direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D6.F5, C30⋊1M4(2), C5⋊C8⋊6D6, C10⋊2(C8⋊S3), C15⋊6(C2×M4(2)), D6.10(C2×F5), C15⋊C8⋊9C22, (S3×Dic5).6C4, C6⋊1(C22.F5), (C22×S3).4F5, C22.22(S3×F5), C6.27(C22×F5), C30.27(C22×C4), Dic5.31(C4×S3), (C2×Dic15).11C4, Dic15.16(C2×C4), (C2×Dic5).151D6, (C3×Dic5).37C23, (S3×Dic5).18C22, Dic5.39(C22×S3), (C6×Dic5).148C22, (C6×C5⋊C8)⋊6C2, (C2×C5⋊C8)⋊4S3, C5⋊3(C2×C8⋊S3), C2.27(C2×S3×F5), (C3×C5⋊C8)⋊9C22, (S3×C2×C10).6C4, C10.27(S3×C2×C4), (C2×C15⋊C8)⋊6C2, C3⋊1(C2×C22.F5), (C2×C6).24(C2×F5), (C2×C10).22(C4×S3), (C2×C30).22(C2×C4), (C2×S3×Dic5).13C2, (S3×C10).15(C2×C4), (C3×Dic5).29(C2×C4), SmallGroup(480,1008)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 564 in 136 conjugacy classes, 54 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×4], C22, C22 [×4], C5, S3 [×2], C6, C6 [×2], C8 [×4], C2×C4 [×6], C23, C10, C10 [×2], C10 [×2], Dic3 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C15, C2×C8 [×2], M4(2) [×4], C22×C4, Dic5 [×2], Dic5 [×2], C2×C10, C2×C10 [×4], C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C5×S3 [×2], C30, C30 [×2], C2×M4(2), C5⋊C8 [×2], C5⋊C8 [×2], C2×Dic5, C2×Dic5 [×5], C22×C10, C8⋊S3 [×4], C2×C3⋊C8, C2×C24, S3×C2×C4, C3×Dic5 [×2], Dic15 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C5⋊C8, C2×C5⋊C8, C22.F5 [×4], C22×Dic5, C2×C8⋊S3, C3×C5⋊C8 [×2], C15⋊C8 [×2], S3×Dic5 [×4], C6×Dic5, C2×Dic15, S3×C2×C10, C2×C22.F5, D6.F5 [×4], C6×C5⋊C8, C2×C15⋊C8, C2×S3×Dic5, C2×D6.F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], M4(2) [×2], C22×C4, F5, C4×S3 [×2], C22×S3, C2×M4(2), C2×F5 [×3], C8⋊S3 [×2], S3×C2×C4, C22.F5 [×2], C22×F5, C2×C8⋊S3, S3×F5, C2×C22.F5, D6.F5 [×2], C2×S3×F5, C2×D6.F5
Generators and relations
G = < a,b,c,d,e | a2=b6=c2=d5=1, e4=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d3 >
►On 240 points
Generators in S240
(1 158)(2 159)(3 160)(4 153)(5 154)(6 155)(7 156)(8 157)(9 37)(10 38)(11 39)(12 40)(13 33)(14 34)(15 35)(16 36)(17 127)(18 128)(19 121)(20 122)(21 123)(22 124)(23 125)(24 126)(25 170)(26 171)(27 172)(28 173)(29 174)(30 175)(31 176)(32 169)(41 190)(42 191)(43 192)(44 185)(45 186)(46 187)(47 188)(48 189)(49 131)(50 132)(51 133)(52 134)(53 135)(54 136)(55 129)(56 130)(57 152)(58 145)(59 146)(60 147)(61 148)(62 149)(63 150)(64 151)(65 116)(66 117)(67 118)(68 119)(69 120)(70 113)(71 114)(72 115)(73 181)(74 182)(75 183)(76 184)(77 177)(78 178)(79 179)(80 180)(81 211)(82 212)(83 213)(84 214)(85 215)(86 216)(87 209)(88 210)(89 222)(90 223)(91 224)(92 217)(93 218)(94 219)(95 220)(96 221)(97 166)(98 167)(99 168)(100 161)(101 162)(102 163)(103 164)(104 165)(105 202)(106 203)(107 204)(108 205)(109 206)(110 207)(111 208)(112 201)(137 233)(138 234)(139 235)(140 236)(141 237)(142 238)(143 239)(144 240)(193 225)(194 226)(195 227)(196 228)(197 229)(198 230)(199 231)(200 232)
