direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C4.D12, C20.65D12, C60.144D4, D6⋊2(C5×Q8), C6.8(D4×C10), D6⋊C4.2C10, C4⋊Dic3⋊6C10, (S3×C10)⋊10Q8, C12.11(C5×D4), C4.13(C5×D12), C6.14(Q8×C10), C10.52(S3×Q8), (C2×Dic6)⋊7C10, C30.295(C2×D4), C10.79(C2×D12), (C2×C20).355D6, C2.10(C10×D12), C15⋊33(C22⋊Q8), C30.112(C2×Q8), (C10×Dic6)⋊23C2, C30.251(C4○D4), (C2×C30).417C23, (C2×C60).336C22, C10.117(D4⋊2S3), (C10×Dic3).146C22, C4⋊C4⋊5(C5×S3), C2.7(C5×S3×Q8), (C3×C4⋊C4)⋊8C10, (C5×C4⋊C4)⋊14S3, C3⋊3(C5×C22⋊Q8), (S3×C2×C4).3C10, (C15×C4⋊C4)⋊26C2, (S3×C2×C20).14C2, C6.26(C5×C4○D4), (C2×C4).12(S3×C10), (C5×D6⋊C4).10C2, (C5×C4⋊Dic3)⋊24C2, C22.52(S3×C2×C10), (C2×C12).25(C2×C10), C2.13(C5×D4⋊2S3), (S3×C2×C10).114C22, (C2×C6).38(C22×C10), (C22×S3).23(C2×C10), (C2×C10).351(C22×S3), (C2×Dic3).10(C2×C10), SmallGroup(480,776)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4.D12
G = < a,b,c,d | a5=b4=c12=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 340 in 148 conjugacy classes, 70 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C20, C20, C2×C10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C22⋊Q8, C2×C20, C2×C20, C2×C20, C5×Q8, C22×C10, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3, C60, C60, S3×C10, S3×C10, C2×C30, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×C20, Q8×C10, C4.D12, C5×Dic6, S3×C20, C10×Dic3, C10×Dic3, C2×C60, C2×C60, S3×C2×C10, C5×C22⋊Q8, C5×C4⋊Dic3, C5×D6⋊C4, C15×C4⋊C4, C10×Dic6, S3×C2×C20, C5×C4.D12
Quotients: C1, C2, C22, C5, S3, D4, Q8, C23, C10, D6, C2×D4, C2×Q8, C4○D4, C2×C10, D12, C22×S3, C5×S3, C22⋊Q8, C5×D4, C5×Q8, C22×C10, C2×D12, D4⋊2S3, S3×Q8, S3×C10, D4×C10, Q8×C10, C5×C4○D4, C4.D12, C5×D12, S3×C2×C10, C5×C22⋊Q8, C10×D12, C5×D4⋊2S3, C5×S3×Q8, C5×C4.D12
►On 240 points
Generators in S240
(1 51 163 34 146)(2 52 164 35 147)(3 53 165 36 148)(4 54 166 25 149)(5 55 167 26 150)(6 56 168 27 151)(7 57 157 28 152)(8 58 158 29 153)(9 59 159 30 154)(10 60 160 31 155)(11 49 161 32 156)(12 50 162 33 145)(13 229 62 78 120)(14 230 63 79 109)(15 231 64 80 110)(16 232 65 81 111)(17 233 66 82 112)(18 234 67 83 113)(19 235 68 84 114)(20 236 69 73 115)(21 237 70 74 116)(22 238 71 75 117)(23 239 72 76 118)(24 240 61 77 119)(37 91 224 194 97)(38 92 225 195 98)(39 93 226 196 99)(40 94 227 197 100)(41 95 228 198 101)(42 96 217 199 102)(43 85 218 200 103)(44 86 219 201 104)(45 87 220 202 105)(46 88 221 203 106)(47 89 222 204 107)(48 90 223 193 108)(121 133 169 181 207)(122 134 170 182 208)(123 135 171 183 209)(124 136 172 184 210)(125 137 173 185 211)(126 138 174 186 212)(127 139 175 187 213)(128 140 176 188 214)(129 141 177 189 215)(130 142 178 190 216)(131 143 179 191 205)(132 144 180 192 206)
(1 205 227 69)(2 70 228 206)(3 207 217 71)(4 72 218 208)(5 209 219 61)(6 62 220 210)(7 211 221 63)(8 64 222 212)(9 213 223 65)(10 66 224 214)(11 215 225 67)(12 68 226 216)(13 45 172 27)(14 28 173 46)(15 47 174 29)(16 30 175 48)(17 37 176 31)(18 32 177 38)(19 39 178 33)(20 34 179 40)(21 41 180 35)(22 36 169 42)(23 43 170 25)(24 26 171 44)(49 129 195 83)(50 84 196 130)(51 131 197 73)(52 74 198 132)(53 121 199 75)(54 76 200 122)(55 123 201 77)(56 78 202 124)(57 125 203 79)(58 80 204 126)(59 127 193 81)(60 82 194 128)(85 182 149 239)(86 240 150 183)(87 184 151 229)(88 230 152 185)(89 186 153 231)(90 232 154 187)(91 188 155 233)(92 234 156 189)(93 190 145 235)(94 236 146 191)(95 192 147 237)(96 238 148 181)(97 140 160 112)(98 113 161 141)(99 142 162 114)(100 115 163 143)(101 144 164 116)(102 117 165 133)(103 134 166 118)(104 119 167 135)(105 136 168 120)(106 109 157 137)(107 138 158 110)(108 111 159 139)
(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)
(1 226 227 12)(2 11 228 225)(3 224 217 10)(4 9 218 223)(5 222 219 8)(6 7 220 221)(13 14 172 173)(15 24 174 171)(16 170 175 23)(17 22 176 169)(18 180 177 21)(19 20 178 179)(25 30 43 48)(26 47 44 29)(27 28 45 46)(31 36 37 42)(32 41 38 35)(33 34 39 40)(49 198 195 52)(50 51 196 197)(53 194 199 60)(54 59 200 193)(55 204 201 58)(56 57 202 203)(61 212 209 64)(62 63 210 211)(65 208 213 72)(66 71 214 207)(67 206 215 70)(68 69 216 205)(73 130 131 84)(74 83 132 129)(75 128 121 82)(76 81 122 127)(77 126 123 80)(78 79 124 125)(85 90 149 154)(86 153 150 89)(87 88 151 152)(91 96 155 148)(92 147 156 95)(93 94 145 146)(97 102 160 165)(98 164 161 101)(99 100 162 163)(103 108 166 159)(104 158 167 107)(105 106 168 157)(109 136 137 120)(110 119 138 135)(111 134 139 118)(112 117 140 133)(113 144 141 116)(114 115 142 143)(181 233 238 188)(182 187 239 232)(183 231 240 186)(184 185 229 230)(189 237 234 192)(190 191 235 236)
G:=sub<Sym(240)| (1,51,163,34,146)(2,52,164,35,147)(3,53,165,36,148)(4,54,166,25,149)(5,55,167,26,150)(6,56,168,27,151)(7,57,157,28,152)(8,58,158,29,153)(9,59,159,30,154)(10,60,160,31,155)(11,49,161,32,156)(12,50,162,33,145)(13,229,62,78,120)(14,230,63,79,109)(15,231,64,80,110)(16,232,65,81,111)(17,233,66,82,112)(18,234,67,83,113)(19,235,68,84,114)(20,236,69,73,115)(21,237,70,74,116)(22,238,71,75,117)(23,239,72,76,118)(24,240,61,77,119)(37,91,224,194,97)(38,92,225,195,98)(39,93,226,196,99)(40,94,227,197,100)(41,95,228,198,101)(42,96,217,199,102)(43,85,218,200,103)(44,86,219,201,104)(45,87,220,202,105)(46,88,221,203,106)(47,89,222,204,107)(48,90,223,193,108)(121,133,169,181,207)(122,134,170,182,208)(123,135,171,183,209)(124,136,172,184,210)(125,137,173,185,211)(126,138,174,186,212)(127,139,175,187,213)(128,140,176,188,214)(129,141,177,189,215)(130,142,178,190,216)(131,143,179,191,205)(132,144,180,192,206), (1,205,227,69)(2,70,228,206)(3,207,217,71)(4,72,218,208)(5,209,219,61)(6,62,220,210)(7,211,221,63)(8,64,222,212)(9,213,223,65)(10,66,224,214)(11,215,225,67)(12,68,226,216)(13,45,172,27)(14,28,173,46)(15,47,174,29)(16,30,175,48)(17,37,176,31)(18,32,177,38)(19,39,178,33)(20,34,179,40)(21,41,180,35)(22,36,169,42)(23,43,170,25)(24,26,171,44)(49,129,195,83)(50,84,196,130)(51,131,197,73)(52,74,198,132)(53,121,199,75)(54,76,200,122)(55,123,201,77)(56,78,202,124)(57,125,203,79)(58,80,204,126)(59,127,193,81)(60,82,194,128)(85,182,149,239)(86,240,150,183)(87,184,151,229)(88,230,152,185)(89,186,153,231)(90,232,154,187)(91,188,155,233)(92,234,156,189)(93,190,145,235)(94,236,146,191)(95,192,147,237)(96,238,148,181)(97,140,160,112)(98,113,161,141)(99,142,162,114)(100,115,163,143)(101,144,164,116)(102,117,165,133)(103,134,166,118)(104,119,167,135)(105,136,168,120)(106,109,157,137)(107,138,158,110)(108,111,159,139), (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), (1,226,227,12)(2,11,228,225)(3,224,217,10)(4,9,218,223)(5,222,219,8)(6,7,220,221)(13,14,172,173)(15,24,174,171)(16,170,175,23)(17,22,176,169)(18,180,177,21)(19,20,178,179)(25,30,43,48)(26,47,44,29)(27,28,45,46)(31,36,37,42)(32,41,38,35)(33,34,39,40)(49,198,195,52)(50,51,196,197)(53,194,199,60)(54,59,200,193)(55,204,201,58)(56,57,202,203)(61,212,209,64)(62,63,210,211)(65,208,213,72)(66,71,214,207)(67,206,215,70)(68,69,216,205)(73,130,131,84)(74,83,132,129)(75,128,121,82)(76,81,122,127)(77,126,123,80)(78,79,124,125)(85,90,149,154)(86,153,150,89)(87,88,151,152)(91,96,155,148)(92,147,156,95)(93,94,145,146)(97,102,160,165)(98,164,161,101)(99,100,162,163)(103,108,166,159)(104,158,167,107)(105,106,168,157)(109,136,137,120)(110,119,138,135)(111,134,139,118)(112,117,140,133)(113,144,141,116)(114,115,142,143)(181,233,238,188)(182,187,239,232)(183,231,240,186)(184,185,229,230)(189,237,234,192)(190,191,235,236)>;
G:=Group( (1,51,163,34,146)(2,52,164,35,147)(3,53,165,36,148)(4,54,166,25,149)(5,55,167,26,150)(6,56,168,27,151)(7,57,157,28,152)(8,58,158,29,153)(9,59,159,30,154)(10,60,160,31,155)(11,49,161,32,156)(12,50,162,33,145)(13,229,62,78,120)(14,230,63,79,109)(15,231,64,80,110)(16,232,65,81,111)(17,233,66,82,112)(18,234,67,83,113)(19,235,68,84,114)(20,236,69,73,115)(21,237,70,74,116)(22,238,71,75,117)(23,239,72,76,118)(24,240,61,77,119)(37,91,224,194,97)(38,92,225,195,98)(39,93,226,196,99)(40,94,227,197,100)(41,95,228,198,101)(42,96,217,199,102)(43,85,218,200,103)(44,86,219,201,104)(45,87,220,202,105)(46,88,221,203,106)(47,89,222,204,107)(48,90,223,193,108)(121,133,169,181,207)(122,134,170,182,208)(123,135,171,183,209)(124,136,172,184,210)(125,137,173,185,211)(126,138,174,186,212)(127,139,175,187,213)(128,140,176,188,214)(129,141,177,189,215)(130,142,178,190,216)(131,143,179,191,205)(132,144,180,192,206), (1,205,227,69)(2,70,228,206)(3,207,217,71)(4,72,218,208)(5,209,219,61)(6,62,220,210)(7,211,221,63)(8,64,222,212)(9,213,223,65)(10,66,224,214)(11,215,225,67)(12,68,226,216)(13,45,172,27)(14,28,173,46)(15,47,174,29)(16,30,175,48)(17,37,176,31)(18,32,177,38)(19,39,178,33)(20,34,179,40)(21,41,180,35)(22,36,169,42)(23,43,170,25)(24,26,171,44)(49,129,195,83)(50,84,196,130)(51,131,197,73)(52,74,198,132)(53,121,199,75)(54,76,200,122)(55,123,201,77)(56,78,202,124)(57,125,203,79)(58,80,204,126)(59,127,193,81)(60,82,194,128)(85,182,149,239)(86,240,150,183)(87,184,151,229)(88,230,152,185)(89,186,153,231)(90,232,154,187)(91,188,155