direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C15⋊8M4(2), C30⋊8M4(2), C23.4(C3⋊F5), (C22×C6).7F5, C15⋊21(C2×M4(2)), (C22×C30).7C4, C6⋊2(C22.F5), C15⋊C8⋊13C22, C6.43(C22×F5), C30.81(C22×C4), C10⋊2(C4.Dic3), (C6×Dic5).23C4, (C2×Dic5).210D6, (C22×C10).9Dic3, Dic5.19(C2×Dic3), (C2×Dic5).14Dic3, (C22×Dic5).12S3, Dic5.54(C22×S3), (C3×Dic5).68C23, C10.12(C22×Dic3), (C6×Dic5).269C22, C5⋊3(C2×C4.Dic3), C22.8(C2×C3⋊F5), C3⋊3(C2×C22.F5), (C2×C15⋊C8)⋊12C2, (C2×C6).50(C2×F5), (C2×C30).44(C2×C4), C2.12(C22×C3⋊F5), (C2×C6×Dic5).17C2, (C3×Dic5).69(C2×C4), (C2×C10).20(C2×Dic3), SmallGroup(480,1071)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 460 in 136 conjugacy classes, 65 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×4], C22, C22 [×2], C22 [×2], C5, C6, C6 [×2], C6 [×2], C8 [×4], C2×C4 [×6], C23, C10, C10 [×2], C10 [×2], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C2×C8 [×2], M4(2) [×4], C22×C4, Dic5 [×2], Dic5 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C3⋊C8 [×4], C2×C12 [×6], C22×C6, C30, C30 [×2], C30 [×2], C2×M4(2), C5⋊C8 [×4], C2×Dic5 [×2], C2×Dic5 [×4], C22×C10, C2×C3⋊C8 [×2], C4.Dic3 [×4], C22×C12, C3×Dic5 [×2], C3×Dic5 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C2×C5⋊C8 [×2], C22.F5 [×4], C22×Dic5, C2×C4.Dic3, C15⋊C8 [×4], C6×Dic5 [×2], C6×Dic5 [×4], C22×C30, C2×C22.F5, C2×C15⋊C8 [×2], C15⋊8M4(2) [×4], C2×C6×Dic5, C2×C15⋊8M4(2)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, Dic3 [×4], D6 [×3], M4(2) [×2], C22×C4, F5, C2×Dic3 [×6], C22×S3, C2×M4(2), C2×F5 [×3], C4.Dic3 [×2], C22×Dic3, C3⋊F5, C22.F5 [×2], C22×F5, C2×C4.Dic3, C2×C3⋊F5 [×3], C2×C22.F5, C15⋊8M4(2) [×2], C22×C3⋊F5, C2×C15⋊8M4(2)
Generators and relations
G = < a,b,c,d | a2=b15=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b2, bd=db, dcd=c5 >
►On 240 points
Generators in S240
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 61)(13 62)(14 63)(15 64)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(28 89)(29 90)(30 76)(31 97)(32 98)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 105)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 115)(47 116)(48 117)(49 118)(50 119)(51 120)(52 106)(53 107)(54 108)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(121 191)(122 192)(123 193)(124 194)(125 195)(126 181)(127 182)(128 183)(129 184)(130 185)(131 186)(132 187)(133 188)(134 189)(135 190)(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)(151 212)(152 213)(153 214)(154 215)(155 216)(156 217)(157 218)(158 219)(159 220)(160 221)(161 222)(162 223)(163 224)(164 225)(165 211)(166 227)(167 228)(168 229)(169 230)(170 231)(171 232)(172 233)(173 234)(174 235)(175 236)(176 237)(177 238)(178 239)(179 240)(180 226)
(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)
(1 226 52 196 28 225 44 191)(2 234 56 198 29 218 33 193)(3 227 60 200 30 211 37 195)(4 235 49 202 16 219 41 182)(5 228 53 204 17 212 45 184)(6 236 57 206 18 220 34 186)(7 229 46 208 19 213 38 188)(8 237 50 210 20 221 42 190)(9 230 54 197 21 214 31 192)(10 238 58 199 22 222 35 194)(11 231 47 201 23 215 39 181)(12 239 51 203 24 223 43 183)(13 232 55 205 25 216 32 185)(14 240 59 207 26 224 36 187)(15 233 48 209 27 217 40 189)(61 178 120 143 85 162 94 128)(62 171 109 145 86 155 98 130)(63 179 113 147 87 163 102 132)(64 172 117 149 88 156 91 134)(65 180 106 136 89 164 95 121)(66 173 110 138 90 157 99 123)(67 166 114 140 76 165 103 125)(68 174 118 142 77 158 92 127)(69 167 107 144 78 151 96 129)(70 175 111 146 79 159 100 131)(71 168 115 148 80 152 104 133)(72 176 119 150 81 160 93 135)(73 169 108 137 82 153 97 122)(74 177 112 139 83 161 101 124)(75 170 116 141 84 154 105 126)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 61)(13 62)(14 63)(15 64)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(28 89)(29 90)(30 76)(31 97)(32 98)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 105)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 115)(47 116)(48 117)(49 118)(50 119)(51 120)(52 106)(53 107)(54 108)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(121 196)(122 197)(123 198)(124 199)(125 200)(126 201)(127 202)(128 203)(129 204)(130 205)(131 206)(132 207)(133 208)(134 209)(135 210)(136 191)(137 192)(138 193)(139 194)(140 195)(141 181)(142 182)(143 183)(144 184)(145 185)(146 186)(147 187)(148 188)(149 189)(150 190)(151 228)(152 229)(153 230)(154 231)(155 232)(156 233)(157 234)(158 235)(159 236)(160 237)(161 238)(162 239)(163 240)(164 226)(165 227)(166 211)(167 212)(168 213)(169 214)(170 215)(171 216)(172 217)(173 218)(174 219)(175 220)(176 221)(177 222)(178 223)(179 224)(180 225)
G:=sub<Sym(240)| (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,61)(13,62)(14,63)(15,64)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,76)(31,97)(32,98)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,105)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(121,191)(122,192)(123,193)(124,194)(125,195)(126,181)(127,182)(128,183)(129,184)(130,185)(131,186)(132,187)(133,188)(134,189)(135,190)(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)(151,212)(152,213)(153,214)(154,215)(155,216)(156,217)(157,218)(158,219)(159,220)(160,221)(161,222)(162,223)(163,224)(164,225)(165,211)(166,227)(167,228)(168,229)(169,230)(170,231)(171,232)(172,233)(173,234)(174,235)(175,236)(176,237)(177,238)(178,239)(179,240)(180,226), (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), (1,226,52,196,28,225,44,191)(2,234,56,198,29,218,33,193)(3,227,60,200,30,211,37,195)(4,235,49,202,16,219,41,182)(5,228,53,204,17,212,45,184)(6,236,57,206,18,220,34,186)(7,229,46,208,19,213,38,188)(8,237,50,210,20,221,42,190)(9,230,54,197,21,214,31,192)(10,238,58,199,22,222,35,194)(11,231,47,201,23,215,39,181)(12,239,51,203,24,223,43,183)(13,232,55,205,25,216,32,185)(14,240,59,207,26,224,36,187)(15,233,48,209,27,217,40,189)(61,178,120,143,85,162,94,128)(62,171,109,145,86,155,98,130)(63,179,113,147,87,163,102,132)(64,172,117,149,88,156,91,134)(65,180,106,136,89,164,95,121)(66,173,110,138,90,157,99,123)(67,166,114,140,76,165,103,125)(68,174,118,142,77,158,92,127)(69,167,107,144,78,151,96,129)(70,175,111,146,79,159,100,131)(71,168,115,148,80,152,104,133)(72,176,119,150,81,160,93,135)(73,169,108,137,82,153,97,122)(74,177,112,139,83,161,101,124)(75,170,116,141,84,154,105,126), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,61)(13,62)(14,63)(15,64)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,76)(31,97)(32,98)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,105)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(121,196)(122,197)(123,198)(124,199)(125,200)(126,201)(127,202)(128,203)(129,204)(130,205)(131,206)(132,207)(133,208)(134,209)(135,210)(136,191)(137,192)(138,193)(139,194)(140,195)(141,181)(142,182)(143,183)(144,184)(145,185)(146,186)(147,187)(148,188)(149,189)(150,190)(151,228)(152,229)(153,230)(154,231)(155,232)(156,233)(157,234)(158,235)(159,236)(160,237)(161,238)(162,239)(163,240)(164,226)(165,227)(166,211)(167,212)(168,213)(169,214)(170,215)(171,216)(172,217)(173,218)(174,219)(175,220)(176,221)(177,222)(178,223)(179,224)(180,225)>;
G:=Group( (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,61)(13,62)(14,63)(15,64)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,76)(31,97)(32,98)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,105)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(121,191)(122,192)(123,193)(124,194)(125,195)(126,181)(127,182)(128,183)(129,184)(130,185)(131,186)(132,187)(133,188)(134,189)(135,190)(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)(151,212)(152,213)(153,214)(154,215)(155,216)(156,217)(157,218)(158,219)(159,220)(160,221)(161,222)(162,223)(163,224)(164,225)(165,211)(166,227)(167,228)(168,229)(169,230)(170,231)(171,232)(172,233)(173,234)(174,235)(175,236)(176,237)(177,238)(178,239)(179,240)(180,226), (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), (1,226,52,196,28,225,44,191)(2,234,56,198,29,218,33,193)(3,227,60,200,30,211,37,195)(4,235,49,202,16,219,41,182)(5,228,53,204,17,212,45,184)(6,236,57,206,18,220,34,186)(7,229,46,208,19,213,38,188)(8,237,50,210,20,221,42,190)(9,230,54,197,21,214,31,192)(10,238,58,199,22,222,35,194)(11,231,47,201,23,215,39,181)(12,239,51,203,24,223,43,183)(13,232,55,205,25,216,32,185)(14,240,59,207,26,224,36,187)(15,233,48,209,27,217,40,189)(61,178,120,143,85,162,94,128)(62,171,109,145,86,155,98,130)(63,179,113,147,87,163,102,132)(64,172,117,149,88,156,91,134)(65,180,106,136,89,164,95,121)(66,173,110,138,90,157,99,123)(67,166,114,140,76,165,103,125)(68,174,118,142,77,158,92,127)(69,167,107,144,78,151,96,129)(70,175,111,146,79,159,100,131)(71,168,115,148,80,152,104,133)(72,176,119,150,81,160,93,135)(73,169,108,137,82,153,97,122)(74,177,112,139,83,161,101,124)(75,170,116,141,84,154,105,126), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,61)(13,62)(14,63)(15,64)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,76)(31,97)(32,98)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,105)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(121,196)(122,197)(123,198)(124,199)(125,200)(126,201)(127,202)(128,203)(129,204)(130,205)(131,206)(132,207)(133,208)(134,209)(135,210)(136,191)(137,192)(138,193)(139,194)(140,195)(141,181)(142,182)(143,183)(144,184)(145,185)(146,186)(147,187)(148,188)(149,189)(150,190)(151,228)(152,229)(153,230)(154,231)(155,232)(156,233)(157,234)(158,235)(159,236)(160,237)(161,238)(162,239)(163,240)(164,226)(165,227)(166,211)(167,212)(168,213)(169,214)(170,215)(171,216)(172,217)(173,218)(174,219)(175,220)(176,221)(177,222)(178,223)(179,224)(180,225) );
G=PermutationGroup([(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,73),(10,74),(11,75),(12,61),(13,62),(14,63),(15,64),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,85),(25,86),(26,87),(27,88),(28,89),(29,90),(30,76),(31,97),(32,98),(33,99),(34,100),(35,101),(36,102),(37,103),(38,104),(39,105),(40,91),(41,92),(42,93),(43,94),(44,95),(45,96),(46,115),(47,116),(48,117),(49,118),(50,119),(51,120),(52,106),(53,107),(54,108),(55,109),(56,110),(57,111),(58,112),(59,113),(60,114),(121,191),(122,192),(123,193),(124,194),(125,195),(126,181),(127,182),(128,183),(129,184),(130,185),(131,186),(132,187),(133,188),(134,189),(135,190),(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),(151,212),(152,213),(153,214),(154,215),(155,216),(156,217),(157,218),(158,219),(159,220),(160,221),(161,222),(162,223),(163,224),(164,225),(165,211),(166,227),(167,228),(168,229),(169,230),(170,231),(171,232),(172,233),(173,234),(174,235),(175,236),(176,237),(177,238),(178,239),(179,240),(180,226)], [(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)], [(1,226,52,196,28,225,44,191),(2,234,56,198,29,218,33,193),(3,227,60,200,30,211,37,195),(4,235,49,202,16,219,41,182),(5,228,53,204,17,212,45,184),(6,236,57,206,18,220,34,186),(7,229,46,208,19,213,38,188),(8,237,50,210,20,221,42,190),(9,230,54,197,21,214,31,192),(10,238,58,199,22,222,35,194),(11,231,47,201,23,215,39,181),(12,239,51,203,24,223,43,183),(13,232,55,205,25,216,32,185),(14,240,59,207,26,224,36,187),(15,233,48,209,27,217,40,189),(61,178,120,143,85,162,94,128),(62,171,109,145,86,155,98,130),(63,179,113,147,87,163,102,132),(64,172,117,149,88,156,91,134),(65,180,106,136,89,164,95,121),(66,173,110,138,90,157,99,123),(67,166,114,140,76,165,103,125),(68,174,118,142,77,158,92,127),(69,167,107,144,78,151,96,129),(70,175,111,146,79,159,100,131),(71,168,115,148,80,152,104,133),(72,176,119,150,81,160,93,135),(73,169,108,137,82,153,97,122),(74,177,112,139,83,161,101,124),(75,170,116,141,84,154,105,126)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,73),(10,74),(11,75),(12,61),(13,62),(14,63),(15,64),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,85),(25,86),(26,87),(27,88),(28,89),(29,90),(30,76),(31,97),(32,98),(33,99),(34,100),(35,101),(36,102),(37,103),(38,104),(39,105),(40,91),(41,92),(42,93),(43,94),(44,95),(45,96),(46,115),(47,116),(48,117),(49,118),(50,119),(51,120),(52,106),(53,107),(54,108),(55,109),(56,110),(57,111),(58,112),(59,113),(60,114),(121,196),(122,197),(123,198),(124,199),(125,200),(126,201),(127,202),(128,203),(129,204),(130,205),(131,206),(132,207),(133,208),(134,209),(135,210),(136,191),(137,192),(138,193),(139,194),(140,195),(141,181),(142,182),(143,183),(144,184),(145,185),(146,186),(147,187),(148,188),(149,189),(150,190),(151,228),(152,229),(153,230),(154,231),(155,232),(156,233),(157,234),(158,235),(159,236),(160,237),(161,238),(162,239),(163,240),(164,226),(165,227),(166,211),(167,212),(168,213),(169,214),(170,215),(171,216),(172,217),(173,218),(174,219),(175,220),(176,221),(177,222),(178,223),(179,224),(180,225)])
Matrix representation ►G ⊆ GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 71 | 225 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 52 | 0 | 0 |
| 0 | 0 | 190 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 0 | 0 | 0 | 0 | 191 | 190 |
| 40 | 64 | 0 | 0 | 0 | 0 |
| 39 | 201 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 150 | 106 | 0 | 0 |
| 0 | 0 | 31 | 91 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,71,0,0,0,0,0,225,0,0,0,0,0,0,51,190,0,0,0,0,52,0,0,0,0,0,0,0,240,191,0,0,0,0,240,190],[40,39,0,0,0,0,64,201,0,0,0,0,0,0,0,0,150,31,0,0,0,0,106,91,0,0,1,0,0,0,0,0,0,1,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240] >;
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | ··· | 6G | 8A | ··· | 8H | 10A | ··· | 10G | 12A | ··· | 12H | 15A | 15B | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | ··· | 6 | 8 | ··· | 8 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 2 | ··· | 2 | 30 | ··· | 30 | 4 | ··· | 4 | 10 | ··· | 10 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | Dic3 | M4(2) | C4.Dic3 | F5 | C2×F5 | C3⋊F5 | C22.F5 | C2×C3⋊F5 | C15⋊8M4(2) |
| kernel | C2×C15⋊8M4(2) | C2×C15⋊C8 | C15⋊8M4(2) | C2×C6×Dic5 | C6×Dic5 | C22×C30 | C22×Dic5 | C2×Dic5 | C2×Dic5 | C22×C10 | C30 | C10 | C22×C6 | C2×C6 | C23 | C6 | C22 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 1 | 3 | 3 | 1 | 4 | 8 | 1 | 3 | 2 | 4 | 6 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_{15}\rtimes_8M_{4(2)} % in TeX
G:=Group("C2xC15:8M4(2)"); // GroupNames label
G:=SmallGroup(480,1071);
// by ID
G=gap.SmallGroup(480,1071);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,80,2693,14118,2379]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^15=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^2,b*d=d*b,d*c*d=c^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