direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C12⋊3D4, C60⋊21D4, C12⋊3(C5×D4), (C6×D4)⋊4C10, (C2×D12)⋊9C10, (D4×C10)⋊13S3, (D4×C30)⋊18C2, C20⋊9(C3⋊D4), Dic3⋊1(C5×D4), C6.52(D4×C10), (C10×D12)⋊25C2, C15⋊10(C4⋊1D4), (C5×Dic3)⋊10D4, (C4×Dic3)⋊6C10, C10.195(S3×D4), (C2×C20).363D6, C30.435(C2×D4), (Dic3×C20)⋊18C2, C23.15(S3×C10), (C22×C10).26D6, (C2×C60).363C22, (C2×C30).434C23, (C22×C30).127C22, (C10×Dic3).231C22, C4⋊1(C5×C3⋊D4), C3⋊2(C5×C4⋊1D4), C2.28(C5×S3×D4), (C2×D4)⋊6(C5×S3), (C2×C3⋊D4)⋊7C10, (C10×C3⋊D4)⋊22C2, (C2×C4).52(S3×C10), C2.16(C10×C3⋊D4), C22.62(S3×C2×C10), (C2×C12).36(C2×C10), C10.137(C2×C3⋊D4), (S3×C2×C10).73C22, (C22×C6).22(C2×C10), (C2×C6).55(C22×C10), (C22×S3).12(C2×C10), (C2×C10).368(C22×S3), (C2×Dic3).39(C2×C10), SmallGroup(480,819)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C12⋊3D4
G = < a,b,c,d | a5=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >
Subgroups: 580 in 216 conjugacy classes, 74 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C10, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C42, C2×D4, C2×D4, C20, C20, C2×C10, C2×C10, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C5×S3, C30, C30, C30, C4⋊1D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C4×Dic3, C2×D12, C2×C3⋊D4, C6×D4, C5×Dic3, C60, S3×C10, C2×C30, C2×C30, C4×C20, D4×C10, D4×C10, C12⋊3D4, C5×D12, C10×Dic3, C5×C3⋊D4, C2×C60, D4×C15, S3×C2×C10, C22×C30, C5×C4⋊1D4, Dic3×C20, C10×D12, C10×C3⋊D4, D4×C30, C5×C12⋊3D4
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C4⋊1D4, C5×D4, C22×C10, S3×D4, C2×C3⋊D4, S3×C10, D4×C10, C12⋊3D4, C5×C3⋊D4, S3×C2×C10, C5×C4⋊1D4, C5×S3×D4, C10×C3⋊D4, C5×C12⋊3D4
►On 240 points
Generators in S240
(1 188 176 119 145)(2 189 177 120 146)(3 190 178 109 147)(4 191 179 110 148)(5 192 180 111 149)(6 181 169 112 150)(7 182 170 113 151)(8 183 171 114 152)(9 184 172 115 153)(10 185 173 116 154)(11 186 174 117 155)(12 187 175 118 156)(13 67 30 202 140)(14 68 31 203 141)(15 69 32 204 142)(16 70 33 193 143)(17 71 34 194 144)(18 72 35 195 133)(19 61 36 196 134)(20 62 25 197 135)(21 63 26 198 136)(22 64 27 199 137)(23 65 28 200 138)(24 66 29 201 139)(37 96 234 53 74)(38 85 235 54 75)(39 86 236 55 76)(40 87 237 56 77)(41 88 238 57 78)(42 89 239 58 79)(43 90 240 59 80)(44 91 229 60 81)(45 92 230 49 82)(46 93 231 50 83)(47 94 232 51 84)(48 95 233 52 73)(97 166 210 128 222)(98 167 211 129 223)(99 168 212 130 224)(100 157 213 131 225)(101 158 214 132 226)(102 159 215 121 227)(103 160 216 122 228)(104 161 205 123 217)(105 162 206 124 218)(106 163 207 125 219)(107 164 208 126 220)(108 165 209 127 221)
(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)
