direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×C4○D20, C30.72C24, C60.282C23, D20⋊11(C2×C6), (C6×D20)⋊30C2, (C2×D20)⋊14C6, (C2×C12)⋊37D10, C30⋊12(C4○D4), (C22×C60)⋊17C2, (C2×C60)⋊48C22, (C22×C20)⋊12C6, (C22×C12)⋊13D5, C10.4(C23×C6), C23.31(C6×D5), C6.72(C23×D5), Dic10⋊11(C2×C6), (C2×Dic10)⋊15C6, (C6×Dic10)⋊31C2, (D5×C12)⋊25C22, (C3×D20)⋊42C22, C20.42(C22×C6), D10.1(C22×C6), (C6×D5).52C23, (C2×C30).382C23, (C22×C6).111D10, C12.240(C22×D5), Dic5.2(C22×C6), (C3×Dic10)⋊38C22, (C3×Dic5).54C23, (C22×C30).167C22, (C6×Dic5).257C22, C5⋊1(C6×C4○D4), (C2×C4×D5)⋊15C6, C4.43(D5×C2×C6), (D5×C2×C12)⋊31C2, (C4×D5)⋊6(C2×C6), C10⋊1(C3×C4○D4), (C2×C4)⋊10(C6×D5), C15⋊21(C2×C4○D4), C5⋊D4⋊6(C2×C6), (C2×C20)⋊13(C2×C6), (C6×C5⋊D4)⋊27C2, (C2×C5⋊D4)⋊12C6, C2.5(D5×C22×C6), C22.5(D5×C2×C6), (C22×C4)⋊8(C3×D5), (C3×C5⋊D4)⋊21C22, (D5×C2×C6).138C22, (C22×C10).54(C2×C6), (C2×C10).65(C22×C6), (C2×Dic5).45(C2×C6), (C22×D5).33(C2×C6), (C2×C6).259(C22×D5), SmallGroup(480,1138)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C4○D20
G = < a,b,c,d | a6=b4=d2=1, c10=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c9 >
Subgroups: 912 in 328 conjugacy classes, 178 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C3×D5, C30, C30, C30, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C22×C12, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, C6×C4○D4, C3×Dic10, D5×C12, C3×D20, C6×Dic5, C3×C5⋊D4, C2×C60, C2×C60, D5×C2×C6, C22×C30, C2×C4○D20, C6×Dic10, D5×C2×C12, C6×D20, C3×C4○D20, C6×C5⋊D4, C22×C60, C6×C4○D20
Quotients: C1, C2, C3, C22, C6, C23, D5, C2×C6, C4○D4, C24, D10, C22×C6, C3×D5, C2×C4○D4, C22×D5, C3×C4○D4, C23×C6, C6×D5, C4○D20, C23×D5, C6×C4○D4, D5×C2×C6, C2×C4○D20, C3×C4○D20, D5×C22×C6, C6×C4○D20
►On 240 points
Generators in S240
(1 97 104 56 181 178)(2 98 105 57 182 179)(3 99 106 58 183 180)(4 100 107 59 184 161)(5 81 108 60 185 162)(6 82 109 41 186 163)(7 83 110 42 187 164)(8 84 111 43 188 165)(9 85 112 44 189 166)(10 86 113 45 190 167)(11 87 114 46 191 168)(12 88 115 47 192 169)(13 89 116 48 193 170)(14 90 117 49 194 171)(15 91 118 50 195 172)(16 92 119 51 196 173)(17 93 120 52 197 174)(18 94 101 53 198 175)(19 95 102 54 199 176)(20 96 103 55 200 177)(21 154 227 212 76 132)(22 155 228 213 77 133)(23 156 229 214 78 134)(24 157 230 215 79 135)(25 158 231 216 80 136)(26 159 232 217 61 137)(27 160 233 218 62 138)(28 141 234 219 63 139)(29 142 235 220 64 140)(30 143 236 201 65 121)(31 144 237 202 66 122)(32 145 238 203 67 123)(33 146 239 204 68 124)(34 147 240 205 69 125)(35 148 221 206 70 126)(36 149 222 207 71 127)(37 150 223 208 72 128)(38 151 224 209 73 129)(39 152 225 210 74 130)(40 153 226 211 75 131)
(1 218 11 208)(2 219 12 209)(3 220 13 210)(4 201 14 211)(5 202 15 212)(6 203 16 213)(7 204 17 214)(8 205 18 215)(9 206 19 216)(10 207 20 217)(21 60 31 50)(22 41 32 51)(23 42 33 52)(24 43 34 53)(25 44 35 54)(26 45 36 55)(27 46 37 56)(28 47 38 57)(29 48 39 58)(30 49 40 59)(61 86 71 96)(62 87 72 97)(63 88 73 98)(64 89 74 99)(65 90 75 100)(66 91 76 81)(67 92 77 82)(68 93 78 83)(69 94 79 84)(70 95 80 85)(101 135 111 125)(102 136 112 126)(103 137 113 127)(104 138 114 128)(105 139 115 129)(106 140 116 130)(107 121 117 131)(108 122 118 132)(109 123 119 133)(110 124 120 134)(141 192 151 182)(142 193 152 183)(143 194 153 184)(144 195 154 185)(145 196 155 186)(146 197 156 187)(147 198 157 188)(148 199 158 189)(149 200 159 190)(150 181 160 191)(161 