direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12.F5, C30⋊3M4(2), (C2×C60).4C4, C6⋊2(C4.F5), C60.51(C2×C4), (D5×C12).7C4, (C4×D5).90D6, C12.51(C2×F5), (C2×C12).10F5, C15⋊11(C2×M4(2)), (C4×D5).4Dic3, (C2×C20).6Dic3, C15⋊C8⋊11C22, C6.33(C22×F5), C30.71(C22×C4), C10⋊1(C4.Dic3), C20.12(C2×Dic3), D10.13(C2×Dic3), (C2×Dic5).207D6, (C22×D5).8Dic3, C10.2(C22×Dic3), Dic5.15(C2×Dic3), (D5×C12).115C22, (C3×Dic5).63C23, Dic5.49(C22×S3), (C6×Dic5).266C22, C3⋊3(C2×C4.F5), C4.12(C2×C3⋊F5), (D5×C2×C6).14C4, (C2×C4×D5).14S3, (C2×C4).7(C3⋊F5), C5⋊1(C2×C4.Dic3), C2.4(C22×C3⋊F5), (D5×C2×C12).17C2, (C2×C15⋊C8)⋊10C2, (C2×C6).44(C2×F5), (C2×C30).38(C2×C4), C22.17(C2×C3⋊F5), (C6×D5).57(C2×C4), (C3×Dic5).65(C2×C4), (C2×C10).14(C2×Dic3), SmallGroup(480,1061)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 524 in 136 conjugacy classes, 65 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, C6, C6 [×2], C6 [×2], C8 [×4], C2×C4, C2×C4 [×5], C23, D5 [×2], C10, C10 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C15, C2×C8 [×2], M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C3⋊C8 [×4], C2×C12, C2×C12 [×5], C22×C6, C3×D5 [×2], C30, C30 [×2], C2×M4(2), C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8 [×2], C4.Dic3 [×4], C22×C12, C3×Dic5 [×2], C60 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, C4.F5 [×4], C2×C5⋊C8 [×2], C2×C4×D5, C2×C4.Dic3, C15⋊C8 [×4], D5×C12 [×4], C6×Dic5, C2×C60, D5×C2×C6, C2×C4.F5, C12.F5 [×4], C2×C15⋊C8 [×2], D5×C2×C12, C2×C12.F5
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, C4.F5 [×2], C22×F5, C2×C4.Dic3, C2×C3⋊F5 [×3], C2×C4.F5, C12.F5 [×2], C22×C3⋊F5, C2×C12.F5
Generators and relations
G = < a,b,c,d | a2=b12=c5=1, d4=b6, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
►On 240 points
Generators in S240
(1 194)(2 195)(3 196)(4 197)(5 198)(6 199)(7 200)(8 201)(9 202)(10 203)(11 204)(12 193)(13 238)(14 239)(15 240)(16 229)(17 230)(18 231)(19 232)(20 233)(21 234)(22 235)(23 236)(24 237)(25 143)(26 144)(27 133)(28 134)(29 135)(30 136)(31 137)(32 138)(33 139)(34 140)(35 141)(36 142)(37 176)(38 177)(39 178)(40 179)(41 180)(42 169)(43 170)(44 171)(45 172)(46 173)(47 174)(48 175)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 61)(57 62)(58 63)(59 64)(60 65)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 97)(82 98)(83 99)(84 100)(85 208)(86 209)(87 210)(88 211)(89 212)(90 213)(91 214)(92 215)(93 216)(94 205)(95 206)(96 207)(109 190)(110 191)(111 192)(112 181)(113 182)(114 183)(115 184)(116 185)(117 186)(118 187)(119 188)(120 189)(121 159)(122 160)(123 161)(124 162)(125 163)(126 164)(127 165)(128 166)(129 167)(130 168)(131 157)(132 158)(145 224)(146 225)(147 226)(148 227)(149 228)(150 217)(151 218)(152 219)(153 220)(154 221)(155 222)(156 223)
(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)
