metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.10D30, Q8.16D30, C60.90C23, C30.66C24, D30.31C23, D60.53C22, C15⋊112- (1+4), Dic15.33C23, Dic30.45C22, C4○D4⋊6D15, C5⋊5(Q8○D12), (Q8×D15)⋊12C2, (C5×D4).34D6, (C2×C4).24D30, (C5×Q8).54D6, (C3×D4).34D10, (C2×C20).174D6, (C3×Q8).37D10, D4⋊2D15⋊12C2, C6.66(C23×D5), (C2×Dic30)⋊17C2, (C2×C12).172D10, (C2×C60).90C22, C10.66(S3×C23), (C2×C30).12C23, D60⋊11C2⋊19C2, C2.14(C23×D15), C4.33(C22×D15), C15⋊7D4.2C22, C3⋊5(D4.10D10), C20.140(C22×S3), (D4×C15).39C22, (C4×D15).29C22, C12.138(C22×D5), (Q8×C15).42C22, C22.4(C22×D15), (C2×Dic15).21C22, (C5×C4○D4)⋊9S3, (C3×C4○D4)⋊5D5, (C15×C4○D4)⋊5C2, (C2×C6).19(C22×D5), (C2×C10).20(C22×S3), SmallGroup(480,1177)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1412 in 292 conjugacy classes, 119 normal (22 characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C5, S3 [×2], C6, C6 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×7], Q8, Q8 [×9], D5 [×2], C10, C10 [×3], Dic3 [×6], C12, C12 [×3], D6 [×2], C2×C6 [×3], C15, C2×Q8 [×5], C4○D4, C4○D4 [×9], Dic5 [×6], C20, C20 [×3], D10 [×2], C2×C10 [×3], Dic6 [×9], C4×S3 [×6], D12, C2×Dic3 [×6], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, D15 [×2], C30, C30 [×3], 2- (1+4), Dic10 [×9], C4×D5 [×6], D20, C2×Dic5 [×6], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C2×Dic6 [×3], C4○D12 [×3], D4⋊2S3 [×6], S3×Q8 [×2], C3×C4○D4, Dic15 [×6], C60, C60 [×3], D30 [×2], C2×C30 [×3], C2×Dic10 [×3], C4○D20 [×3], D4⋊2D5 [×6], Q8×D5 [×2], C5×C4○D4, Q8○D12, Dic30 [×9], C4×D15 [×6], D60, C2×Dic15 [×6], C15⋊7D4 [×6], C2×C60 [×3], D4×C15 [×3], Q8×C15, D4.10D10, C2×Dic30 [×3], D60⋊11C2 [×3], D4⋊2D15 [×6], Q8×D15 [×2], C15×C4○D4, D4.10D30
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], D15, 2- (1+4), C22×D5 [×7], S3×C23, D30 [×7], C23×D5, Q8○D12, C22×D15 [×7], D4.10D10, C23×D15, D4.10D30
Generators and relations
G = < a,b,c,d | a4=b2=1, c30=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c29 >
►On 240 points
Generators in S240
(1 151 31 121)(2 122 32 152)(3 153 33 123)(4 124 34 154)(5 155 35 125)(6 126 36 156)(7 157 37 127)(8 128 38 158)(9 159 39 129)(10 130 40 160)(11 161 41 131)(12 132 42 162)(13 163 43 133)(14 134 44 164)(15 165 45 135)(16 136 46 166)(17 167 47 137)(18 138 48 168)(19 169 49 139)(20 140 50 170)(21 171 51 141)(22 142 52 172)(23 173 53 143)(24 144 54 174)(25 175 55 145)(26 146 56 176)(27 177 57 147)(28 148 58 178)(29 179 59 149)(30 150 60 180)(61 206 91 236)(62 237 92 207)(63 208 93 238)(64 239 94 209)(65 210 95 240)(66 181 96 211)(67 212 97 182)(68 183 98 213)(69 214 99 184)(70 185 100 215)(71 216 101 186)(72 187 102 217)(73 218 103 188)(74 189 104 219)(75 220 105 190)(76 191 106 221)(77 222 107 192)(78 193 108 223)(79 224 109 194)(80 195 110 225)(81 226 111 196)(82 197 112 227)(83 228 113 198)(84 199 114 229)(85 230 115 200)(86 201 116 231)(87 232 117 202)(88 203 118 233)(89 234 119 204)(90 205 120 235)
(1 66)(2 97)(3 68)(4 99)(5 70)(6 101)(7 72)(8 103)(9 74)(10 105)(11 76)(12 107)(13 78)(14 109)(15 80)(16 111)(17 82)(18 113)(19 84)(20 115)(21 86)(22 117)(23 88)(24 119)(25 90)(26 61)(27 92)(28 63)(29 94)(30 65)(31 96)(32 67)(33 98)(34 69)(35 100)(36 71)(37 102)(38 73)(39 104)(40 75)(41 106)(42 77)(43 108)(44 79)(45 110)(46 81)(47 112)(48 83)(49 114)(50 85)(51 116)(52 87)(53 118)(54 89)(55 120)(56 91)(57 62)(58 93)(59 64)(60 95)(121 181)(122 212)(123 183)(124 214)(125 185)(126 216)(127 187)(128 218)(129 189)(130 220)(131 191)(132 222)(133 193)(134 224)(135 195)(136 226)(137 197)(138 228)(139 199)(140 230)(141 201)(142 232)(143 203)(144 234)(145 205)(146 236)(147 207)(148 238)(149 209)(150 240)(151 211)(152 182)(153 213)(154 184)(155 215)(156 186)(157 217)(158 188)(159 219)(160 190)(161 221)(162 192)(163 223)(164 194)(165 225)(166 196)(167 227)(168 198)(169 229)(170 200)(171 231)(172 202)(173 233)(174 204)(175 235)(176 206)(177 237)(178 208)(179 239)(180 210)
(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)
(1 195 31 225)(2 224 32 194)(3 193 33 223)(4 222 34 192)(5 191 35 221)(6 220 36 190)(7 189 37 219)(8 218 38 188)(9 187 39 217)(10 216 40 186)(11 185 41 215)(12 214 42 184)(13 183 43 213)(14 212 44 182)(15 181 45 211)(16 210 46 240)(17 239 47 209)(18 208 48 238)(19 237 49 207)(20 206 50 236)(21 235 51 205)(22 204 52 234)(23 233 53 203)(24 202 54 232)(25 231 55 201)(26 200 56 230)(27 229 57 199)(28 198 58 228)(29 227 59 197)(30 196 60 226)(61 170 91 140)(62 139 92 169)(63 168 93 138)(64 137 94 167)(65 166 95 136)(66 135 96 165)(67 164 97 134)(68 133 98 163)(69 162 99 132)(70 131 100 161)(71 160 101 130)(72 129 102 159)(73 158 103 128)(74 127 104 157)(75 156 105 126)(76 125 106 155)(77 154 107 124)(78 123 108 153)(79 152 109 122)(80 121 110 151)(81 150 111 180)(82 179 112 149)(83 148 113 178)(84 177 114 147)(85 146 115 176)(86 175 116 145)(87 144 117 174)(88 173 118 143)(89 142 119 172)(90 171 120 141)
G:=sub<Sym(240)| (1,151,31,121)(2,122,32,152)(3,153,33,123)(4,124,34,154)(5,155,35,125)(6,126,36,156)(7,157,37,127)(8,128,38,158)(9,159,39,129)(10,130,40,160)(11,161,41,131)(12,132,42,162)(13,163,43,133)(14,134,44,164)(15,165,45,135)(16,136,46,166)(17,167,47,137)(18,138,48,168)(19,169,49,139)(20,140,50,170)(21,171,51,141)(22,142,52,172)(23,173,53,143)(24,144,54,174)(25,175,55,145)(26,146,56,176)(27,177,57,147)(28,148,58,178)(29,179,59,149)(30,150,60,180)(61,206,91,236)(62,237,92,207)(63,208,93,238)(64,239,94,209)(65,210,95,240)(66,181,96,211)(67,212,97,182)(68,183,98,213)(69,214,99,184)(70,185,100,215)(71,216,101,186)(72,187,102,217)(73,218,103,188)(74,189,104,219)(75,220,105,190)(76,191,106,221)(77,222,107,192)(78,193,108,223)(79,224,109,194)(80,195,110,225)(81,226,111,196)(82,197,112,227)(83,228,113,198)(84,199,114,229)(85,230,115,200)(86,201,116,231)(87,232,117,202)(88,203,118,233)(89,234,119,204)(90,205,120,235), (1,66)(2,97)(3,68)(4,99)(5,70)(6,101)(7,72)(8,103)(9,74)(10,105)(11,76)(12,107)(13,78)(14,109)(15,80)(16,111)(17,82)(18,113)(19,84)(20,115)(21,86)(22,117)(23,88)(24,119)(25,90)(26,61)(27,92)(28,63)(29,94)(30,65)(31,96)(32,67)(33,98)(34,69)(35,100)(36,71)(37,102)(38,73)(39,104)(40,75)(41,106)(42,77)(43,108)(44,79)(45,110)(46,81)(47,112)(48,83)(49,114)(50,85)(51,116)(52,87)(53,118)(54,89)(55,120)(56,91)(57,62)(58,93)(59,64)(60,95)(121,181)(122,212)(123,183)(124,214)(125,185)(126,216)(127,187)(128,218)(129,189)(130,220)(131,191)(132,222)(133,193)(134,224)(135,195)(136,226)(137,197)(138,228)(139,199)(140,230)(141,201)(142,232)(143,203)(144,234)(145,205)(146,236)(147,207)(148,238)(149,209)(150,240)(151,211)(152,182)(153,213)(154,184)(155,215)(156,186)(157,217)(158,188)(159,219)(160,190)(161,221)(162,192)(163,223)(164,194)(165,225)(166,196)(167,227)(168,198)(169,229)(170,200)(171,231)(172,202)(173,233)(174,204)(175,235)(176,206)(177,237)(178,208)(179,239)(180,210), (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), (1,195,31,225)(2,224,32,194)(3,193,33,223)(4,222,34,192)(5,191,35,221)(6,220,36,190)(7,189,37,219)(8,218,38,188)(9,187,39,217)(10,216,40,186)(11,185,41,215)(12,214,42,184)(13,183,43,213)(14,212,44,182)(15,181,45,211)(16,210,46,240)(17,239,47,209)(18,208,48,238)(19,237,49,207)(20,206,50,236)(21,235,51,205)(22,204,52,234)(23,233,53,203)(24,202,54,232)(25,231,55,201)(26,200,56,230)(27,229,57,199)(28,198,58,228)(29,227,59,197)(30,196,60,226)(61,170,91,140)(62,139,92,169)(63,168,93,138)(64,137,94,167)(65,166,95,136)(66,135,96,165)(67,164,97,134)(68,133,98,163)(69,162,99,132)(70,131,100,161)(71,160,101,130)(72,129,102,159)(73,158,103,128)(74,127,104,157)(75,156,105,126)(76,125,106,155)(77,154,107,124)(78,123,108,153)(79,152,109,122)(80,121,110,151)(81,150,111,180)(82,179,112,149)(83,148,113,178)(84,177,114,147)(85,146,115,176)(86,175,116,145)(87,144,117,174)(88,173,118,143)(89,142,119,172)(90,171,120,141)>;
G:=Group( (1,151,31,121)(2,122,32,152)(3,153,33,123)(4,124,34,154)(5,155,35,125)(6,126,36,156)(7,157,37,127)(8,128,38,158)(9,159,39,129)(10,130,40,160)(11,161,41,131)(12,132,42,162)(13,163,43,133)(14,134,44,164)(15,165,45,135)(16,136,46,166)(17,167,47,137)(18,138,48,168)(19,169,49,139)(20,140,50,170)(21,171,51,141)(22,142,52,172)(23,173,53,143)(24,144,54,174)(25,175,55,145)(26,146,56,176)(27,177,57,147)(28,148,58,178)(29,179,59,149)(30,150,60,180)(61,206,91,236)(62,237,92,207)(63,208,93,238)(64,239,94,209)(65,210,95,240)(66,181,96,211)(67,212,97,182)(68,183,98,213)(69,214,99,184)(70,185,100,215)(71,216,101,186)(72,187,102,217)(73,218,103,188)(74,189,104,219)(75,220,105,190)(76,191,106,221)(77,222,107,192)(78,193,108,223)(79,224,109,194)(80,195,110,225)(81,226,111,196)(82,197,112,227)(83,228,113,198)(84,199,114,229)(85,230,115,200)(86,201,116,231)(87,232,117,202)(88,203,118,233)(89,234,119,204)(90,205,120,235), (1,66)(2,97)(3,68)(4,99)(5,70)(6,101)(7,72)(8,103)(9,74)(10,105)(11,76)(12,107)(13,78)(14,109)(15,80)(16,111)(17,82)(18,113)(19,84)(20,115)(21,86)(22,117)(23,88)(24,119)(25,90)(26,61)(27,92)(28,63)(29,94)(30,65)(31,96)(32,67)(33,98)(34,69)(35,100)(36,71)(37,102)(38,73)(39,104)(40,75)(41,106)(42,77)(43,108)(44,79)(45,110)(46,81)(47,112)(48,83)(49,114)(50,85)(51,116)(52,87)(53,118)(54,89)(55,120)(56,91)(57,62)(58,93)(59,64)(60,95)(121,181)(122,212)(123,183)(124,214)(125,185)(126,216)(127,187)(128,218)(129,189)(130,220)(131,191)(132,222)(133,193)(134,224)(135,195)(136,226)(137,197)(138,228)(139,199)(140,230)(141,201)(142,232)(143,203)(144,234)(145,205)(146,236)(147,207)(148,238)(149,209)(150,240)(151,211)(152,182)(153,213)(154,184)(155,215)(156,186)(157,217)(158,188)(159,219)(160,190)(161,221)(162,192)(163,223)(164,194)(165,225)(166,196)(167,227)(168,198)(169,229)(170,200)(171,231)(172,202)(173,233)(174,204)(175,235)(176,206)(177,237)(178,208)(179,239)(180,210), (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), (1,195,31,225)(2,224,32,194)(3,193,33,223)(4,222,34,192)(5,191,35,221)(6,220,36,190)(7,189,37,219)(8,218,38,188)(9,187,39,217)(10,216,40,186)(11,185,41,215)(12,214,42,184)(13,183,43,213)(14,212,44,182)(15,181,45,211)(16,210,46,240)(17,239,47,209)(18,208,48,238)(19,237,49,207)(20,206,50,236)(21,235,51,205)(22,204,52,234)(23,233,53,203)(24,202,54,232)(25,231,55,201)(26,200,56,230)(27,229,57,199)(28,198,58,228)(29,227,59,197)(30,196,60,226)(61,170,91,140)(62,139,92,169)(63,168,93,138)(64,137,94,167)(65,166,95,136)(66,135,96,165)(67,164,97,134)(68,133,98,163)(69,162,99,132)(70,131,100,161)(71,160,101,130)(72,129,102,159)(73,158,103,128)(74,127,104,157)(75,156,105,126)(76,125,106,155)(77,154,107,124)(78,123,108,153)(79,152,109,122)(80,121,110,151)(81,150,111,180)(82,179,112,149)(83,148,113,178)(84,177,114,147)(85,146,115,176)(86,175,116,145)(87,144,117,174)(88,173,118,143)(89,142,119,172)(90,171,120,141) );
G=PermutationGroup([(1,151,31,121),(2,122,32,152),(3,153,33,123),(4,124,34,154),(5,155,35,125),(6,126,36,156),(7,157,37,127),(8,128,38,158),(9,159,39,129),(10,130,40,160),(11,161,41,131),(12,132,42,162),(13,163,43,133),(14,134,44,164),(15,165,45,135),(16,136,46,166),(17,167,47,137),(18,138,48,168),(19,169,49,139),(20,140,50,170),(21,171,51,141),(22,142,52,172),(23,173,53,143),(24,144,54,174),(25,175,55,145),(26,146,56,176),(27,177,57,147),(28,148,58,178),(29,179,59,149),(30,150,60,180),(61,206,91,236),(62,237,92,207),(63,208,93,238),(64,239,94,209),(65,210,95,240),(66,181,96,211),(67,212,97,182),(68,183,98,213),(69,214,99,184),(70,185,100,215),(71,216,101,186),(72,187,102,217),(73,218,103,188),(74,189,104,219),(75,220,105,190),(76,191,106,221),(77,222,107,192),(78,193,108,223),(79,224,109,194),(80,195,110,225),(81,226,111,196),(82,197,112,227),(83,228,113,198),(84,199,114,229),(85,230,115,200),(86,201,116,231),(87,232,117,202),(88,203,118,233),(89,234,119,204),(90,205,120,235)], [(1,66),(2,97),(3,68),(4,99),(5,70),(6,101),(7,72),(8,103),(9,74),(10,105),(11,76),(12,107),(13,78),(14,109),(15,80),(16,111),(17,82),(18,113),(19,84),(20,115),(21,86),(22,117),(23,88),(24,119),(25,90),(26,61),(27,92),(28,63),(29,94),(30,65),(31,96),(32,67),(33,98),(34,69),(35,100),(36,71),(37,102),(38,73),(39,104),(40,75),(41,106),(42,77),(43,108),(44,79),(45,110),(46,81),(47,112),(48,83),(49,114),(50,85),(51,116),(52,87),(53,118),(54,89),(55,120),(56,91),(57,62),(58,93),(59,64),(60,95),(121,181),(122,212),(123,183),(124,214),(125,185),(126,216),(127,187),(128,218),(129,189),(130,220),(131,191),(132,222),(133,193),(134,224),(135,195),(136,226),(137,197),(138,228),(139,199),(140,230),(141,201),(142,232),(143,203),(144,234),(145,205),(146,236),(147,207),(148,238),(149,209),(150,240),(151,211),(152,182),(153,213),(154,184),(155,215),(156,186),(157,217),(158,188),(159,219),(160,190),(161,221),(162,192),(163,223),(164,194),(165,225),(166,196),(167,227),(168,198),(169,229),(170,200),(171,231),(172,202),(173,233),(174,204),(175,235),(176,206),(177,237),(178,208),(179,239),(180,210)], [(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)], [(1,195,31,225),(2,224,32,194),(3,193,33,223),(4,222,34,192),(5,191,35,221),(6,220,36,190),(7,189,37,219),(8,218,38,188),(9,187,39,217),(10,216,40,186),(11,185,41,215),(12,214,42,184),(13,183,43,213),(14,212,44,182),(15,181,45,211),(16,210,46,240),(17,239,47,209),(18,208,48,238),(19,237,49,207),(20,206,50,236),(21,235,51,205),(22,204,52,234),(23,233,53,203),(24,202,54,232),(25,231,55,201),(26,200,56,230),(27,229,57,199),(28,198,58,228),(29,227,59,197),(30,196,60,226),(61,170,91,140),(62,139,92,169),(63,168,93,138),(64,137,94,167),(65,166,95,136),(66,135,96,165),(67,164,97,134),(68,133,98,163),(69,162,99,132),(70,131,100,161),(71,160,101,130),(72,129,102,159),(73,158,103,128),(74,127,104,157),(75,156,105,126),(76,125,106,155),(77,154,107,124),(78,123,108,153),(79,152,109,122),(80,121,110,151),(81,150,111,180),(82,179,112,149),(83,148,113,178),(84,177,114,147),(85,146,115,176),(86,175,116,145),(87,144,117,174),(88,173,118,143),(89,142,119,172),(90,171,120,141)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 60 | 0 | 14 | 14 |
| 0 | 60 | 47 | 0 |
| 0 | 35 | 1 | 0 |
| 26 | 26 | 0 | 1 |
| 38 | 35 | 2 | 5 |
| 26 | 23 | 2 | 58 |
| 43 | 31 | 3 | 26 |
| 49 | 12 | 55 | 58 |
| 25 | 53 | 1 | 10 |
| 8 | 11 | 11 | 52 |
| 17 | 31 | 58 | 8 |
| 29 | 47 | 31 | 28 |
| 59 | 57 | 13 | 48 |
| 23 | 2 | 45 | 58 |
| 26 | 50 | 21 | 23 |
| 24 | 50 | 38 | 40 |
G:=sub<GL(4,GF(61))| [60,0,0,26,0,60,35,26,14,47,1,0,14,0,0,1],[38,26,43,49,35,23,31,12,2,2,3,55,5,58,26,58],[25,8,17,29,53,11,31,47,1,11,58,31,10,52,8,28],[59,23,26,24,57,2,50,50,13,45,21,38,48,58,23,40] >;
87 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 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 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 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 | 2 | 2 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 30 | ··· | 30 | 2 | 2 | 2 | 4 | 4 | 4 | 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 |
87 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | D15 | D30 | D30 | D30 | 2- (1+4) | Q8○D12 | D4.10D10 | D4.10D30 |
| kernel | D4.10D30 | C2×Dic30 | D60⋊11C2 | D4⋊2D15 | Q8×D15 | C15×C4○D4 | C5×C4○D4 | C3×C4○D4 | C2×C20 | C5×D4 | C5×Q8 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C15 | C5 | C3 | C1 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 1 | 2 | 3 | 3 | 1 | 6 | 6 | 2 | 4 | 12 | 12 | 4 | 1 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
D_4._{10}D_{30} % in TeX
G:=Group("D4.10D30"); // GroupNames label
G:=SmallGroup(480,1177);
// by ID
G=gap.SmallGroup(480,1177);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,219,100,675,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^30=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^29>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