,233)(92,234,156,189)(93,190,145,235)(94,236,146,191)(95,192,147,237)(96,238,148,181)(97,140,160,112)(98,113,161,141)(99,142,162,114)(100,115,163,143)(101,144,164,116)(102,117,165,133)(103,134,166,118)(104,119,167,135)(105,136,168,120)(106,109,157,137)(107,138,158,110)(108,111,159,139), (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), (1,226,227,12)(2,11,228,225)(3,224,217,10)(4,9,218,223)(5,222,219,8)(6,7,220,221)(13,14,172,173)(15,24,174,171)(16,170,175,23)(17,22,176,169)(18,180,177,21)(19,20,178,179)(25,30,43,48)(26,47,44,29)(27,28,45,46)(31,36,37,42)(32,41,38,35)(33,34,39,40)(49,198,195,52)(50,51,196,197)(53,194,199,60)(54,59,200,193)(55,204,201,58)(56,57,202,203)(61,212,209,64)(62,63,210,211)(65,208,213,72)(66,71,214,207)(67,206,215,70)(68,69,216,205)(73,130,131,84)(74,83,132,129)(75,128,121,82)(76,81,122,127)(77,126,123,80)(78,79,124,125)(85,90,149,154)(86,153,150,89)(87,88,151,152)(91,96,155,148)(92,147,156,95)(93,94,145,146)(97,102,160,165)(98,164,161,101)(99,100,162,163)(103,108,166,159)(104,158,167,107)(105,106,168,157)(109,136,137,120)(110,119,138,135)(111,134,139,118)(112,117,140,133)(113,144,141,116)(114,115,142,143)(181,233,238,188)(182,187,239,232)(183,231,240,186)(184,185,229,230)(189,237,234,192)(190,191,235,236) );
G=PermutationGroup([[(1,51,163,34,146),(2,52,164,35,147),(3,53,165,36,148),(4,54,166,25,149),(5,55,167,26,150),(6,56,168,27,151),(7,57,157,28,152),(8,58,158,29,153),(9,59,159,30,154),(10,60,160,31,155),(11,49,161,32,156),(12,50,162,33,145),(13,229,62,78,120),(14,230,63,79,109),(15,231,64,80,110),(16,232,65,81,111),(17,233,66,82,112),(18,234,67,83,113),(19,235,68,84,114),(20,236,69,73,115),(21,237,70,74,116),(22,238,71,75,117),(23,239,72,76,118),(24,240,61,77,119),(37,91,224,194,97),(38,92,225,195,98),(39,93,226,196,99),(40,94,227,197,100),(41,95,228,198,101),(42,96,217,199,102),(43,85,218,200,103),(44,86,219,201,104),(45,87,220,202,105),(46,88,221,203,106),(47,89,222,204,107),(48,90,223,193,108),(121,133,169,181,207),(122,134,170,182,208),(123,135,171,183,209),(124,136,172,184,210),(125,137,173,185,211),(126,138,174,186,212),(127,139,175,187,213),(128,140,176,188,214),(129,141,177,189,215),(130,142,178,190,216),(131,143,179,191,205),(132,144,180,192,206)], [(1,205,227,69),(2,70,228,206),(3,207,217,71),(4,72,218,208),(5,209,219,61),(6,62,220,210),(7,211,221,63),(8,64,222,212),(9,213,223,65),(10,66,224,214),(11,215,225,67),(12,68,226,216),(13,45,172,27),(14,28,173,46),(15,47,174,29),(16,30,175,48),(17,37,176,31),(18,32,177,38),(19,39,178,33),(20,34,179,40),(21,41,180,35),(22,36,169,42),(23,43,170,25),(24,26,171,44),(49,129,195,83),(50,84,196,130),(51,131,197,73),(52,74,198,132),(53,121,199,75),(54,76,200,122),(55,123,201,77),(56,78,202,124),(57,125,203,79),(58,80,204,126),(59,127,193,81),(60,82,194,128),(85,182,149,239),(86,240,150,183),(87,184,151,229),(88,230,152,185),(89,186,153,231),(90,232,154,187),(91,188,155,233),(92,234,156,189),(93,190,145,235),(94,236,146,191),(95,192,147,237),(96,238,148,181),(97,140,160,112),(98,113,161,141),(99,142,162,114),(100,115,163,143),(101,144,164,116),(102,117,165,133),(103,134,166,118),(104,119,167,135),(105,136,168,120),(106,109,157,137),(107,138,158,110),(108,111,159,139)], [(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)], [(1,226,227,12),(2,11,228,225),(3,224,217,10),(4,9,218,223),(5,222,219,8),(6,7,220,221),(13,14,172,173),(15,24,174,171),(16,170,175,23),(17,22,176,169),(18,180,177,21),(19,20,178,179),(25,30,43,48),(26,47,44,29),(27,28,45,46),(31,36,37,42),(32,41,38,35),(33,34,39,40),(49,198,195,52),(50,51,196,197),(53,194,199,60),(54,59,200,193),(55,204,201,58),(56,57,202,203),(61,212,209,64),(62,63,210,211),(65,208,213,72),(66,71,214,207),(67,206,215,70),(68,69,216,205),(73,130,131,84),(74,83,132,129),(75,128,121,82),(76,81,122,127),(77,126,123,80),(78,79,124,125),(85,90,149,154),(86,153,150,89),(87,88,151,152),(91,96,155,148),(92,147,156,95),(93,94,145,146),(97,102,160,165),(98,164,161,101),(99,100,162,163),(103,108,166,159),(104,158,167,107),(105,106,168,157),(109,136,137,120),(110,119,138,135),(111,134,139,118),(112,117,140,133),(113,144,141,116),(114,115,142,143),(181,233,238,188),(182,187,239,232),(183,231,240,186),(184,185,229,230),(189,237,234,192),(190,191,235,236)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 10M | ··· | 10T | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 20Q | ··· | 20X | 20Y | ··· | 20AF | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 6 | ··· | 6 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | Q8 | D6 | C4○D4 | D12 | C5×S3 | C5×D4 | C5×Q8 | S3×C10 | C5×C4○D4 | C5×D12 | D4⋊2S3 | S3×Q8 | C5×D4⋊2S3 | C5×S3×Q8 |
| kernel | C5×C4.D12 | C5×C4⋊Dic3 | C5×D6⋊C4 | C15×C4⋊C4 | C10×Dic6 | S3×C2×C20 | C4.D12 | C4⋊Dic3 | D6⋊C4 | C3×C4⋊C4 | C2×Dic6 | S3×C2×C4 | C5×C4⋊C4 | C60 | S3×C10 | C2×C20 | C30 | C20 | C4⋊C4 | C12 | D6 | C2×C4 | C6 | C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 8 | 8 | 4 | 4 | 4 | 1 | 2 | 2 | 3 | 2 | 4 | 4 | 8 | 8 | 12 | 8 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C5×C4.D12 ►in GL4(𝔽61) generated by
| 34 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 50 | 0 | 0 | 0 |
| 28 | 11 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 40 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 23 | 23 |
| 0 | 0 | 38 | 46 |
| 1 | 40 | 0 | 0 |
| 3 | 60 | 0 | 0 |
| 0 | 0 | 38 | 46 |
| 0 | 0 | 23 | 23 |
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,1,0,0,0,0,1],[50,28,0,0,0,11,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,40,60,0,0,0,0,23,38,0,0,23,46],[1,3,0,0,40,60,0,0,0,0,38,23,0,0,46,23] >;
C5×C4.D12 in GAP, Magma, Sage, TeX
C_5\times C_4.D_{12} % in TeX
G:=Group("C5xC4.D12"); // GroupNames label
G:=SmallGroup(480,776);
// by ID
G=gap.SmallGroup(480,776);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,926,891,646,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^12=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations