direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D20⋊8C4, D20⋊8C12, C5⋊4(D4×C12), C4⋊1(D5×C12), C12⋊9(C4×D5), C15⋊38(C4×D4), C20⋊5(C2×C12), C60⋊27(C2×C4), (C3×D20)⋊17C4, D10⋊4(C2×C12), Dic5⋊5(C3×D4), (C4×Dic5)⋊3C6, (C2×D20).8C6, C6.177(D4×D5), C10.23(C6×D4), (C3×Dic5)⋊20D4, (C6×D20).19C2, C30.336(C2×D4), D10⋊C4⋊12C6, (C12×Dic5)⋊15C2, (C2×C12).280D10, C30.263(C4○D4), C30.182(C22×C4), (C2×C60).398C22, C10.24(C22×C12), (C2×C30).351C23, C6.49(Q8⋊2D5), (C6×Dic5).283C22, (C5×C4⋊C4)⋊4C6, C2.4(C3×D4×D5), C4⋊C4⋊8(C3×D5), (C2×C4×D5)⋊12C6, (C3×C4⋊C4)⋊17D5, (D5×C2×C12)⋊28C2, (C15×C4⋊C4)⋊13C2, C2.13(D5×C2×C12), C6.107(C2×C4×D5), (C6×D5)⋊24(C2×C4), (C2×C4).41(C6×D5), C22.18(D5×C2×C6), (C2×C20).56(C2×C6), C10.33(C3×C4○D4), C2.2(C3×Q8⋊2D5), (C3×D10⋊C4)⋊28C2, (D5×C2×C6).128C22, (C2×C10).34(C22×C6), (C2×Dic5).61(C2×C6), (C22×D5).25(C2×C6), (C2×C6).347(C22×D5), SmallGroup(480,686)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D20⋊8C4
G = < a,b,c,d | a3=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b11, dcd-1=b10c >
Subgroups: 624 in 188 conjugacy classes, 86 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, D5, C10, C12, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C3×D5, C30, C4×D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C3×Dic5, C3×Dic5, C60, C60, C6×D5, C6×D5, C2×C30, C4×Dic5, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, D4×C12, D5×C12, C3×D20, C6×Dic5, C2×C60, C2×C60, D5×C2×C6, D20⋊8C4, C12×Dic5, C3×D10⋊C4, C15×C4⋊C4, D5×C2×C12, C6×D20, C3×D20⋊8C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, D5, C12, C2×C6, C22×C4, C2×D4, C4○D4, D10, C2×C12, C3×D4, C22×C6, C3×D5, C4×D4, C4×D5, C22×D5, C22×C12, C6×D4, C3×C4○D4, C6×D5, C2×C4×D5, D4×D5, Q8⋊2D5, D4×C12, D5×C12, D5×C2×C6, D20⋊8C4, D5×C2×C12, C3×D4×D5, C3×Q8⋊2D5, C3×D20⋊8C4
►On 240 points
Generators in S240
(1 112 133)(2 113 134)(3 114 135)(4 115 136)(5 116 137)(6 117 138)(7 118 139)(8 119 140)(9 120 121)(10 101 122)(11 102 123)(12 103 124)(13 104 125)(14 105 126)(15 106 127)(16 107 128)(17 108 129)(18 109 130)(19 110 131)(20 111 132)(21 56 158)(22 57 159)(23 58 160)(24 59 141)(25 60 142)(26 41 143)(27 42 144)(28 43 145)(29 44 146)(30 45 147)(31 46 148)(32 47 149)(33 48 150)(34 49 151)(35 50 152)(36 51 153)(37 52 154)(38 53 155)(39 54 156)(40 55 157)(61 206 187)(62 207 188)(63 208 189)(64 209 190)(65 210 191)(66 211 192)(67 212 193)(68 213 194)(69 214 195)(70 215 196)(71 216 197)(72 217 198)(73 218 199)(74 219 200)(75 220 181)(76 201 182)(77 202 183)(78 203 184)(79 204 185)(80 205 186)(81 165 234)(82 166 235)(83 167 236)(84 168 237)(85 169 238)(86 170 239)(87 171 240)(88 172 221)(89 173 222)(90 174 223)(91 175 224)(92 176 225)(93 177 226)(94 178 227)(95 179 228)(96 180 229)(97 161 230)(98 162 231)(99 163 232)(100 164 233)