(1 115 39 5 119 35)(2 116 40 6 120 36)(3 117 33 7 113 37)(4 118 34 8 114 38)(9 160 66 13 156 70)(10 153 67 14 157 71)(11 154 68 15 158 72)(12 155 69 16 159 65)(17 187 225 21 191 229)(18 188 226 22 192 230)(19 189 227 23 185 231)(20 190 228 24 186 232)(25 166 223 29 162 219)(26 167 224 30 163 220)(27 168 217 31 164 221)(28 161 218 32 165 222)(41 196 126 45 200 122)(42 197 127 46 193 123)(43 198 128 47 194 124)(44 199 121 48 195 125)(49 106 137 53 110 141)(50 107 138 54 111 142)(51 108 139 55 112 143)(52 109 140 56 105 144)(57 76 213 61 80 209)(58 77 214 62 73 210)(59 78 215 63 74 211)(60 79 216 64 75 212)(81 146 178 85 150 182)(82 147 179 86 151 183)(83 148 180 87 152 184)(84 149 181 88 145 177)(89 173 100 93 169 104)(90 174 101 94 170 97)(91 175 102 95 171 98)(92 176 103 96 172 99)(129 201 239 133 205 235)(130 202 240 134 206 236)(131 203 233 135 207 237)(132 204 234 136 208 238)
(1 35)(2 40)(3 37)(4 34)(5 39)(6 36)(7 33)(8 38)(9 160)(10 157)(11 154)(12 159)(13 156)(14 153)(15 158)(16 155)(18 22)(20 24)(25 223)(26 220)(27 217)(28 222)(29 219)(30 224)(31 221)(32 218)(41 196)(42 193)(43 198)(44 195)(45 200)(46 197)(47 194)(48 199)(49 53)(51 55)(57 61)(59 63)(66 70)(68 72)(73 214)(74 211)(75 216)(76 213)(77 210)(78 215)(79 212)(80 209)(81 182)(82 179)(83 184)(84 181)(85 178)(86 183)(87 180)(88 177)(89 173)(90 170)(91 175)(92 172)(93 169)(94 174)(95 171)(96 176)(98 102)(100 104)(105 140)(106 137)(107 142)(108 139)(109 144)(110 141)(111 138)(112 143)(113 117)(115 119)(122 126)(124 128)(129 133)(131 135)(146 150)(148 152)(161 165)(163 167)(185 227)(186 232)(187 229)(188 226)(189 231)(190 228)(191 225)(192 230)(201 239)(202 236)(203 233)(204 238)(205 235)(206 240)(207 237)(208 234)
(1 173 137 209 228)(2 210 174 229 138)(3 230 211 139 175)(4 140 231 176 212)(5 169 141 213 232)(6 214 170 225 142)(7 226 215 143 171)(8 144 227 172 216)(9 43 182 205 224)(10 206 44 217 183)(11 218 207 184 45)(12 177 219 46 208)(13 47 178 201 220)(14 202 48 221 179)(15 222 203 180 41)(16 181 223 42 204)(17 54 116 58 101)(18 59 55 102 117)(19 103 60 118 56)(20 119 104 49 61)(21 50 120 62 97)(22 63 51 98 113)(23 99 64 114 52)(24 115 100 53 57)(25 193 238 155 84)(26 156 194 85 239)(27 86 157 240 195)(28 233 87 196 158)(29 197 234 159 88)(30 160 198 81 235)(31 82 153 236 199)(32 237 83 200 154)(33 188 78 112 95)(34 105 189 96 79)(35 89 106 80 190)(36 73 90 191 107)(37 192 74 108 91)(38 109 185 92 75)(39 93 110 76 186)(40 77 94 187 111)(65 145 162 127 136)(66 128 146 129 163)(67 130 121 164 147)(68 165 131 148 122)(69 149 166 123 132)(70 124 150 133 167)(71 134 125 168 151)(72 161 135 152 126)
(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)
G:=sub<Sym(240)| (1,158)(2,159)(3,160)(4,153)(5,154)(6,155)(7,156)(8,157)(9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(17,127)(18,128)(19,121)(20,122)(21,123)(22,124)(23,125)(24,126)(25,170)(26,171)(27,172)(28,173)(29,174)(30,175)(31,176)(32,169)(41,190)(42,191)(43,192)(44,185)(45,186)(46,187)(47,188)(48,189)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,129)(56,130)(57,152)(58,145)(59,146)(60,147)(61,148)(62,149)(63,150)(64,151)(65,116)(66,117)(67,118)(68,119)(69,120)(70,113)(71,114)(72,115)(73,181)(74,182)(75,183)(76,184)(77,177)(78,178)(79,179)(80,180)(81,211)(82,212)(83,213)(84,214)(85,215)(86,216)(87,209)(88,210)(89,222)(90,223)(91,224)(92,217)(93,218)(94,219)(95,220)(96,221)(97,166)(98,167)(99,168)(100,161)(101,162)(102,163)(103,164)(104,165)(105,202)(106,203)(107,204)(108,205)(109,206)(110,207)(111,208)(112,201)(137,233)(138,234)(139,235)(140,236)(141,237)(142,238)(143,239)(144,240)(193,225)(194,226)(195,227)(196,228)(197,229)(198,230)(199,231)(200,232), (1,115,39,5,119,35)(2,116,40,6,120,36)(3,117,33,7,113,37)(4,118,34,8,114,38)(9,160,66,13,156,70)(10,153,67,14,157,71)(11,154,68,15,158,72)(12,155,69,16,159,65)(17,187,225,21,191,229)(18,188,226,22,192,230)(19,189,227,23,185,231)(20,190,228,24,186,232)(25,166,223,29,162,219)(26,167,224,30,163,220)(27,168,217,31,164,221)(28,161,218,32,165,222)(41,196,126,45,200,122)(42,197,127,46,193,123)(43,198,128,47,194,124)(44,199,121,48,195,125)(49,106,137,53,110,141)(50,107,138,54,111,142)(51,108,139,55,112,143)(52,109,140,56,105,144)(57,76,213,61,80,209)(58,77,214,62,73,210)(59,78,215,63,74,211)(60,79,216,64,75,212)(81,146,178,85,150,182)(82,147,179,86,151,183)(83,148,180,87,152,184)(84,149,181,88,145,177)(89,173,100,93,169,104)(90,174,101,94,170,97)(91,175,102,95,171,98)(92,176,103,96,172,99)(129,201,239,133,205,235)(130,202,240,134,206,236)(131,203,233,135,207,237)(132,204,234,136,208,238), (1,35)(2,40)(3,37)(4,34)(5,39)(6,36)(7,33)(8,38)(9,160)(10,157)(11,154)(12,159)(13,156)(14,153)(15,158)(16,155)(18,22)(20,24)(25,223)(26,220)(27,217)(28,222)(29,219)(30,224)(31,221)(32,218)(41,196)(42,193)(43,198)(44,195)(45,200)(46,197)(47,194)(48,199)(49,53)(51,55)(57,61)(59,63)(66,70)(68,72)(73,214)(74,211)(75,216)(76,213)(77,210)(78,215)(79,212)(80,209)(81,182)(82,179)(83,184)(84,181)(85,178)(86,183)(87,180)(88,177)(89,173)(90,170)(91,175)(92,172)(93,169)(94,174)(95,171)(96,176)(98,102)(100,104)(105,140)(106,137)(107,142)(108,139)(109,144)(110,141)(111,138)(112,143)(113,117)(115,119)(122,126)(124,128)(129,133)(131,135)(146,150)(148,152)(161,165)(163,167)(185,227)(186,232)(187,229)(188,226)(189,231)(190,228)(191,225)(192,230)(201,239)(202,236)(203,233)(204,238)(205,235)(206,240)(207,237)(208,234), (1,173,137,209,228)(2,210,174,229,138)(3,230,211,139,175)(4,140,231,176,212)(5,169,141,213,232)(6,214,170,225,142)(7,226,215,143,171)(8,144,227,172,216)(9,43,182,205,224)(10,206,44,217,183)(11,218,207,184,45)(12,177,219,46,208)(13,47,178,201,220)(14,202,48,221,179)(15,222,203,180,41)(16,181,223,42,204)(17,54,116,58,101)(18,59,55,102,117)(19,103,60,118,56)(20,119,104,49,61)(21,50,120,62,97)(22,63,51,98,113)(23,99,64,114,52)(24,115,100,53,57)(25,193,238,155,84)(26,156,194,85,239)(27,86,157,240,195)(28,233,87,196,158)(29,197,234,159,88)(30,160,198,81,235)(31,82,153,236,199)(32,237,83,200,154)(33,188,78,112,95)(34,105,189,96,79)(35,89,106,80,190)(36,73,90,191,107)(37,192,74,108,91)(38,109,185,92,75)(39,93,110,76,186)(40,77,94,187,111)(65,145,162,127,136)(66,128,146,129,163)(67,130,121,164,147)(68,165,131,148,122)(69,149,166,123,132)(70,124,150,133,167)(71,134,125,168,151)(72,161,135,152,126), (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)>;