(1 212 132 150 100)(2 213 121 151 101)(3 214 122 152 102)(4 215 123 153 103)(5 216 124 154 104)(6 205 125 155 105)(7 206 126 156 106)(8 207 127 145 107)(9 208 128 146 108)(10 209 129 147 97)(11 210 130 148 98)(12 211 131 149 99)(13 134 55 43 114)(14 135 56 44 115)(15 136 57 45 116)(16 137 58 46 117)(17 138 59 47 118)(18 139 60 48 119)(19 140 49 37 120)(20 141 50 38 109)(21 142 51 39 110)(22 143 52 40 111)(23 144 53 41 112)(24 133 54 42 113)(25 69 179 192 235)(26 70 180 181 236)(27 71 169 182 237)(28 72 170 183 238)(29 61 171 184 239)(30 62 172 185 240)(31 63 173 186 229)(32 64 174 187 230)(33 65 175 188 231)(34 66 176 189 232)(35 67 177 190 233)(36 68 178 191 234)(73 195 90 159 218)(74 196 91 160 219)(75 197 92 161 220)(76 198 93 162 221)(77 199 94 163 222)(78 200 95 164 223)(79 201 96 165 224)(80 202 85 166 225)(81 203 86 167 226)(82 204 87 168 227)(83 193 88 157 228)(84 194 89 158 217)
(1 141 203 26 7 135 197 32)(2 140 204 25 8 134 198 31)(3 139 193 36 9 133 199 30)(4 138 194 35 10 144 200 29)(5 137 195 34 11 143 201 28)(6 136 196 33 12 142 202 27)(13 221 63 121 19 227 69 127)(14 220 64 132 20 226 70 126)(15 219 65 131 21 225 71 125)(16 218 66 130 22 224 72 124)(17 217 67 129 23 223 61 123)(18 228 68 128 24 222 62 122)(37 82 192 207 43 76 186 213)(38 81 181 206 44 75 187 212)(39 80 182 205 45 74 188 211)(40 79 183 216 46 73 189 210)(41 78 184 215 47 84 190 209)(42 77 185 214 48 83 191 208)(49 168 235 145 55 162 229 151)(50 167 236 156 56 161 230 150)(51 166 237 155 57 160 231 149)(52 165 238 154 58 159 232 148)(53 164 239 153 59 158 233 147)(54 163 240 152 60 157 234 146)(85 169 105 116 91 175 99 110)(86 180 106 115 92 174 100 109)(87 179 107 114 93 173 101 120)(88 178 108 113 94 172 102 119)(89 177 97 112 95 171 103 118)(90 176 98 111 96 170 104 117)
G:=sub<Sym(240)| (1,194)(2,195)(3,196)(4,197)(5,198)(6,199)(7,200)(8,201)(9,202)(10,203)(11,204)(12,193)(13,238)(14,239)(15,240)(16,229)(17,230)(18,231)(19,232)(20,233)(21,234)(22,235)(23,236)(24,237)(25,143)(26,144)(27,133)(28,134)(29,135)(30,136)(31,137)(32,138)(33,139)(34,140)(35,141)(36,142)(37,176)(38,177)(39,178)(40,179)(41,180)(42,169)(43,170)(44,171)(45,172)(46,173)(47,174)(48,175)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,61)(57,62)(58,63)(59,64)(60,65)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,97)(82,98)(83,99)(84,100)(85,208)(86,209)(87,210)(88,211)(89,212)(90,213)(91,214)(92,215)(93,216)(94,205)(95,206)(96,207)(109,190)(110,191)(111,192)(112,181)(113,182)(114,183)(115,184)(116,185)(117,186)(118,187)(119,188)(120,189)(121,159)(122,160)(123,161)(124,162)(125,163)(126,164)(127,165)(128,166)(129,167)(130,168)(131,157)(132,158)(145,224)(146,225)(147,226)(148,227)(149,228)(150,217)(151,218)(152,219)(153,220)(154,221)(155,222)(156,223), (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), (1,212,132,150,100)(2,213,121,151,101)(3,214,122,152,102)(4,215,123,153,103)(5,216,124,154,104)(6,205,125,155,105)(7,206,126,156,106)(8,207,127,145,107)(9,208,128,146,108)(10,209,129,147,97)(11,210,130,148,98)(12,211,131,149,99)(13,134,55,43,114)(14,135,56,44,115)(15,136,57,45,116)(16,137,58,46,117)(17,138,59,47,118)(18,139,60,48,119)(19,140,49,37,120)(20,141,50,38,109)(21,142,51,39,110)(22,143