(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 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 44)(42 43)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 86)(82 85)(83 84)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(101 102)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)(121 124)(122 123)(125 140)(126 139)(127 138)(128 137)(129 136)(130 135)(131 134)(132 133)(141 148)(142 147)(143 146)(144 145)(149 160)(150 159)(151 158)(152 157)(153 156)(154 155)(161 174)(162 173)(163 172)(164 171)(165 170)(166 169)(167 168)(175 180)(176 179)(177 178)(181 182)(183 200)(184 199)(185 198)(186 197)(187 196)(188 195)(189 194)(190 193)(191 192)(201 220)(202 219)(203 218)(204 217)(205 216)(206 215)(207 214)(208 213)(209 212)(210 211)(221 232)(222 231)(223 230)(224 229)(225 228)(226 227)(233 240)(234 239)(235 238)(236 237)
(1 178 53 182)(2 169 54 193)(3 180 55 184)(4 171 56 195)(5 162 57 186)(6 173 58 197)(7 164 59 188)(8 175 60 199)(9 166 41 190)(10 177 42 181)(11 168 43 192)(12 179 44 183)(13 170 45 194)(14 161 46 185)(15 172 47 196)(16 163 48 187)(17 174 49 198)(18 165 50 189)(19 176 51 200)(20 167 52 191)(21 214 136 87)(22 205 137 98)(23 216 138 89)(24 207 139 100)(25 218 140 91)(26 209 121 82)(27 220 122 93)(28 211 123 84)(29 202 124 95)(30 213 125 86)(31 204 126 97)(32 215 127 88)(33 206 128 99)(34 217 129 90)(35 208 130 81)(36 219 131 92)(37 210 132 83)(38 201 133 94)(39 212 134 85)(40 203 135 96)(61 107 232 150)(62 118 233 141)(63 109 234 152)(64 120 235 143)(65 111 236 154)(66 102 237 145)(67 113 238 156)(68 104 239 147)(69 115 240 158)(70 106 221 149)(71 117 222 160)(72 108 223 151)(73 119 224 142)(74 110 225 153)(75 101 226 144)(76 112 227 155)(77 103 228 146)(78 114 229 157)(79 105 230 148)(80 116 231 159)
G:=sub<Sym(240)| (1,112,133)(2,113,134)(3,114,135)(4,115,136)(5,116,137)(6,117,138)(7,118,139)(8,119,140)(9,120,121)(10,101,122)(11,102,123)(12,103,124)(13,104,125)(14,105,126)(15,106,127)(16,107,128)(17,108,129)(18,109,130)(19,110,131)(20,111,132)(21,56,158)(22,57,159)(23,58,160)(24,59,141)(25,60,142)(26,41,143)(27,42,144)(28,43,145)(29,44,146)(30,45,147)(31,46,148)(32,47,149)(33,48,150)(34,49,151)(35,50,152)(36,51,153)(37,52,154)(38,53,155)(39,54,156)(40,55,157)(61,206,187)(62,207,188)(63,208,189)(64,209,190)(65,210,191)(66,211,192)(67,212,193)(68,213,194)(69,214,195)(70,215,196)(71,216,197)(72,217,198)(73,218,199)(74,219,200)(75,220,181)(76,201,182)(77,202,183)(78,203,184)(79,204,185)(80,205,186)(81,165,234)(82,166,235)(83,167,236)(84,168,237)(85,169,238)(86,170,239)(87,171,240)(88,172,221)(89,173,222)(90,174,223)(91,175,224)(92,176,225)(93,177,226)(94,178,227)(95,179,228)(96,180,229)(97,161,230)(98,162,231)(99,163,232)(100,164,233), (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)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,44)(42,43)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(121,124)(122,123)(125,140)(126,139)(127,138)(128,137)(129,136)(130,135)(131,134)(132,133)(141,148)(142,147)(143