236 171 226)(162 237 172 227)(163 238 173 228)(164 239 174 229)(165 240 175 230)(166 221 176 231)(167 222 177 232)(168 223 178 233)(169 224 179 234)(170 225 180 235)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(33 40)(34 39)(35 38)(36 37)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 62)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 100)(94 99)(95 98)(96 97)(101 106)(102 105)(103 104)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)(121 134)(122 133)(123 132)(124 131)(125 130)(126 129)(127 128)(135 140)(136 139)(137 138)(141 158)(142 157)(143 156)(144 155)(145 154)(146 153)(147 152)(148 151)(149 150)(159 160)(161 174)(162 173)(163 172)(164 171)(165 170)(166 169)(167 168)(175 180)(176 179)(177 178)(181 200)(182 199)(183 198)(184 197)(185 196)(186 195)(187 194)(188 193)(189 192)(190 191)(201 214)(202 213)(203 212)(204 211)(205 210)(206 209)(207 208)(215 220)(216 219)(217 218)(221 224)(222 223)(225 240)(226 239)(227 238)(228 237)(229 236)(230 235)(231 234)(232 233)
G:=sub<Sym(240)| (1,97,104,56,181,178)(2,98,105,57,182,179)(3,99,106,58,183,180)(4,100,107,59,184,161)(5,81,108,60,185,162)(6,82,109,41,186,163)(7,83,110,42,187,164)(8,84,111,43,188,165)(9,85,112,44,189,166)(10,86,113,45,190,167)(11,87,114,46,191,168)(12,88,115,47,192,169)(13,89,116,48,193,170)(14,90,117,49,194,171)(15,91,118,50,195,172)(16,92,119,51,196,173)(17,93,120,52,197,174)(18,94,101,53,198,175)(19,95,102,54,199,176)(20,96,103,55,200,177)(21,154,227,212,76,132)(22,155,228,213,77,133)(23,156,229,214,78,134)(24,157,230,215,79,135)(25,158,231,216,80,136)(26,159,232,217,61,137)(27,160,233,218,62,138)(28,141,234,219,63,139)(29,142,235,220,64,140)(30,143,236,201,65,121)(31,144,237,202,66,122)(32,145,238,203,67,123)(33,146,239,204,68,124)(34,147,240,205,69,125)(35,148,221,206,70,126)(36,149,222,207,71,127)(37,150,223,208,72,128)(38,151,224,209,73,129)(39,152,225,210,74,130)(40,153,226,211,75,131), (1,218,11,208)(2,219,12,209)(3,220,13,210)(4,201,14,211)(5,202,15,212)(6,203,16,213)(7,204,17,214)(8,205,18,215)(9,206,19,216)(10,207,20,217)(21,60,31,50)(22,41,32,51)(23,42,33,52)(24,43,34,53)(25,44,35,54)(26,45,36,55)(27,46,37,56)(28,47,38,57)(29,48,39,58)(30,49,40,59)(61,86,71,96)(62,87,72,97)(63,88,73,98)(64,89,74,99)(65,90,75,100)(66,91,76,81)(67,92,77,82)(68,93,78,83)(69,94,79,84)(70,95,80,85)(101,135,111,125)(102,136,112,126)(103,137,113,127)(104,138,114,128)(105,139,115,129)(106,140,116,130)(107,121,117,131)(108,122,118,132)(109,123,119,133)(110,124,120,134)(141,192,151,182)(142,193,152,183)(143,194,153,184)(144,195,154,185)(145,196,155,186)(146,197,156,187)(147,198,157,188)(148,199,158,189)(149,200,159,190)(150,181,160,191)(161,236,171,226)(162,237,172,227)(163,238,173,228)(164,239,174,229)(165,240,175,230)(166,221,176,231)(167,222,177,232)(168,223,178,233)(169,224,179,234)(170,225,180,235), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,100)(94,99)(95,98)(96,97)(101,106)(102,105)(103,104)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(135,140)(136,139)(137,138)(141,158)(142,157)(143,156)(144,155)(145,154)(146,153)(147,152)(148,151)(149,150)(159,160)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)(175,180)(176,179)(177,178)(181,200)(182,199)(183,198)(184,197)(185,196)(186,195)(187,194)(188,193)(189,192)(190,191)(201,214)(202,213)(203,212)(204,211)(205,210)(206,209)(207,208)(215,220)(216,219)(217,218)(221,224)(222,223)(225,240)(226,239)(227,238)(228,237)(229,236)(230,235)(231,234)(232,233)>;
G:=Group( (1,97,104,56,181,178)(2,98,105,57,182,179)(3,99,106,58,183,180)(4,100,107,59,184,161)(5,81,108,60,185,162)(6,82,109,41,186,163)(7,83,110,42,187,164)(8,84,111,43,188,165)(9,85,112,44,189,166)(10,86,113,45,190,167)(11,87,114,46,191,168)(12,88,115,47,192,169)(13,89,116,48,193,170)(14,90,117,49,194,171)(15,91,118,50,195,172)(16,92,119,51,196,173)(17,93,120,52,197,174)(18,94,101,53,198,175)(19,95,102,54,199,176)(20,96,103,55,200,177)(21,154,227,212,76,132)(22,155,228,213,77,133)(23,156,229,214,78,134)(24,157,230,215,79,135)(25,158,231,216,80,136)(26,159,232,217,61,137)(27,160,233,218,62,138)(28,141,234,219,63,139)(29,142,235,220,64,140)(30,143,236,201,65,121)(31,144,237,202,66,122)(32,145,238,203,67,123)(33,146,239,204,68,124)(34,147,240,205,69,125)(35,148,221,206,70,126)(36,149,222,207,71,127)(37,150,223,208,72,128)(38,151,224,209,73,129)(39,152,225,210,74,130)(40,153,226,211,75,131), (1,218,11,208)(2,219,12,209)(3,220,13,210)(4,201,14,211)(5,202,15,212)(6,203,16,213)(7,204,17,214)(8,205,18,215)(9,206,19,216)(10,207,20,217)(21,60,31,50)(22,41,32,51)(23,42,33,52)(24,43,34,53)(25,44,35,54)(26,45,36,55)(27,46,37,56)(28,47,38,57)(29,48,39,58)(30,49,40,59)(61,86,71,96)(62,87,72,97)(63,88,73,98)(64,89,74,99)(65,90,75,100)(66,91,76,81)(67,92,77,82)(68,93,78,83)(69,94,79,84)(70,95,80,85)(101,135,111,125)(102,136,112,126)(103,137,113,127)(104,138,114,128)(105,139,115,129)(106,140,116,130)(107,121,117,131)(108,122,118,132)(109,123,119,133)(110,124,120,134)(141,192,151,182)(142,193,152,183)(143,194,153,184)(144,195,154,185)(145,196,155,186)(146,197,156,187)(147,198,157,188)(148,199,158,189)(149,200,159,190)(150,181,160,191)(161,236,171,226)(162,237,172,227)(163,238,173,228)(164,239,174,229)(165,240,175,230)(166,221,176,231)(167,222,177,232)(168,223,178,233)(169,224,179,234)(170,225,180,235), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,100)(94,99)(95,98)(96,97)(101,106)(102,105)(103,104)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(135,140)(136,139)(137,138)(141,158)(142,157)(143,156)(144,155)(145,154)(146,153)(147,152)(148,151)(149,150)(159,160)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)(175,180)(176,179)(177,178)(181,200)(182,199)(183,198)(184,197)(185,196)(186,195)(187,194)(188,193)(189,192)(190,191)(201,214)(202,213)(203,212)(204,211)(205,210)(206,209)(207,208)(215,220)(216,219)(217,218)(221,224)(222,223)(225,240)(226,239)(227,238)(228,237)(229,236)(230,235)(231,234)(232,233) );
G=PermutationGroup([[(1,97,104,56,181,178),(2,98,105,57,182,179),(3,99,106,58,183,180),(4,100,107,59,184,161),(5,81,108,60,185,162),(6,82,109,41,186,163),(7,83,110,42,187,164),(8,84,111,43,188,165),(9,85,112,44,189,166),(10,86,113,45,190,167),(11,87,114,46,191,168),(12,88,115,47,192,169),(13,89,116,48,193,170),(14,90,117,49,194,171),(15,91,118,50,195,172),(16,92,119,51,196,173),(17,93,120,52,197,174),(18,94,101,53,198,175),(19,95,102,54,199,176),(20,96,103,55,200,177),(21,154,227,212,76,132),(22,155,228,213,77,133),(23,156,229,214,78,134),(24,157,230,215,79,135),(25,158,231,216,80,136),(26,159,232,217,61,137),(27,160,233,218,62,138),(28,141,234,219,63,139),(29,142,235,220,64,140),(30,143,236,201,65,121),(31,144,237,202,66,122),(32,145,238,203,67,123),(33,146,239,204,68,124),(34,147,240,205,69,125),(35,148,221,206,70,126),(36,149,222,207,71,127),(37,150,223,208,72,128),(38,151,224,209,73,129),(39,152,225,210,74,130),(40,153,226,211,75,131)], [(1,218,11,208),(2,219,12,209),(3,220,13,210),(4,201,14,211),(5,202,15,212),(6,203,16,213),(7,204,17,214),(8,205,18,215),(9,206,19,216),(10,207,20,217),(21,60,31,50),(22,41,32,51),(23,42,33,52),(24,43,34,53),(25,44,35,54),(26,45,36,55),(27,46,37,56),(28,47,38,57),(29,48,39,58),(30,49,40,59),(61,86,71,96),(62,87,72,97),(63,88,73,98),(64,89,74,99),(65,90,75,100),(66,91,76,81),(67,92,77,82),(68,93,78,83),(69,94,79,84),(70,95,80,85),(101,135,111,125),(102,136,112,126),(103,137,113,127),(104,138,114,128),(105,139,115,129),(106,140,116,130),(107,121,117,131),(108,122,118,132),(109,123,119,133),(110,124,120,134),(141,192,151,182),(142,193,152,183),(143,194,153,184),(144,195,154,185),(145,196,155,186),(146,197,156,187),(147,198,157,188),(148,199,158,189),(149,200,159,190),(150,181,160,191),(161,236,171,226),(162,237,172,227),(163,238,173,228),(164,239,174,229),(165,240,175,230),(166,221,176,231),(167,222,177,232),(168,223,178,233),(169,224,179,234),(170,225,180,235)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(33,40),(34,39),(35,38),(36,37),(41,50),(42,49),(43,48),(44,47),(45,46),(51,60),(52,59),(53,58),(54,57),(55,56),(61,62),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(93,100),(94,99),(95,98),(96,97),(101,106),(102,105),(103,104),(107,120),(108,119),(109,118),(110,117),(111,116),(112,115),(113,114),(121,134),(122,133),(123,132),(124,131),(125,130),(126,129),(127,128),(135,140),(136,139),(137,138),(141,158),(142,157),(143,156),(144,155),(145,154),(146,153),(147,152),(148,151),(149,150),(159,160),(161,174),(162,173),(163,172),(164,171),(165,170),(166,169),(167,168),(175,180),(176,179),(177,178),(181,200),(182,199),(183,198),(184,197),(185,196),(186,195),(187,194),(188,193),(189,192),(190,191),(201,214),(202,213),(203,212),(204,211),(205,210),(206,209),(207,208),(215,220),(216,219),(217,218),(221,224),(222,223),(225,240),(226,239),(227,238),(228,237),(229,236),(230,235),(231,234),(232,233)]])
156 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | ··· | 6R | 10A | ··· | 10N | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 12M | ··· | 12T | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | D5 | C4○D4 | D10 | D10 | C3×D5 | C3×C4○D4 | C6×D5 | C6×D5 | C4○D20 | C3×C4○D20 |
| kernel | C6×C4○D20 | C6×Dic10 | D5×C2×C12 | C6×D20 | C3×C4○D20 | C6×C5⋊D4 | C22×C60 | C2×C4○D20 | C2×Dic10 | C2×C4×D5 | C2×D20 | C4○D20 | C2×C5⋊D4 | C22×C20 | C22×C12 | C30 | C2×C12 | C22×C6 | C22×C4 | C10 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 2 | 2 | 4 | 2 | 16 | 4 | 2 | 2 | 4 | 12 | 2 | 4 | 8 | 24 | 4 | 16 | 32 |
Matrix representation of C6×C4○D20 ►in GL3(𝔽61) generated by
| 48 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 60 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 60 | 0 | 0 |
| 0 | 2 | 32 |
| 0 | 29 | 7 |
| 1 | 0 | 0 |
| 0 | 7 | 29 |
| 0 | 32 | 54 |
G:=sub<GL(3,GF(61))| [48,0,0,0,1,0,0,0,1],[60,0,0,0,11,0,0,0,11],[60,0,0,0,2,29,0,32,7],[1,0,0,0,7,32,0,29,54] >;
C6×C4○D20 in GAP, Magma, Sage, TeX
C_6\times C_4\circ D_{20} % in TeX
G:=Group("C6xC4oD20"); // GroupNames label
G:=SmallGroup(480,1138);
// by ID
G=gap.SmallGroup(480,1138);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,268,1571,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^4=d^2=1,c^10=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^9>;
// generators/relations