,52,40,111)(23,144,53,41,112)(24,133,54,42,113)(25,69,179,192,235)(26,70,180,181,236)(27,71,169,182,237)(28,72,170,183,238)(29,61,171,184,239)(30,62,172,185,240)(31,63,173,186,229)(32,64,174,187,230)(33,65,175,188,231)(34,66,176,189,232)(35,67,177,190,233)(36,68,178,191,234)(73,195,90,159,218)(74,196,91,160,219)(75,197,92,161,220)(76,198,93,162,221)(77,199,94,163,222)(78,200,95,164,223)(79,201,96,165,224)(80,202,85,166,225)(81,203,86,167,226)(82,204,87,168,227)(83,193,88,157,228)(84,194,89,158,217), (1,141,203,26,7,135,197,32)(2,140,204,25,8,134,198,31)(3,139,193,36,9,133,199,30)(4,138,194,35,10,144,200,29)(5,137,195,34,11,143,201,28)(6,136,196,33,12,142,202,27)(13,221,63,121,19,227,69,127)(14,220,64,132,20,226,70,126)(15,219,65,131,21,225,71,125)(16,218,66,130,22,224,72,124)(17,217,67,129,23,223,61,123)(18,228,68,128,24,222,62,122)(37,82,192,207,43,76,186,213)(38,81,181,206,44,75,187,212)(39,80,182,205,45,74,188,211)(40,79,183,216,46,73,189,210)(41,78,184,215,47,84,190,209)(42,77,185,214,48,83,191,208)(49,168,235,145,55,162,229,151)(50,167,236,156,56,161,230,150)(51,166,237,155,57,160,231,149)(52,165,238,154,58,159,232,148)(53,164,239,153,59,158,233,147)(54,163,240,152,60,157,234,146)(85,169,105,116,91,175,99,110)(86,180,106,115,92,174,100,109)(87,179,107,114,93,173,101,120)(88,178,108,113,94,172,102,119)(89,177,97,112,95,171,103,118)(90,176,98,111,96,170,104,117)>;
G:=Group( (1,194)(2,195)(3,196)(4,197)(5,198)(6,199)(7,200)(8,201)(9,202)(10,203)(11,204)(12,193)(13,238)(14,239)(15,240)(16,229)(17,230)(18,231)(19,232)(20,233)(21,234)(22,235)(23,236)(24,237)(25,143)(26,144)(27,133)(28,134)(29,135)(30,136)(31,137)(32,138)(33,139)(34,140)(35,141)(36,142)(37,176)(38,177)(39,178)(40,179)(41,180)(42,169)(43,170)(44,171)(45,172)(46,173)(47,174)(48,175)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,61)(57,62)(58,63)(59,64)(60,65)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,97)(82,98)(83,99)(84,100)(85,208)(86,209)(87,210)(88,211)(89,212)(90,213)(91,214)(92,215)(93,216)(94,205)(95,206)(96,207)(109,190)(110,191)(111,192)(112,181)(113,182)(114,183)(115,184)(116,185)(117,186)(118,187)(119,188)(120,189)(121,159)(122,160)(123,161)(124,162)(125,163)(126,164)(127,165)(128,166)(129,167)(130,168)(131,157)(132,158)(145,224)(146,225)(147,226)(148,227)(149,228)(150,217)(151,218)(152,219)(153,220)(154,221)(155,222)(156,223), (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), (1,212,132,150,100)(2,213,121,151,101)(3,214,122,152,102)(4,215,123,153,103)(5,216,124,154,104)(6,205,125,155,105)(7,206,126,156,106)(8,207,127,145,107)(9,208,128,146,108)(10,209,129,147,97)(11,210,130,148,98)(12,211,131,149,99)(13,134,55,43,114)(14,135,56,44,115)(15,136,57,45,116)(16,137,58,46,117)(17,138,59,47,118)(18,139,60,48,119)(19,140,49,37,120)(20,141,50,38,109)(21,142,51,39,110)(22,143,52,40,111)(23,144,53,41,112)(24,133,54,42,113)(25,69,179,192,235)(26,70,180,181,236)(27,71,169,182,237)(28,72,170,183,238)(29,61,171,184,239)(30,62,172,185,240)(31,63,173,186,229)(32,64,174