,146)(144,145)(149,160)(150,159)(151,158)(152,157)(153,156)(154,155)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)(175,180)(176,179)(177,178)(181,182)(183,200)(184,199)(185,198)(186,197)(187,196)(188,195)(189,194)(190,193)(191,192)(201,220)(202,219)(203,218)(204,217)(205,216)(206,215)(207,214)(208,213)(209,212)(210,211)(221,232)(222,231)(223,230)(224,229)(225,228)(226,227)(233,240)(234,239)(235,238)(236,237), (1,178,53,182)(2,169,54,193)(3,180,55,184)(4,171,56,195)(5,162,57,186)(6,173,58,197)(7,164,59,188)(8,175,60,199)(9,166,41,190)(10,177,42,181)(11,168,43,192)(12,179,44,183)(13,170,45,194)(14,161,46,185)(15,172,47,196)(16,163,48,187)(17,174,49,198)(18,165,50,189)(19,176,51,200)(20,167,52,191)(21,214,136,87)(22,205,137,98)(23,216,138,89)(24,207,139,100)(25,218,140,91)(26,209,121,82)(27,220,122,93)(28,211,123,84)(29,202,124,95)(30,213,125,86)(31,204,126,97)(32,215,127,88)(33,206,128,99)(34,217,129,90)(35,208,130,81)(36,219,131,92)(37,210,132,83)(38,201,133,94)(39,212,134,85)(40,203,135,96)(61,107,232,150)(62,118,233,141)(63,109,234,152)(64,120,235,143)(65,111,236,154)(66,102,237,145)(67,113,238,156)(68,104,239,147)(69,115,240,158)(70,106,221,149)(71,117,222,160)(72,108,223,151)(73,119,224,142)(74,110,225,153)(75,101,226,144)(76,112,227,155)(77,103,228,146)(78,114,229,157)(79,105,230,148)(80,116,231,159)>;
G:=Group( (1,112,133)(2,113,134)(3,114,135)(4,115,136)(5,116,137)(6,117,138)(7,118,139)(8,119,140)(9,120,121)(10,101,122)(11,102,123)(12,103,124)(13,104,125)(14,105,126)(15,106,127)(16,107,128)(17,108,129)(18,109,130)(19,110,131)(20,111,132)(21,56,158)(22,57,159)(23,58,160)(24,59,141)(25,60,142)(26,41,143)(27,42,144)(28,43,145)(29,44,146)(30,45,147)(31,46,148)(32,47,149)(33,48,150)(34,49,151)(35,50,152)(36,51,153)(37,52,154)(38,53,155)(39,54,156)(40,55,157)(61,206,187)(62,207,188)(63,208,189)(64,209,190)(65,210,191)(66,211,192)(67,212,193)(68,213,194)(69,214,195)(70,215,196)(71,216,197)(72,217,198)(73,218,199)(74,219,200)(75,220,181)(76,201,182)(77,202,183)(78,203,184)(79,204,185)(80,205,186)(81,165,234)(82,166,235)(83,167,236)(84,168,237)(85,169,238)(86,170,239)(87,171,240)(88,172,221)(89,173,222)(90,174,223)(91,175,224)(92,176,225)(93,177,226)(94,178,227)(95,179,228)(96,180,229)(97,161,230)(98,162,231)(99,163,232)(100,164,233), (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)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,44)(42,43)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(121,124)(122,123)(125,140)(126,139)(127,138)(128,137)(129,136)(130,135)(131,134)(132,133)(141,148)(142,147)(143,146)(144,145)(149,160)(150,159)(151,158)(152,157)(153,156)(154,155)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)(175,180)(176,179)(177,178)(181,182)(183,200)(184,199)(185,198)(186,197)(187,196)(188,195)(189,194)(190,193)(191,192)(201,220)(202,219)(203,218)(204,217)(205,216)(206,215)(207,214)(208,213)(209,212)(210,211)(221,232)(222,231)(223,230)(224,229)(225,228)(226,227)(233,240)(234,239)(235,238)(236,237), (1,178,53,182)(2,169,54,193)(3,180,55,184)(4,171,56,195)(5,162,57,186)(6,173,58,197)(7,164,59,188)(8,175,60,199)(9,166,41,190)(10,177,42,181)(11,168,43,192)(12,179,44,183)(13,170,45,194)(14,161,46,185)(15,172,47,196)(16,163,48,187)(17,174,49,198)(18,165,50,189)(19,176,51,200)(20,167,52,191)(21,214,136,87)(22,205,137,98)(23,216,138,89)(24,207,139,100)(25,218,140,91)(26,209,121,82)(27,220,122,93)(28,211,123,84)(29,202,124,95)(30,213,125,86)(31,204,126,97)(32,215,127,88)(33,206,128,99)(34,217,129,90)(35,208,130,81)(36,219,131,92)(37,210,132,83)(38,201,133,94)(39,212,134,85)(40,203,135,96)(61,107,232,150)(62,118,233,141)(63,109,234,152)(64,120,235,143)(65,111,236,154)(66,102,237,145)(67,113,238,156)(68,104,239,147)(69,115,240,158)(70,106,221,149)(71,117,222,160)(72,108,223,151)(73,119,224,142)(74,110,225,153)(75,101,226,144)(76,112,227,155)(77,103,228,146)(78,114,229,157)(79,105,230,148)(80,116,231,159) );
G=PermutationGroup([[(1,112,133),(2,113,134),(3,114,135),(4,115,136),(5,116,137),(6,117,138),(7,118,139),(8,119,140),(9,120,121),(10,101,122),(11,102,123),(12,103,124),(13,104,125),(14,105,126),(15,106,127),(16,107,128),(17,108,129),(18,109,130),(19,110,131),(20,111,132),(21,56,158),(22,57,159),(23,58,160),(24,59,141),(25,60,142),(26,41,143),(27,42,144),(28,43,145),(29,44,146),(30,45,147),(31,46,148),(32,47,149),(33,48,150),(34,49,151),(35,50,152),(36,51,153),(37,52,154),(38,53,155),(39,54,156),(40,55,157),(61,206,187),(62,207,188),(63,208,189),(64,209,190),(65,210,191),(66,211,192),(67,212,193),(68,213,194),(69,214,195),(70,215,196),(71,216,197),(72,217,198),(73,218,199),(74,219,200),(75,220,181),(76,201,182),(77,202,183),(78,203,184),(79,204,185),(80,205,186),(81,165,234),(82,166,235),(83,167,236),(84,168,237),(85,169,238),(86,170,239),(87,171,240),(88,172,221),(89,173,222),(90,174,223),(91,175,224),(92,176,225),(93,177,226),(94,178,227),(95,179,228),(96,180,229),(97,161,230),(98,162,231),(99,163,232),(100,164,233)], [(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),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(35,40),(36,39),(37,38),(41,44),(42,43),(45,60),(46,59),(47,58),(48,57),(49,56),(50,55),(51,54),(52,53),(61,70),(62,69),(63,68),(64,67),(65,66),(71,80),(72,79),(73,78),(74,77),(75,76),(81,86),(82,85),(83,84),(87,100),(88,99),(89,98),(90,97),(91,96),(92,95),(93,94),(101,102),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112),(121,124),(122,123),(125,140),(126,139),(127,138),(128,137),(129,136),(130,135),(131,134),(132,133),(141,148),(142,147),(143,146),(144,145),(149,160),(150,159),(151,158),(152,157),(153,156),(154,155),(161,174),(162,173),(163,172),(164,171),(165,170),(166,169),(167,168),(175,180),(176,179),(177,178),(181,182),(183,200),(184,199),(185,198),(186,197),(187,196),(188,195),(189,194),(190,193),(191,192),(201,220),(202,219),(203,218),(204,217),(205,216),(206,215),(207,214),(208,213),(209,212),(210,211),(221,232),(222,231),(223,230),(224,229),(225,228),(226,227),(233,240),(234,239),(235,238),(236,237)], [(1,178,53,182),(2,169,54,193),(3,180,55,184),(4,171,56,195),(5,162,57,186),(6,173,58,197),(7,164,59,188),(8,175,60,199),(9,166,41,190),(10,177,42,181),(11,168,43,192),(12,179,44,183),(13,170,45,194),(14,161,46,185),(15,172,47,196),(16,163,48,187),(17,174,49,198),(18,165,50,189),(19,176,51,200),(20,167,52,191),(21,214,136,87),(22,205,137,98),(23,216,138,89),(24,207,139,100),(25,218,140,91),(26,209,121,82),(27,220,122,93),(28,211,123,84),(29,202,124,95),(30,213,125,86),(31,204,126,97),(32,215,127,88),(33,206,128,99),(34,217,129,90),(35,208,130,81),(36,219,131,92),(37,210,132,83),(38,201,133,94),(39,212,134,85),(40,203,135,96),(61,107,232,150),(62,118,233,141),(63,109,234,152),(64,120,235,143),(65,111,236,154),(66,102,237,145),(67,113,238,156),(68,104,239,147),(69,115,240,158),(70,106,221,149),(71,117,222,160),(72,108,223,151),(73,119,224,142),(74,110,225,153),(75,101,226,144),(76,112,227,155),(77,103,228,146),(78,114,229,157),(79,105,230,148),(80,116,231,159)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | ··· | 6F | 6G | ··· | 6N | 10A | ··· | 10F | 12A | ··· | 12L | 12M | ··· | 12T | 12U | 12V | 12W | 12X | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 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 | 10 | 10 | 10 | 10 | 1 | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
120 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | D4 | D5 | C4○D4 | D10 | C3×D4 | C3×D5 | C4×D5 | C3×C4○D4 | C6×D5 | D5×C12 | D4×D5 | Q8⋊2D5 | C3×D4×D5 | C3×Q8⋊2D5 |
| kernel | C3×D20⋊8C4 | C12×Dic5 | C3×D10⋊C4 | C15×C4⋊C4 | D5×C2×C12 | C6×D20 | D20⋊8C4 | C3×D20 | C4×Dic5 | D10⋊C4 | C5×C4⋊C4 | C2×C4×D5 | C2×D20 | D20 | C3×Dic5 | C3×C4⋊C4 | C30 | C2×C12 | Dic5 | C4⋊C4 | C12 | C10 | C2×C4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 8 | 2 | 4 | 2 | 4 | 2 | 16 | 2 | 2 | 2 | 6 | 4 | 4 | 8 | 4 | 12 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C3×D20⋊8C4 ►in GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 59 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 18 | 60 |
| 0 | 0 | 19 | 60 |
| 60 | 59 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 17 |
| 0 | 0 | 18 | 0 |
| 60 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[60,1,0,0,59,1,0,0,0,0,18,19,0,0,60,60],[60,0,0,0,59,1,0,0,0,0,0,18,0,0,17,0],[60,1,0,0,0,1,0,0,0,0,11,0,0,0,0,11] >;
C3×D20⋊8C4 in GAP, Magma, Sage, TeX
C_3\times D_{20}\rtimes_8C_4 % in TeX
G:=Group("C3xD20:8C4"); // GroupNames label
G:=SmallGroup(480,686);
// by ID
G=gap.SmallGroup(480,686);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,344,555,142,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11,d*c*d^-1=b^10*c>;
// generators/relations