(1 164 62 45)(2 157 63 38)(3 162 64 43)(4 167 65 48)(5 160 66 41)(6 165 67 46)(7 158 68 39)(8 163 69 44)(9 168 70 37)(10 161 71 42)(11 166 72 47)(12 159 61 40)(13 83 150 108)(14 76 151 101)(15 81 152 106)(16 74 153 99)(17 79 154 104)(18 84 155 97)(19 77 156 102)(20 82 145 107)(21 75 146 100)(22 80 147 105)(23 73 148 98)(24 78 149 103)(25 92 188 208)(26 85 189 213)(27 90 190 206)(28 95 191 211)(29 88 192 216)(30 93 181 209)(31 86 182 214)(32 91 183 207)(33 96 184 212)(34 89 185 205)(35 94 186 210)(36 87 187 215)(49 119 220 135)(50 112 221 140)(51 117 222 133)(52 110 223 138)(53 115 224 143)(54 120 225 136)(55 113 226 141)(56 118 227 134)(57 111 228 139)(58 116 217 144)(59 109 218 137)(60 114 219 142)(121 196 237 175)(122 201 238 180)(123 194 239 173)(124 199 240 178)(125 204 229 171)(126 197 230 176)(127 202 231 169)(128 195 232 174)(129 200 233 179)(130 193 234 172)(131 198 235 177)(132 203 236 170)
(1 4)(2 3)(5 12)(6 11)(7 10)(8 9)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)(37 163)(38 162)(39 161)(40 160)(41 159)(42 158)(43 157)(44 168)(45 167)(46 166)(47 165)(48 164)(49 223)(50 222)(51 221)(52 220)(53 219)(54 218)(55 217)(56 228)(57 227)(58 226)(59 225)(60 224)(61 66)(62 65)(63 64)(67 72)(68 71)(69 70)(73 107)(74 106)(75 105)(76 104)(77 103)(78 102)(79 101)(80 100)(81 99)(82 98)(83 97)(84 108)(85 206)(86 205)(87 216)(88 215)(89 214)(90 213)(91 212)(92 211)(93 210)(94 209)(95 208)(96 207)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)(121 238)(122 237)(123 236)(124 235)(125 234)(126 233)(127 232)(128 231)(129 230)(130 229)(131 240)(132 239)(133 140)(134 139)(135 138)(136 137)(141 144)(142 143)(145 148)(146 147)(149 156)(150 155)(151 154)(152 153)(169 174)(170 173)(171 172)(175 180)(176 179)(177 178)(181 186)(182 185)(183 184)(187 192)(188 191)(189 190)(193 204)(194 203)(195 202)(196 201)(197 200)(198 199)
G:=sub<Sym(240)| (1,188,176,119,145)(2,189,177,120,146)(3,190,178,109,147)(4,191,179,110,148)(5,192,180,111,149)(6,181,169,112,150)(7,182,170,113,151)(8,183,171,114,152)(9,184,172,115,153)(10,185,173,116,154)(11,186,174,117,155)(12,187,175,118,156)(13,67,30,202,140)(14,68,31,203,141)(15,69,32,204,142)(16,70,33,193,143)(17,71,34,194,144)(18,72,35,195,133)(19,61,36,196,134)(20,62,25,197,135)(21,63,26,198,136)(22,64,27,199,137)(23,65,28,200,138)(24,66,29,201,139)(37,96,234,53,74)(38,85,235,54,75)(39,86,236,55,76)(40,87,237,56,77)(41,88,238,57,78)(42,89,239,58,79)(43,90,240,59,80)(44,91,229,60,81)(45,92,230,49,82)(46,93,231,50,83)(47,94,232,51,84)(48,95,233,52,73)(97,166,210,128,222)(98,167,211,129,223)(99,168,212,130,224)(100,157,213,131,225)(101,158,214,132,226)(102,159,215,121,227)(103,160,216,122,228)(104,161,205,123,217)(105,162,206,124,218)(106,163,207,125,219)(107,164,208,126,220)(108,165,209,127,221), (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), (1,164,62,45)(2,157,63,38)(3,162,64,43)(4,167,65,48)(5,160,66,41)(6,165,67,46)(7,158,68,39)(8,163,69,44)(9,168,70,37)(10,161,71,42)(11,166,72,47)(12,159,61,40)(13,83,150,108)(14,76,151,101)(15,81,152,106)(16,74,153,99)(17,79,154,104)(18,84,155,97)(19,77,156,102)(20,82,145,107)(21,75,146,100)(22,80,147,105)(23,73,148,98)(24,78,149,103)(25,92,188,208)(26,85,189,213)(27,90,190,206)(28