,187,230)(33,65,175,188,231)(34,66,176,189,232)(35,67,177,190,233)(36,68,178,191,234)(73,195,90,159,218)(74,196,91,160,219)(75,197,92,161,220)(76,198,93,162,221)(77,199,94,163,222)(78,200,95,164,223)(79,201,96,165,224)(80,202,85,166,225)(81,203,86,167,226)(82,204,87,168,227)(83,193,88,157,228)(84,194,89,158,217), (1,141,203,26,7,135,197,32)(2,140,204,25,8,134,198,31)(3,139,193,36,9,133,199,30)(4,138,194,35,10,144,200,29)(5,137,195,34,11,143,201,28)(6,136,196,33,12,142,202,27)(13,221,63,121,19,227,69,127)(14,220,64,132,20,226,70,126)(15,219,65,131,21,225,71,125)(16,218,66,130,22,224,72,124)(17,217,67,129,23,223,61,123)(18,228,68,128,24,222,62,122)(37,82,192,207,43,76,186,213)(38,81,181,206,44,75,187,212)(39,80,182,205,45,74,188,211)(40,79,183,216,46,73,189,210)(41,78,184,215,47,84,190,209)(42,77,185,214,48,83,191,208)(49,168,235,145,55,162,229,151)(50,167,236,156,56,161,230,150)(51,166,237,155,57,160,231,149)(52,165,238,154,58,159,232,148)(53,164,239,153,59,158,233,147)(54,163,240,152,60,157,234,146)(85,169,105,116,91,175,99,110)(86,180,106,115,92,174,100,109)(87,179,107,114,93,173,101,120)(88,178,108,113,94,172,102,119)(89,177,97,112,95,171,103,118)(90,176,98,111,96,170,104,117) );
G=PermutationGroup([(1,194),(2,195),(3,196),(4,197),(5,198),(6,199),(7,200),(8,201),(9,202),(10,203),(11,204),(12,193),(13,238),(14,239),(15,240),(16,229),(17,230),(18,231),(19,232),(20,233),(21,234),(22,235),(23,236),(24,237),(25,143),(26,144),(27,133),(28,134),(29,135),(30,136),(31,137),(32,138),(33,139),(34,140),(35,141),(36,142),(37,176),(38,177),(39,178),(40,179),(41,180),(42,169),(43,170),(44,171),(45,172),(46,173),(47,174),(48,175),(49,66),(50,67),(51,68),(52,69),(53,70),(54,71),(55,72),(56,61),(57,62),(58,63),(59,64),(60,65),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,97),(82,98),(83,99),(84,100),(85,208),(86,209),(87,210),(88,211),(89,212),(90,213),(91,214),(92,215),(93,216),(94,205),(95,206),(96,207),(109,190),(110,191),(111,192),(112,181),(113,182),(114,183),(115,184),(116,185),(117,186),(118,187),(119,188),(120,189),(121,159),(122,160),(123,161),(124,162),(125,163),(126,164),(127,165),(128,166),(129,167),(130,168),(131,157),(132,158),(145,224),(146,225),(147,226),(148,227),(149,228),(150,217),(151,218),(152,219),(153,220),(154,221),(155,222),(156,223)], [(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)], [(1,212,132,150,100),(2,213,121,151,101),(3,214,122,152,102),(4,215,123,153,103),(5,216,124,154,104),(6,205,125,155,105),(7,206,126,156,106),(8,207,127,145,107),(9,208,128,146,108),(10,209,129,147,97),(11,210,130,148,98),(12,211,131,149,99),(13,134,55,43,114),(14,135,56,44,115),(15,136,57,45,116),(16,137,58,46,117),(17,138,59,47,118),(18,139,60,48,119),(19,140,49,37,120),(20,141,50,38,109),(21,142,51,39,110),(22,143,52,40,111),(23,144,53,41,112),(24,133,54,42,113),(25,69,179,192,235),(26,70,180,181,236),(27,71,169,182,237),(28,72,170,183,238),(29,61,171,184,239),(30,62,172,185,240),(31,63,173,186,229),(32,64,174,187,230),(33,65,175,188,231),(34,66,176,189,232),(35,67,177,190,233),(36,68,178,191,234),(73,195,90,159,218),(74,196,91,160,219),(75,197,92,161,220),(76,198,93,162,221),(77,199,94,163,222),(78,200,95,164,223),(79,201,96,165,224),(80,202,85,166,225),(81,203,86,167,226),(82,204,87,168,227),(83,193,88,157,228),(84,194,89,158,217)], [(1,141,203,26,7,135,197,32),(2,140,204,25,8,134,198,31),(3,139,193,36,9,133,199,30),(4,138,194,35,10,144,200,29),(5,137,195,34,11,143,201,28),(6,136,196,33,12,142,202,27),(13,221,63,121,19,227,69,127),(14,220,64,132,20,226,70,126),(15,219,65,131,21,225,71,125),(16,218,66,130,22,224,72,124),(17,217,67,129,23,223,61,123),(18,228,68,128,24,222,62,122),(37,82,192,207,43,76,186,213),(38,81,181,206,44,75,187,212),(39,80,182,205,45,74,188,211),(40,79,183,216,46,73,189,210),(41,78,184,215,47,84,190,209),(42,77,185,214,48,83,191,208),(49,168,235,145,55,162,229,151),(50,167,236,156,56,161,230,150),(51,166,237,155,57,160,231,149),(52,165,238,154,58,159,232,148),(53,164,239,153,59,158,233,147),(54,163,240,152,60,157,234,146),(85,169,105,116,91,175,99,110),(86,180,106,115,92,174,100,109),(87,179,107,114,93,173,101,120),(88,178,108,113,94,172,102,119),(89,177,97,112,95,171,103,118),(90,176,98,111,96,170,104,117)])
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 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 84 | 0 | 224 | 224 |
| 0 | 0 | 17 | 101 | 17 | 0 |
| 0 | 0 | 0 | 17 | 101 | 17 |
| 0 | 0 | 224 | 224 | 0 | 84 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 81 | 21 | 0 | 0 | 0 | 0 |
| 181 | 160 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 54 | 211 | 84 |
| 0 | 0 | 157 | 30 | 187 | 214 |
| 0 | 0 | 157 | 184 | 211 | 127 |
| 0 | 0 | 27 | 184 | 57 | 214 |
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],[1,240,0,0,0,0,1,0,0,0,0,0,0,0,84,17,0,224,0,0,0,101,17,224,0,0,224,17,101,0,0,0,224,0,17,84],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[81,181,0,0,0,0,21,160,0,0,0,0,0,0,27,157,157,27,0,0,54,30,184,184,0,0,211,187,211,57,0,0,84,214,127,214] >;
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | ··· | 8H | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 4 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 30 | ··· | 30 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | - | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | Dic3 | D6 | D6 | Dic3 | Dic3 | M4(2) | C4.Dic3 | F5 | C2×F5 | C2×F5 | C3⋊F5 | C4.F5 | C2×C3⋊F5 | C2×C3⋊F5 | C12.F5 |
| kernel | C2×C12.F5 | C12.F5 | C2×C15⋊C8 | D5×C2×C12 | D5×C12 | C2×C60 | D5×C2×C6 | C2×C4×D5 | C4×D5 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C10 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 8 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_{12}.F_5 % in TeX
G:=Group("C2xC12.F5"); // GroupNames label
G:=SmallGroup(480,1061);
// by ID
G=gap.SmallGroup(480,1061);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,100,80,2693,14118,2379]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^5=1,d^4=b^6,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