G:=Group( (1,158)(2,159)(3,160)(4,153)(5,154)(6,155)(7,156)(8,157)(9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(17,127)(18,128)(19,121)(20,122)(21,123)(22,124)(23,125)(24,126)(25,170)(26,171)(27,172)(28,173)(29,174)(30,175)(31,176)(32,169)(41,190)(42,191)(43,192)(44,185)(45,186)(46,187)(47,188)(48,189)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,129)(56,130)(57,152)(58,145)(59,146)(60,147)(61,148)(62,149)(63,150)(64,151)(65,116)(66,117)(67,118)(68,119)(69,120)(70,113)(71,114)(72,115)(73,181)(74,182)(75,183)(76,184)(77,177)(78,178)(79,179)(80,180)(81,211)(82,212)(83,213)(84,214)(85,215)(86,216)(87,209)(88,210)(89,222)(90,223)(91,224)(92,217)(93,218)(94,219)(95,220)(96,221)(97,166)(98,167)(99,168)(100,161)(101,162)(102,163)(103,164)(104,165)(105,202)(106,203)(107,204)(108,205)(109,206)(110,207)(111,208)(112,201)(137,233)(138,234)(139,235)(140,236)(141,237)(142,238)(143,239)(144,240)(193,225)(194,226)(195,227)(196,228)(197,229)(198,230)(199,231)(200,232), (1,115,39,5,119,35)(2,116,40,6,120,36)(3,117,33,7,113,37)(4,118,34,8,114,38)(9,160,66,13,156,70)(10,153,67,14,157,71)(11,154,68,15,158,72)(12,155,69,16,159,65)(17,187,225,21,191,229)(18,188,226,22,192,230)(19,189,227,23,185,231)(20,190,228,24,186,232)(25,166,223,29,162,219)(26,167,224,30,163,220)(27,168,217,31,164,221)(28,161,218,32,165,222)(41,196,126,45,200,122)(42,197,127,46,193,123)(43,198,128,47,194,124)(44,199,121,48,195,125)(49,106,137,53,110,141)(50,107,138,54,111,142)(51,108,139,55,112,143)(52,109,140,56,105,144)(57,76,213,61,80,209)(58,77,214,62,73,210)(59,78,215,63,74,211)(60,79,216,64,75,212)(81,146,178,85,150,182)(82,147,179,86,151,183)(83,148,180,87,152,184)(84,149,181,88,145,177)(89,173,100,93,169,104)(90,174,101,94,170,97)(91,175,102,95,171,98)(92,176,103,96,172,99)(129,201,239,133,205,235)(130,202,240,134,206,236)(131,203,233,135,207,237)(132,204,234,136,208,238), (1,35)(2,40)(3,37)(4,34)(5,39)(6,36)(7,33)(8,38)(9,160)(10,157)(11,154)(12,159)(13,156)(14,153)(15,158)(16,155)(18,22)(20,24)(25,223)(26,220)(27,217)(28,222)(29,219)(30,224)(31,221)(32,218)(41,196)(42,193)(43,198)(44,195)(45,200)(46,197)(47,194)(48,199)(49,53)(51,55)(57,61)(59,63)(66,70)(68,72)(73,214)(74,211)(75,216)(76,213)(77,210)(78,215)(79,212)(80,209)(81,182)(82,179)(83,184)(84,181)(85,178)(86,183)(87,180)(88,177)(89,173)(90,170)(91,175)(92,172)(93,169)(94,174)(95,171)(96,176)(98,102)(100,104)(105,140)(106,137)(107,142)(108,139)(109,144)(110,141)(111,138)(112,143)(113,117)(115,119)(122,126)(124,128)(129,133)(131,135)(146,150)(148,152)(161,165)(163,167)(185,227)(186,232)(187,229)(188,226)(189,231)(190,228)(191,225)(192,230)(201,239)(202,236)(203,233)(204,238)(205,235)(206,240)(207,237)(208,234), (1,173,137,209,228)(2,210,174,229,138)(3,230,211,139,175)(4,140,231,176,212)(5,169,141,213,232)(6,214,170,225,142)(7,226,215,143,171)(8,144,227,172,216)(9,43,182,205,224)(10,206,44,217,183)(11,218,207,184,45)(12,177,219,46,208)(13,47,178,201,220)(14,202,48,221,179)(15,222,203,180,41)(16,181,223,42,204)(17,54,116,58,101)(18,59,55,102,117)(19,103,60,118,56)(20,119,104,49,61)(21,50,120,62,97)(22,63,51,98,113)(23,99,64,114,52)(24,115,100,53,57)(25,193,238,155,84)(26,156,194,85,239)(27,86,157,240,195)(28,233,87,196,158)(29,197,234,159,88)(30,160,198,81,235)(31,82,153,236,199)(32,237,83,200,154)(33,188,78,112,95)(34,105,189,96,79)(35,89,106,80,190)(36,73,90,191,107)(37,192,74,108,91)(38,109,185,92,75)(39,93,110,76,186)(40,77,94,187,111)(65,145,162,127,136)(66,128,146,129,163)(67,130,121,164,147)(68,165,131,148,122)(69,149,166,123,132)(70,124,150,133,167)(71,134,125,168,151)(72,161,135,152,126), (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) );
G=PermutationGroup([(1,158),(2,159),(3,160),(4,153),(5,154),(6,155),(7,156),(8,157),(9,37),(10,38),(11,39),(12,40),(13,33),(14,34),(15,35),(16,36),(17,127),(18,128),(19,121),(20,122),(21,123),(22,124),(23,125),(24,126),(25,170),(26,171),(27,172),(28,173),(29,174),(30,175),(31,176),(32,169),(41,190),(42,191),(43,192),(44,185),(45,186),(46,187),(47,188),(48,189),(49,131),(50,132),(51,133),(52,134),(53,135),(54,136),(55,129),(56,130),(57,152),(58,145),(59,146),(60,147),(61,148),(62,149),(63,150),(64,151),(65,116),(66,117),(67,118),(68,119),(69,120),(70,113),(71,114),(72,115),(73,181),(74,182),(75,183),(76,184),(77,177),(78,178),(79,179),(80,180),(81,211),(82,212),(83,213),(84,214),(85,215),(86,216),(87,209),(88,210),(89,222),(90,223),(91,224),(92,217),(93,218),(94,219),(95,220),(96,221),(97,166),(98,167),(99,168),(100,161),(101,162),(102,163),(103,164),(104,165),(105,202),(106,203),(107,204),(108,205),(109,206),(110,207),(111,208),(112,201),(137,233),(138,234),(139,235),(140,236),(141,237),(142,238),(143,239),(144,240),(193,225),(194,226),(195,227),(196,228),(197,229),(198,230),(199,231),(200,232)], [(1,115,39,5,119,35),(2,116,40,6,120,36),(3,117,33,7,113,37),(4,118,34,8,114,38),(9,160,66,13,156,70),(10,153,67,14,157,71),(11,154,68,15,158,72),(12,155,69,16,159,65),(17,187,225,21,191,229),(18,188,226,22,192,230),(19,189,227,23,185,231),(20,190,228,24,186,232),(25,166,223,29,162,219),(26,167,224,30,163,220),(27,168,217,31,164,221),(28,161,218,32,165,222),(41,196,126,45,200,122),(42,197,127,46,193,123),(43,198,128,47,194,124),(44,199,121,48,195,125),(49,106,137,53,110,141),(50,107,138,54,111,142),(51,108,139,55,112,143),(52,109,140,56,105,144),(57,76,213,61,80,209),(58,77,214,62,73,210),(59,78,215,63,74,211),(60,79,216,64,75,212),(81,146,178,85,150,182),(82,147,179,86,151,183),(83,148,180,87,152,184),(84,149,181,88,145,177),(89,173,100,93,169,104),(90,174,101,94,170,97),(91,175,102,95,171,98),(92,176,103,96,172,99),(129,201,239,133,205,235),(130,202,240,134,206,236),(131,203,233,135,207,237),(132,204,234,136,208,238)], [(1,35),(2,40),(3,37),(4,34),(5,39),(6,36),(7,33),(8,38),(9,160),(10,157),(11,154),(12,159),(13,156),(14,153),(15,158),(16,155),(18,22),(20,24),(25,223),(26,220),(27,217),(28,222),(29,219),(30,224),(31,221),(32,218),(41,196),(42,193),(43,198),(44,195),(45,200),(46,197),(47,194),(48,199),(49,53),(51,55),(57,61),(59,63),(66,70),(68,72),(73,214),(74,211),(75,216),(76,213),(77,210),(78,215),(79,212),(80,209),(81,182),(82,179),(83,184),(84,181),(85,178),(86,183),(87,180),(88,177),(89,173),(90,170),(91,175),(92,172),(93,169),(94,174),(95,171),(96,176),(98,102),(100,104),(105,140),(106,137),(107,142),(108,139),(109,144),(110,141),(111,138),(112,143),(113,117),(115,119),(122,126),(124,128),(129,133),(131,135),(146,150),(148,152),(161,165),(163,167),(185,227),(186,232),(187,229),(188,226),(189,231),(190,228),(191,225),(192,230),(201,239),(202,236),(203,233),(204,238),(205,235),(206,240),(207,237),(208,234)], [(1,173,137,209,228),(2,210,174,229,138),(3,230,211,139,175),(4,140,231,176,212),(5,169,141,213,232),(6,214,170,225,142),(7,226,215,143,171),(8,144,227,172,216),(9,43,182,205,224),(10,206,44,217,183),(11,218,207,184,45),(12,177,219,46,208),(13,47,178,201,220),(14,202,48,221,179),(15,222,203,180,41),(16,181,223,42,204),(17,54,116,58,101),(18,59,55,102,117),(19,103,60,118,56),(20,119,104,49,61),(21,50,120,62,97),(22,63,51,98,113),(23,99,64,114,52),(24,115,100,53,57),(25,193,238,155,84),(26,156,194,85,239),(27,86,157,240,195),(28,233,87,196,158),(29,197,234,159,88),(30,160,198,81,235),(31,82,153,236,199),(32,237,83,200,154),(33,188,78,112,95),(34,105,189,96,79),(35,89,106,80,190),(36,73,90,191,107),(37,192,74,108,91),(38,109,185,92,75),(39,93,110,76,186),(40,77,94,187,111),(65,145,162,127,136),(66,128,146,129,163),(67,130,121,164,147),(68,165,131,148,122),(69,149,166,123,132),(70,124,150,133,167),(71,134,125,168,151),(72,161,135,152,126)], [(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)])
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 |
| 240 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 152 | 133 | 1 | 0 |
| 0 | 0 | 180 | 195 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 51 | 0 | 0 |
| 0 | 0 | 188 | 63 | 189 | 190 |
| 0 | 0 | 119 | 142 | 52 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 11 | 139 | 239 |
| 0 | 0 | 2 | 220 | 4 | 2 |
| 0 | 0 | 77 | 161 | 15 | 19 |
| 0 | 0 | 124 | 133 | 114 | 226 |
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],[240,240,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,240,0,0,0,0,0,1,0,0,0,0,0,0,240,0,152,180,0,0,0,240,133,195,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,188,119,0,0,240,51,63,142,0,0,0,0,189,52,0,0,0,0,190,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,2,77,124,0,0,11,220,161,133,0,0,139,4,15,114,0,0,239,2,19,226] >;
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 12A | 12B | 12C | 12D | 15 | 24A | ··· | 24H | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 24 | ··· | 24 | 30 | 30 | 30 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 5 | 5 | 5 | 5 | 30 | 30 | 4 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 10 | 10 | 10 | 10 | 8 | 10 | ··· | 10 | 8 | 8 | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | D6 | M4(2) | C4×S3 | C4×S3 | C8⋊S3 | F5 | C2×F5 | C2×F5 | C22.F5 | S3×F5 | D6.F5 | C2×S3×F5 |
| kernel | C2×D6.F5 | D6.F5 | C6×C5⋊C8 | C2×C15⋊C8 | C2×S3×Dic5 | S3×Dic5 | C2×Dic15 | S3×C2×C10 | C2×C5⋊C8 | C5⋊C8 | C2×Dic5 | C30 | Dic5 | C2×C10 | C10 | C22×S3 | D6 | C2×C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 4 | 2 | 2 | 8 | 1 | 2 | 1 | 4 | 1 | 2 | 1 |
In GAP, Magma, Sage, TeX
C_2\times D_6.F_5
% in TeX
G:=Group("C2xD6.F5"); // GroupNames label
G:=SmallGroup(480,1008);
// by ID
G=gap.SmallGroup(480,1008);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,80,1356,9414,2379]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^2=d^5=1,e^4=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=d^3>;
// generators/relations
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