,95,191,211)(29,88,192,216)(30,93,181,209)(31,86,182,214)(32,91,183,207)(33,96,184,212)(34,89,185,205)(35,94,186,210)(36,87,187,215)(49,119,220,135)(50,112,221,140)(51,117,222,133)(52,110,223,138)(53,115,224,143)(54,120,225,136)(55,113,226,141)(56,118,227,134)(57,111,228,139)(58,116,217,144)(59,109,218,137)(60,114,219,142)(121,196,237,175)(122,201,238,180)(123,194,239,173)(124,199,240,178)(125,204,229,171)(126,197,230,176)(127,202,231,169)(128,195,232,174)(129,200,233,179)(130,193,234,172)(131,198,235,177)(132,203,236,170), (1,4)(2,3)(5,12)(6,11)(7,10)(8,9)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,163)(38,162)(39,161)(40,160)(41,159)(42,158)(43,157)(44,168)(45,167)(46,166)(47,165)(48,164)(49,223)(50,222)(51,221)(52,220)(53,219)(54,218)(55,217)(56,228)(57,227)(58,226)(59,225)(60,224)(61,66)(62,65)(63,64)(67,72)(68,71)(69,70)(73,107)(74,106)(75,105)(76,104)(77,103)(78,102)(79,101)(80,100)(81,99)(82,98)(83,97)(84,108)(85,206)(86,205)(87,216)(88,215)(89,214)(90,213)(91,212)(92,211)(93,210)(94,209)(95,208)(96,207)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(121,238)(122,237)(123,236)(124,235)(125,234)(126,233)(127,232)(128,231)(129,230)(130,229)(131,240)(132,239)(133,140)(134,139)(135,138)(136,137)(141,144)(142,143)(145,148)(146,147)(149,156)(150,155)(151,154)(152,153)(169,174)(170,173)(171,172)(175,180)(176,179)(177,178)(181,186)(182,185)(183,184)(187,192)(188,191)(189,190)(193,204)(194,203)(195,202)(196,201)(197,200)(198,199)>;
G:=Group( (1,188,176,119,145)(2,189,177,120,146)(3,190,178,109,147)(4,191,179,110,148)(5,192,180,111,149)(6,181,169,112,150)(7,182,170,113,151)(8,183,171,114,152)(9,184,172,115,153)(10,185,173,116,154)(11,186,174,117,155)(12,187,175,118,156)(13,67,30,202,140)(14,68,31,203,141)(15,69,32,204,142)(16,70,33,193,143)(17,71,34,194,144)(18,72,35,195,133)(19,61,36,196,134)(20,62,25,197,135)(21,63,26,198,136)(22,64,27,199,137)(23,65,28,200,138)(24,66,29,201,139)(37,96,234,53,74)(38,85,235,54,75)(39,86,236,55,76)(40,87,237,56,77)(41,88,238,57,78)(42,89,239,58,79)(43,90,240,59,80)(44,91,229,60,81)(45,92,230,49,82)(46,93,231,50,83)(47,94,232,51,84)(48,95,233,52,73)(97,166,210,128,222)(98,167,211,129,223)(99,168,212,130,224)(100,157,213,131,225)(101,158,214,132,226)(102,159,215,121,227)(103,160,216,122,228)(104,161,205,123,217)(105,162,206,124,218)(106,163,207,125,219)(107,164,208,126,220)(108,165,209,127,221), (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), (1,164,62,45)(2,157,63,38)(3,162,64,43)(4,167,65,48)(5,160,66,41)(6,165,67,46)(7,158,68,39)(8,163,69,44)(9,168,70,37)(10,161,71,42)(11,166,72,47)(12,159,61,40)(13,83,150,108)(14,76,151,101)(15,81,152,106)(16,74,153,99)(17,79,154,104)(18,84,155,97)(19,77,156,102)(20,82,145,107)(21,75,146,100)(22,80,147,105)(23,73,148,98)(24,78,149,103)(25,92,188,208)(26,85,189,213)(27,90,190,206)(28,95,191,211)(29,88,192,216)(30,93,181,209)(31,86,182,214)(32,91,183,207)(33,96,184,212)(34,89,185,205)(35,94,186,210)(36,87,187,215)(49,119,220,135)(50,112,221,140)(51,117,222,133)(52,110,223,138)(53,115,224,143)(54,120,225,136)(55,113,226,141)(56,118,227,134)(57,111,228,139)(58,116,217,144)(59,109,218,137)(60,114,219,142)(121,196,237,175)(122,201,238,180)(123,194,239,173)(124,199,240,178)(125,204,229,171)(126,197,230,176)(127,202,231,169)(128,195,232,174)(129,200,233,179)(130,193,234,172)(131,198,235,177)(132,203,236,170), (1,4)(2,3)(5,12)(6,11)(7,10)(8,9)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,163)(38,162)(39,161)(40,160)(41,159)(42,158)(43,157)(44,168)(45,167)(46,166)(47,165)(48,164)(49,223)(50,222)(51,221)(52,220)(53,219)(54,218)(55,217)(56,228)(57,227)(58,226)(59,225)(60,224)(61,66)(62,65)(63,64)(67,72)(68,71)(69,70)(73,107)(74,106)(75,105)(76,104)(77,103)(78,102)(79,101)(80,100)(81,99)(82,98)(83,97)(84,108)(85,206)(86,205)(87,216)(88,215)(89,214)(90,213)(91,212)(92,211)(93,210)(94,209)(95,208)(96,207)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(121,238)(122,237)(123,236)(124,235)(125,234)(126,233)(127,232)(128,231)(129,230)(130,229)(131,240)(132,239)(133,140)(134,139)(135,138)(136,137)(141,144)(142,143)(145,148)(146,147)(149,156)(150,155)(151,154)(152,153)(169,174)(170,173)(171,172)(175,180)(176,179)(177,178)(181,186)(182,185)(183,184)(187,192)(188,191)(189,190)(193,204)(194,203)(195,202)(196,201)(197,200)(198,199) );
G=PermutationGroup([[(1,188,176,119,145),(2,189,177,120,146),(3,190,178,109,147),(4,191,179,110,148),(5,192,180,111,149),(6,181,169,112,150),(7,182,170,113,151),(8,183,171,114,152),(9,184,172,115,153),(10,185,173,116,154),(11,186,174,117,155),(12,187,175,118,156),(13,67,30,202,140),(14,68,31,203,141),(15,69,32,204,142),(16,70,33,193,143),(17,71,34,194,144),(18,72,35,195,133),(19,61,36,196,134),(20,62,25,197,135),(21,63,26,198,136),(22,64,27,199,137),(23,65,28,200,138),(24,66,29,201,139),(37,96,234,53,74),(38,85,235,54,75),(39,86,236,55,76),(40,87,237,56,77),(41,88,238,57,78),(42,89,239,58,79),(43,90,240,59,80),(44,91,229,60,81),(45,92,230,49,82),(46,93,231,50,83),(47,94,232,51,84),(48,95,233,52,73),(97,166,210,128,222),(98,167,211,129,223),(99,168,212,130,224),(100,157,213,131,225),(101,158,214,132,226),(102,159,215,121,227),(103,160,216,122,228),(104,161,205,123,217),(105,162,206,124,218),(106,163,207,125,219),(107,164,208,126,220),(108,165,209,127,221)], [(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)], [(1,164,62,45),(2,157,63,38),(3,162,64,43),(4,167,65,48),(5,160,66,41),(6,165,67,46),(7,158,68,39),(8,163,69,44),(9,168,70,37),(10,161,71,42),(11,166,72,47),(12,159,61,40),(13,83,150,108),(14,76,151,101),(15,81,152,106),(16,74,153,99),(17,79,154,104),(18,84,155,97),(19,77,156,102),(20,82,145,107),(21,75,146,100),(22,80,147,105),(23,73,148,98),(24,78,149,103),(25,92,188,208),(26,85,189,213),(27,90,190,206),(28,95,191,211),(29,88,192,216),(30,93,181,209),(31,86,182,214),(32,91,183,207),(33,96,184,212),(34,89,185,205),(35,94,186,210),(36,87,187,215),(49,119,220,135),(50,112,221,140),(51,117,222,133),(52,110,223,138),(53,115,224,143),(54,120,225,136),(55,113,226,141),(56,118,227,134),(57,111,228,139),(58,116,217,144),(59,109,218,137),(60,114,219,142),(121,196,237,175),(122,201,238,180),(123,194,239,173),(124,199,240,178),(125,204,229,171),(126,197,230,176),(127,202,231,169),(128,195,232,174),(129,200,233,179),(130,193,234,172),(131,198,235,177),(132,203,236,170)], [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22),(25,28),(26,27),(29,36),(30,35),(31,34),(32,33),(37,163),(38,162),(39,161),(40,160),(41,159),(42,158),(43,157),(44,168),(45,167),(46,166),(47,165),(48,164),(49,223),(50,222),(51,221),(52,220),(53,219),(54,218),(55,217),(56,228),(57,227),(58,226),(59,225),(60,224),(61,66),(62,65),(63,64),(67,72),(68,71),(69,70),(73,107),(74,106),(75,105),(76,104),(77,103),(78,102),(79,101),(80,100),(81,99),(82,98),(83,97),(84,108),(85,206),(86,205),(87,216),(88,215),(89,214),(90,213),(91,212),(92,211),(93,210),(94,209),(95,208),(96,207),(109,120),(110,119),(111,118),(112,117),(113,116),(114,115),(121,238),(122,237),(123,236),(124,235),(125,234),(126,233),(127,232),(128,231),(129,230),(130,229),(131,240),(132,239),(133,140),(134,139),(135,138),(136,137),(141,144),(142,143),(145,148),(146,147),(149,156),(150,155),(151,154),(152,153),(169,174),(170,173),(171,172),(175,180),(176,179),(177,178),(181,186),(182,185),(183,184),(187,192),(188,191),(189,190),(193,204),(194,203),(195,202),(196,201),(197,200),(198,199)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20X | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 12 | ··· | 12 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | C3⋊D4 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | C5×C3⋊D4 | S3×D4 | C5×S3×D4 |
| kernel | C5×C12⋊3D4 | Dic3×C20 | C10×D12 | C10×C3⋊D4 | D4×C30 | C12⋊3D4 | C4×Dic3 | C2×D12 | C2×C3⋊D4 | C6×D4 | D4×C10 | C5×Dic3 | C60 | C2×C20 | C22×C10 | C20 | C2×D4 | Dic3 | C12 | C2×C4 | C23 | C4 | C10 | C2 |
| # reps | 1 | 1 | 1 | 4 | 1 | 4 | 4 | 4 | 16 | 4 | 1 | 4 | 2 | 1 | 2 | 4 | 4 | 16 | 8 | 4 | 8 | 16 | 2 | 8 |
Matrix representation of C5×C12⋊3D4 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 58 | 0 |
| 0 | 0 | 0 | 58 |
| 0 | 1 | 0 | 0 |
| 60 | 1 | 0 | 0 |
| 0 | 0 | 1 | 59 |
| 0 | 0 | 1 | 60 |
| 43 | 9 | 0 | 0 |
| 52 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 60 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 59 |
| 0 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,58,0,0,0,0,58],[0,60,0,0,1,1,0,0,0,0,1,1,0,0,59,60],[43,52,0,0,9,18,0,0,0,0,1,0,0,0,0,1],[0,60,0,0,60,0,0,0,0,0,1,0,0,0,59,60] >;
C5×C12⋊3D4 in GAP, Magma, Sage, TeX
C_5\times C_{12}\rtimes_3D_4 % in TeX
G:=Group("C5xC12:3D4"); // GroupNames label
G:=SmallGroup(480,819);
// by ID
G=gap.SmallGroup(480,819);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,568,926,891,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations