direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C4⋊D12, C20⋊6D12, C60⋊25D4, C12⋊4(C5×D4), C4⋊1(C5×D12), (C4×C12)⋊4C10, (C4×C60)⋊16C2, (C4×C20)⋊14S3, C42⋊6(C5×S3), C6.3(D4×C10), (C2×D12)⋊1C10, C15⋊7(C4⋊1D4), C2.5(C10×D12), (C10×D12)⋊17C2, (C2×C20).429D6, C10.74(C2×D12), C30.290(C2×D4), (C2×C30).394C23, (C2×C60).526C22, C3⋊1(C5×C4⋊1D4), (C2×C4).78(S3×C10), C22.36(S3×C2×C10), (C2×C12).91(C2×C10), (S3×C2×C10).63C22, (C22×S3).1(C2×C10), (C2×C6).15(C22×C10), (C2×C10).328(C22×S3), SmallGroup(480,753)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4⋊D12
G = < a,b,c,d | a5=b4=c12=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 660 in 216 conjugacy classes, 82 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, D4, C23, C10, C10, C12, D6, C2×C6, C15, C42, C2×D4, C20, C2×C10, C2×C10, D12, C2×C12, C22×S3, C5×S3, C30, C4⋊1D4, C2×C20, C5×D4, C22×C10, C4×C12, C2×D12, C60, S3×C10, C2×C30, C4×C20, D4×C10, C4⋊D12, C5×D12, C2×C60, S3×C2×C10, C5×C4⋊1D4, C4×C60, C10×D12, C5×C4⋊D12
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, D12, C22×S3, C5×S3, C4⋊1D4, C5×D4, C22×C10, C2×D12, S3×C10, D4×C10, C4⋊D12, C5×D12, S3×C2×C10, C5×C4⋊1D4, C10×D12, C5×C4⋊D12
►On 240 points
Generators in S240
(1 187 195 148 210)(2 188 196 149 211)(3 189 197 150 212)(4 190 198 151 213)(5 191 199 152 214)(6 192 200 153 215)(7 181 201 154 216)(8 182 202 155 205)(9 183 203 156 206)(10 184 204 145 207)(11 185 193 146 208)(12 186 194 147 209)(13 66 31 38 122)(14 67 32 39 123)(15 68 33 40 124)(16 69 34 41 125)(17 70 35 42 126)(18 71 36 43 127)(19 72 25 44 128)(20 61 26 45 129)(21 62 27 46 130)(22 63 28 47 131)(23 64 29 48 132)(24 65 30 37 121)(49 169 220 86 80)(50 170 221 87 81)(51 171 222 88 82)(52 172 223 89 83)(53 173 224 90 84)(54 174 225 91 73)(55 175 226 92 74)(56 176 227 93 75)(57 177 228 94 76)(58 178 217 95 77)(59 179 218 96 78)(60 180 219 85 79)(97 167 115 233 134)(98 168 116 234 135)(99 157 117 235 136)(100 158 118 236 137)(101 159 119 237 138)(102 160 120 238 139)(103 161 109 239 140)(104 162 110 240 141)(105 163 111 229 142)(106 164 112 230 143)(107 165 113 231 144)(108 166 114 232 133)
(1 34 73 236)(2 35 74 237)(3 36 75 238)(4 25 76 239)(5 26 77 240)(6 27 78 229)(7 28 79 230)(8 29 80 231)(9 30 81 232)(10 31 82 233)(11 32 83 234)(12 33 84 235)(13 222 167 145)(14 223 168 146)(15 224 157 147)(16 225 158 148)(17 226 159 149)(18 227 160 150)(19 228 161 151)(20 217 162 152)(21 218 163 153)(22 219 164 154)(23 220 165 155)(24 221 166 156)(37 50 133 183)(38 51 134 184)(39 52 135 185)(40 53 136 186)(41 54 137 187)(42 55 138 188)(43 56 139 189)(44 57 140 190)(45 58 141 191)(46 59 142 192)(47 60 143 181)(48 49 144 182)(61 95 110 214)(62 96 111 215)(63 85 112 216)(64 86 113 205)(65 87 114 206)(66 88 115 207)(67 89 116 208)(68 90 117 209)(69 91 118 210)(70 92 119 211)(71 93 120 212)(72 94 109 213)(97 204 122 171)(98 193 123 172)(99 194 124 173)(100 195 125 174)(101 196 126 175)(102 197 127 176)(103 198 128 177)(104 199 129 178)(105 200 130 179)(106 201 131 180)(107 202 132 169)(108 203 121 170)
(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)
(1 238)(2 237)(3 236)(4 235)(5 234)(6 233)(7 232)(8 231)(9 230)(10 229)(11 240)(12 239)(13 218)(14 217)(15 228)(16 227)(17 226)(18 225)(19 224)(20 223)(21 222)(22 221)(23 220)(24 219)(25 84)(26 83)(27 82)(28 81)(29 80)(30 79)(31 78)(32 77)(33 76)(34 75)(35 74)(36 73)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(61 89)(62 88)(63 87)(64 86)(65 85)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)(97 200)(98 199)(99 198)(100 197)(101 196)(102 195)(103 194)(104 193)(105 204)(106 203)(107 202)(108 201)(109 209)(110 208)(111 207)(112 206)(113 205)(114 216)(115 215)(116 214)(117 213)(118 212)(119 211)(120 210)(121 180)(122 179)(123 178)(124 177)(125 176)(126 175)(127 174)(128 173)(129 172)(130 171)(131 170)(132 169)(133 181)(134 192)(135 191)(136 190)(137 189)(138 188)(139 187)(140 186)(141 185)(142 184)(143 183)(144 182)(145 163)(146 162)(147 161)(148 160)(149 159)(150 158)(151 157)(152 168)(153 167)(154 166)(155 165)(156 164)
G:=sub<Sym(240)| (1,187,195,148,210)(2,188,196,149,211)(3,189,197,150,212)(4,190,198,151,213)(5,191,199,152,214)(6,192,200,153,215)(7,181,201,154,216)(8,182,202,155,205)(9,183,203,156,206)(10,184,204,145,207)(11,185,193,146,208)(12,186,194,147,209)(13,66,31,38,122)(14,67,32,39,123)(15,68,33,40,124)(16,69,34,41,125)(17,70,35,42,126)(18,71,36,43,127)(19,72,25,44,128)(20,61,26,45,129)(21,62,27,46,130)(22,63,28,47,131)(23,64,29,48,132)(24,65,30,37,121)(49,169,220,86,80)(50,170,221,87,81)(51,171,222,88,82)(52,172,223,89,83)(53,173,224,90,84)(54,174,225,91,73)(55,175,226,92,74)(56,176,227,93,75)(57,177,228,94,76)(58,178,217,95,77)(59,179,218,96,78)(60,180,219,85,79)(97,167,115,233,134)(98,168,116,234,135)(99,157,117,235,136)(100,158,118,236,137)(101,159,119,237,138)(102,160,120,238,139)(103,161,109,239,140)(104,162,110,240,141)(105,163,111,229,142)(106,164,112,230,143)(107,165,113,231,144)(108,166,114,232,133), (1,34,73,236)(2,35,74,237)(3,36,75,238)(4,25,76,239)(5,26,77,240)(6,27,78,229)(7,28,79,230)(8,29,80,231)(9,30,81,232)(10,31,82,233)(11,32,83,234)(12,33,84,235)(13,222,167,145)(14,223,168,146)(15,224,157,147)(16,225,158,148)(17,226,159,149)(18,227,160,150)(19,228,161,151)(20,217,162,152)(21,218,163,153)(22,219,164,154)(23,220,165,155)(24,221,166,156)(37,50,133,183)(38,51,134,184)(39,52,135,185)(40,53,136,186)(41,54,137,187)(42,55,138,188)(43,56,139,189)(44,57,140,190)(45,58,141,191)(46,59,142,192)(47,60,143,181)(48,49,144,182)(61,95,110,214)(62,96,111,215)(63,85,112,216)(64,86,113,205)(65,87,114,206)(66,88,115,207)(67,89,116,208)(68,90,117,209)(69,91,118,210)(70,92,119,211)(71,93,120,212)(72,94,109,213)(97,204,122,171)(98,193,123,172)(99,194,124,173)(100,195,125,174)(101,196,126,175)(102,197,127,176)(103,198,128,177)(104,199,129,178)(105,200,130,179)(106,201,131,180)(107,202,132,169)(108,203,121,170), (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), (1,238)(2,237)(3,236)(4,235)(5,234)(6,233)(7,232)(8,231)(9,230)(10,229)(11,240)(12,239)(13,218)(14,217)(15,228)(16,227)(17,226)(18,225)(19,224)(20,223)(21,222)(22,221)(23,220)(24,219)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(61,89)(62,88)(63,87)(64,86)(65,85)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(97,200)(98,199)(99,198)(100,197)(101,196)(102,195)(103,194)(104,193)(105,204)(106,203)(107,202)(108,201)(109,209)(110,208)(111,207)(112,206)(113,205)(114,216)(115,215)(116,214)(117,213)(118,212)(119,211)(120,210)(121,180)(122,179)(123,178)(124,177)(125,176)(126,175)(127,174)(128,173)(129,172)(130,171)(131,170)(132,169)(133,181)(134,192)(135,191)(136,190)(137,189)(138,188)(139,187)(140,186)(141,185)(142,184)(143,183)(144,182)(145,163)(146,162)(147,161)(148,160)(149,159)(150,158)(151,157)(152,168)(153,167)(154,166)(155,165)(156,164)>;
G:=Group( (1,187,195,148,210)(2,188,196,149,211)(3,189,197,150,212)(4,190,198,151,213)(5,191,199,152,214)(6,192,200,153,215)(7,181,201,154,216)(8,182,202,155,205)(9,183,203,156,206)(10,184,204,145,207)(11,185,193,146,208)(12,186,194,147,209)(13,66,31,38,122)(14,67,32,39,123)(15,68,33,40,124)(16,69,34,41,125)(17,70,35,42,126)(18,71,36,43,127)(19,72,25,44,128)(20,61,26,45,129)(21,62,27,46,130)(22,63,28,47,131)(23,64,29,48,132)(24,65,30,37,121)(49,169,220,86,80)(50,170,221,87,81)(51,171,222,88,82)(52,172,223,89,83)(53,173,224,90,84)(54,174,225,91,73)(55,175,226,92,74)(56,176,227,93,75)(57,177,228,94,76)(58,178,217,95,77)(59,179,218,96,78)(60,180,219,85,79)(97,167,115,233,134)(98,168,116,234,135)(99,157,117,235,136)(100,158,118,236,137)(101,159,119,237,138)(102,160,120,238,139)(103,161,109,239,140)(104,162,110,240,141)(105,163,111,229,142)(106,164,112,230,143)(107,165,113,231,144)(108,166,114,232,133), (1,34,73,236)(2,35,74,237)(3,36,75,238)(4,25,76,239)(5,26,77,240)(6,27,78,229)(7,28,79,230)(8,29,80,231)(9,30,81,232)(10,31,82,233)(11,32,83,234)(12,33,84,235)(13,222,167,145)(14,223,168,146)(15,224,157,147)(16,225,158,148)(17,226,159,149)(18,227,160,150)(19,228,161,151)(20,217,162,152)(21,218,163,153)(22,219,164,154)(23,220,165,155)(24,221,166,156)(37,50,133,183)(38,51,134,184)(39,52,135,185)(40,53,136,186)(41,54,137,187)(42,55,138,188)(43,56,139,189)(44,57,140,190)(45,58,141,191)(46,59,142,192)(47,60,143,181)(48,49,144,182)(61,95,110,214)(62,96,111,215)(63,85,112,216)(64,86,113,205)(65,87,114,206)(66,88,115,207)(67,89,116,208)(68,90,117,209)(69,91,118,210)(70,92,119,211)(71,93,120,212)(72,94,109,213)(97,204,122,171)(98,193,123,172)(99,194,124,173)(100,195,125,174)(101,196,126,175)(102,197,127,176)(103,198,128,177)(104,199,129,178)(105,200,130,179)(106,201,131,180)(107,202,132,169)(108,203,121,170), (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), (1,238)(2,237)(3,236)(4,235)(5,234)(6,233)(7,232)(8,231)(9,230)(10,229)(11,240)(12,239)(13,218)(14,217)(15,228)(16,227)(17,226)(18,225)(19,224)(20,223)(21,222)(22,221)(23,220)(24,219)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(61,89)(62,88)(63,87)(64,86)(65,85)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(97,200)(98,199)(99,198)(100,197)(101,196)(102,195)(103,194)(104,193)(105,204)(106,203)(107,202)(108,201)(109,209)(110,208)(111,207)(112,206)(113,205)(114,216)(115,215)(116,214)(117,213)(118,212)(119,211)(120,210)(121,180)(122,179)(123,178)(124,177)(125,176)(126,175)(127,174)(128,173)(129,172)(130,171)(131,170)(132,169)(133,181)(134,192)(135,191)(136,190)(137,189)(138,188)(139,187)(140,186)(141,185)(142,184)(143,183)(144,182)(145,163)(146,162)(147,161)(148,160)(149,159)(150,158)(151,157)(152,168)(153,167)(154,166)(155,165)(156,164) );
G=PermutationGroup([[(1,187,195,148,210),(2,188,196,149,211),(3,189,197,150,212),(4,190,198,151,213),(5,191,199,152,214),(6,192,200,153,215),(7,181,201,154,216),(8,182,202,155,205),(9,183,203,156,206),(10,184,204,145,207),(11,185,193,146,208),(12,186,194,147,209),(13,66,31,38,122),(14,67,32,39,123),(15,68,33,40,124),(16,69,34,41,125),(17,70,35,42,126),(18,71,36,43,127),(19,72,25,44,128),(20,61,26,45,129),(21,62,27,46,130),(22,63,28,47,131),(23,64,29,48,132),(24,65,30,37,121),(49,169,220,86,80),(50,170,221,87,81),(51,171,222,88,82),(52,172,223,89,83),(53,173,224,90,84),(54,174,225,91,73),(55,175,226,92,74),(56,176,227,93,75),(57,177,228,94,76),(58,178,217,95,77),(59,179,218,96,78),(60,180,219,85,79),(97,167,115,233,134),(98,168,116,234,135),(99,157,117,235,136),(100,158,118,236,137),(101,159,119,237,138),(102,160,120,238,139),(103,161,109,239,140),(104,162,110,240,141),(105,163,111,229,142),(106,164,112,230,143),(107,165,113,231,144),(108,166,114,232,133)], [(1,34,73,236),(2,35,74,237),(3,36,75,238),(4,25,76,239),(5,26,77,240),(6,27,78,229),(7,28,79,230),(8,29,80,231),(9,30,81,232),(10,31,82,233),(11,32,83,234),(12,33,84,235),(13,222,167,145),(14,223,168,146),(15,224,157,147),(16,225,158,148),(17,226,159,149),(18,227,160,150),(19,228,161,151),(20,217,162,152),(21,218,163,153),(22,219,164,154),(23,220,165,155),(24,221,166,156),(37,50,133,183),(38,51,134,184),(39,52,135,185),(40,53,136,186),(41,54,137,187),(42,55,138,188),(43,56,139,189),(44,57,140,190),(45,58,141,191),(46,59,142,192),(47,60,143,181),(48,49,144,182),(61,95,110,214),(62,96,111,215),(63,85,112,216),(64,86,113,205),(65,87,114,206),(66,88,115,207),(67,89,116,208),(68,90,117,209),(69,91,118,210),(70,92,119,211),(71,93,120,212),(72,94,109,213),(97,204,122,171),(98,193,123,172),(99,194,124,173),(100,195,125,174),(101,196,126,175),(102,197,127,176),(103,198,128,177),(104,199,129,178),(105,200,130,179),(106,201,131,180),(107,202,132,169),(108,203,121,170)], [(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)], [(1,238),(2,237),(3,236),(4,235),(5,234),(6,233),(7,232),(8,231),(9,230),(10,229),(11,240),(12,239),(13,218),(14,217),(15,228),(16,227),(17,226),(18,225),(19,224),(20,223),(21,222),(22,221),(23,220),(24,219),(25,84),(26,83),(27,82),(28,81),(29,80),(30,79),(31,78),(32,77),(33,76),(34,75),(35,74),(36,73),(37,60),(38,59),(39,58),(40,57),(41,56),(42,55),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(61,89),(62,88),(63,87),(64,86),(65,85),(66,96),(67,95),(68,94),(69,93),(70,92),(71,91),(72,90),(97,200),(98,199),(99,198),(100,197),(101,196),(102,195),(103,194),(104,193),(105,204),(106,203),(107,202),(108,201),(109,209),(110,208),(111,207),(112,206),(113,205),(114,216),(115,215),(116,214),(117,213),(118,212),(119,211),(120,210),(121,180),(122,179),(123,178),(124,177),(125,176),(126,175),(127,174),(128,173),(129,172),(130,171),(131,170),(132,169),(133,181),(134,192),(135,191),(136,190),(137,189),(138,188),(139,187),(140,186),(141,185),(142,184),(143,183),(144,182),(145,163),(146,162),(147,161),(148,160),(149,159),(150,158),(151,157),(152,168),(153,167),(154,166),(155,165),(156,164)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 10M | ··· | 10AB | 12A | ··· | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | D4 | D6 | D12 | C5×S3 | C5×D4 | S3×C10 | C5×D12 |
| kernel | C5×C4⋊D12 | C4×C60 | C10×D12 | C4⋊D12 | C4×C12 | C2×D12 | C4×C20 | C60 | C2×C20 | C20 | C42 | C12 | C2×C4 | C4 |
| # reps | 1 | 1 | 6 | 4 | 4 | 24 | 1 | 6 | 3 | 12 | 4 | 24 | 12 | 48 |
Matrix representation of C5×C4⋊D12 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 34 |
| 8 | 3 | 0 | 0 |
| 19 | 53 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 8 | 3 | 0 | 0 |
| 19 | 53 | 0 | 0 |
| 0 | 0 | 23 | 23 |
| 0 | 0 | 38 | 46 |
| 53 | 58 | 0 | 0 |
| 21 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 60 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,34,0,0,0,0,34],[8,19,0,0,3,53,0,0,0,0,60,0,0,0,0,60],[8,19,0,0,3,53,0,0,0,0,23,38,0,0,23,46],[53,21,0,0,58,8,0,0,0,0,1,1,0,0,0,60] >;
C5×C4⋊D12 in GAP, Magma, Sage, TeX
C_5\times C_4\rtimes D_{12} % in TeX
G:=Group("C5xC4:D12"); // GroupNames label
G:=SmallGroup(480,753);
// by ID
G=gap.SmallGroup(480,753);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,568,926,226,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations