(145,214,175,184)(146,185,176,215)(147,216,177,186)(148,187,178,217)(149,218,179,188)(150,189,180,219), (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), (1,140,31,170)(2,169,32,139)(3,138,33,168)(4,167,34,137)(5,136,35,166)(6,165,36,135)(7,134,37,164)(8,163,38,133)(9,132,39,162)(10,161,40,131)(11,130,41,160)(12,159,42,129)(13,128,43,158)(14,157,44,127)(15,126,45,156)(16,155,46,125)(17,124,47,154)(18,153,48,123)(19,122,49,152)(20,151,50,121)(21,180,51,150)(22,149,52,179)(23,178,53,148)(24,147,54,177)(25,176,55,146)(26,145,56,175)(27,174,57,144)(28,143,58,173)(29,172,59,142)(30,141,60,171)(61,194,91,224)(62,223,92,193)(63,192,93,222)(64,221,94,191)(65,190,95,220)(66,219,96,189)(67,188,97,218)(68,217,98,187)(69,186,99,216)(70,215,100,185)(71,184,101,214)(72,213,102,183)(73,182,103,212)(74,211,104,181)(75,240,105,210)(76,209,106,239)(77,238,107,208)(78,207,108,237)(79,236,109,206)(80,205,110,235)(81,234,111,204)(82,203,112,233)(83,232,113,202)(84,201,114,231)(85,230,115,200)(86,199,116,229)(87,228,117,198)(88,197,118,227)(89,226,119,196)(90,195,120,225) );
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)], [(1,61,31,91),(2,92,32,62),(3,63,33,93),(4,94,34,64),(5,65,35,95),(6,96,36,66),(7,67,37,97),(8,98,38,68),(9,69,39,99),(10,100,40,70),(11,71,41,101),(12,102,42,72),(13,73,43,103),(14,104,44,74),(15,75,45,105),(16,106,46,76),(17,77,47,107),(18,108,48,78),(19,79,49,109),(20,110,50,80),(21,81,51,111),(22,112,52,82),(23,83,53,113),(24,114,54,84),(25,85,55,115),(26,116,56,86),(27,87,57,117),(28,118,58,88),(29,89,59,119),(30,120,60,90),(121,190,151,220),(122,221,152,191),(123,192,153,222),(124,223,154,193),(125,194,155,224),(126,225,156,195),(127,196,157,226),(128,227,158,197),(129,198,159,228),(130,229,160,199),(131,200,161,230),(132,231,162,201),(133,202,163,232),(134,233,164,203),(135,204,165,234),(136,235,166,205),(137,206,167,236),(138,237,168,207),(139,208,169,238),(140,239,170,209),(141,210,171,240),(142,181,172,211),(143,212,173,182),(144,183,174,213),(145,214,175,184),(146,185,176,215),(147,216,177,186),(148,187,178,217),(149,218,179,188),(150,189,180,219)], [(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)], [(1,140,31,170),(2,169,32,139),(3,138,33,168),(4,167,34,137),(5,136,35,166),(6,165,36,135),(7,134,37,164),(8,163,38,133),(9,132,39,162),(10,161,40,131),(11,130,41,160),(12,159,42,129),(13,128,43,158),(14,157,44,127),(15,126,45,156),(16,155,46,125),(17,124,47,154),(18,153,48,123),(19,122,49,152),(20,151,50,121),(21,180,51,150),(22,149,52,179),(23,178,53,148),(24,147,54,177),(25,176,55,146),(26,145,56,175),(27,174,57,144),(28,143,58,173),(29,172,59,142),(30,141,60,171),(61,194,91,224),(62,223,92,193),(63,192,93,222),(64,221,94,191),(65,190,95,220),(66,219,96,189),(67,188,97,218),(68,217,98,187),(69,186,99,216),(70,215,100,185),(71,184,101,214),(72,213,102,183),(73,182,103,212),(74,211,104,181),(75,240,105,210),(76,209,106,239),(77,238,107,208),(78,207,108,237),(79,236,109,206),(80,205,110,235),(81,234,111,204),(82,203,112,233),(83,232,113,202),(84,201,114,231),(85,230,115,200),(86,199,116,229),(87,228,117,198),(88,197,118,227),(89,226,119,196),(90,195,120,225)]])
81 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | ··· | 10F | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 60 | 2 | 2 | 2 | 4 | 4 | 60 | 2 | 2 | 2 | 2 | 2 | 60 | 60 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 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 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | C3⋊D4 | C3⋊D4 | D15 | C5⋊D4 | C5⋊D4 | D30 | D30 | C15⋊7D4 | C15⋊7D4 | C8.C22 | Q8.11D6 | C20.C23 | Q8.11D30 |
| kernel | Q8.11D30 | C60.7C4 | Q8⋊2D15 | C15⋊7Q16 | D60⋊11C2 | Q8×C30 | Q8×C10 | C60 | C2×C30 | C6×Q8 | C2×C20 | C5×Q8 | C2×C12 | C3×Q8 | C20 | C2×C10 | C2×Q8 | C12 | C2×C6 | C2×C4 | Q8 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of Q8.11D30 ►in GL4(𝔽241) generated by
| 1 | 240 | 0 | 0 |
| 2 | 240 | 0 | 0 |
| 0 | 0 | 240 | 1 |
| 0 | 0 | 239 | 1 |
| 137 | 136 | 0 | 0 |
| 64 | 104 | 0 | 0 |
| 0 | 0 | 73 | 32 |
| 0 | 0 | 210 | 168 |
| 81 | 160 | 0 | 0 |
| 162 | 160 | 0 | 0 |
| 0 | 0 | 122 | 119 |
| 0 | 0 | 3 | 119 |
| 0 | 0 | 122 | 119 |
| 0 | 0 | 3 | 119 |
| 81 | 160 | 0 | 0 |
| 162 | 160 | 0 | 0 |
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