direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C20.53D4, C60.231D4, (C2×C30).9Q8, C5⋊2C8.1C12, C4.13(D5×C12), C20.53(C3×D4), C12.83(C4×D5), C30.51(C4⋊C4), C20.26(C2×C12), C60.160(C2×C4), (C2×C6).4Dic10, C15⋊14(C8.C4), C4.Dic5.2C6, (C2×C12).352D10, (C3×M4(2)).1D5, M4(2).1(C3×D5), (C5×M4(2)).1C6, C12.121(C5⋊D4), (C2×C60).275C22, (C15×M4(2)).1C2, C22.1(C3×Dic10), C6.18(C10.D4), (C2×C10).(C3×Q8), C5⋊4(C3×C8.C4), C10.15(C3×C4⋊C4), (C2×C5⋊2C8).4C6, (C3×C5⋊2C8).4C4, (C2×C4).31(C6×D5), C4.28(C3×C5⋊D4), (C6×C5⋊2C8).16C2, (C2×C20).11(C2×C6), C2.5(C3×C10.D4), (C3×C4.Dic5).6C2, SmallGroup(480,100)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C20.53D4
G = < a,b,c,d | a3=b20=1, c4=b10, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b9, dcd-1=b10c3 >
Subgroups: 128 in 60 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C2×C4, C10, C10, C12, C2×C6, C15, C2×C8, M4(2), M4(2), C20, C2×C10, C24, C2×C12, C30, C30, C8.C4, C5⋊2C8, C5⋊2C8, C40, C2×C20, C2×C24, C3×M4(2), C3×M4(2), C60, C2×C30, C2×C5⋊2C8, C4.Dic5, C5×M4(2), C3×C8.C4, C3×C5⋊2C8, C3×C5⋊2C8, C120, C2×C60, C20.53D4, C6×C5⋊2C8, C3×C4.Dic5, C15×M4(2), C3×C20.53D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, D5, C12, C2×C6, C4⋊C4, D10, C2×C12, C3×D4, C3×Q8, C3×D5, C8.C4, Dic10, C4×D5, C5⋊D4, C3×C4⋊C4, C6×D5, C10.D4, C3×C8.C4, C3×Dic10, D5×C12, C3×C5⋊D4, C20.53D4, C3×C10.D4, C3×C20.53D4
►On 240 points
Generators in S240
(1 58 38)(2 59 39)(3 60 40)(4 41 21)(5 42 22)(6 43 23)(7 44 24)(8 45 25)(9 46 26)(10 47 27)(11 48 28)(12 49 29)(13 50 30)(14 51 31)(15 52 32)(16 53 33)(17 54 34)(18 55 35)(19 56 36)(20 57 37)(61 106 83)(62 107 84)(63 108 85)(64 109 86)(65 110 87)(66 111 88)(67 112 89)(68 113 90)(69 114 91)(70 115 92)(71 116 93)(72 117 94)(73 118 95)(74 119 96)(75 120 97)(76 101 98)(77 102 99)(78 103 100)(79 104 81)(80 105 82)(121 161 156)(122 162 157)(123 163 158)(124 164 159)(125 165 160)(126 166 141)(127 167 142)(128 168 143)(129 169 144)(130 170 145)(131 171 146)(132 172 147)(133 173 148)(134 174 149)(135 175 150)(136 176 151)(137 177 152)(138 178 153)(139 179 154)(140 180 155)(181 221 211)(182 222 212)(183 223 213)(184 224 214)(185 225 215)(186 226 216)(187 227 217)(188 228 218)(189 229 219)(190 230 220)(191 231 201)(192 232 202)(193 233 203)(194 234 204)(195 235 205)(196 236 206)(197 237 207)(198 238 208)(199 239 209)(200 240 210)
(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)
(1 136 16 131 11 126 6 121)(2 125 17 140 12 135 7 130)(3 134 18 129 13 124 8 139)(4 123 19 138 14 133 9 128)(5 132 20 127 15 122 10 137)(21 158 36 153 31 148 26 143)(22 147 37 142 32 157 27 152)(23 156 38 151 33 146 28 141)(24 145 39 160 34 155 29 150)(25 154 40 149 35 144 30 159)(41 163 56 178 51 173 46 168)(42 172 57 167 52 162 47 177)(43 161 58 176 53 171 48 166)(44 170 59 165 54 180 49 175)(45 179 60 174 55 169 50 164)(61 183 66 188 71 193 76 198)(62 192 67 197 72 182 77 187)(63 181 68 186 73 191 78 196)(64 190 69 195 74 200 79 185)(65 199 70 184 75 189 80 194)(81 215 86 220 91 205 96 210)(82 204 87 209 92 214 97 219)(83 213 88 218 93 203 98 208)(84 202 89 207 94 212 99 217)(85 211 90 216 95 201 100 206)(101 238 106 223 111 228 116 233)(102 227 107 232 112 237 117 222)(103 236 108 221 113 226 118 231)(104 225 109 230 114 235 119 240)(105 234 110 239 115 224 120 229)
(1 191 6 196 11 181 16 186)(2 200 7 185 12 190 17 195)(3 189 8 194 13 199 18 184)(4 198 9 183 14 188 19 193)(5 187 10 192 15 197 20 182)(21 208 26 213 31 218 36 203)(22 217 27 202 32 207 37 212)(23 206 28 211 33 216 38 201)(24 215 29 220 34 205 39 210)(25 204 30 209 35 214 40 219)(41 238 46 223 51 228 56 233)(42 227 47 232 52 237 57 222)(43 236 48 221 53 226 58 231)(44 225 49 230 54 235 59 240)(45 234 50 239 55 224 60 229)(61 133 66 138 71 123 76 128)(62 122 67 127 72 132 77 137)(63 131 68 136 73 121 78 126)(64 140 69 125 74 130 79 135)(65 129 70 134 75 139 80 124)(81 150 86 155 91 160 96 145)(82 159 87 144 92 149 97 154)(83 148 88 153 93 158 98 143)(84 157 89 142 94 147 99 152)(85 146 90 151 95 156 100 141)(101 168 106 173 111 178 116 163)(102 177 107 162 112 167 117 172)(103 166 108 171 113 176 118 161)(104 175 109 180 114 165 119 170)(105 164 110 169 115 174 120 179)
G:=sub<Sym(240)| (1,58,38)(2,59,39)(3,60,40)(4,41,21)(5,42,22)(6,43,23)(7,44,24)(8,45,25)(9,46,26)(10,47,27)(11,48,28)(12,49,29)(13,50,30)(14,51,31)(15,52,32)(16,53,33)(17,54,34)(18,55,35)(19,56,36)(20,57,37)(61,106,83)(62,107,84)(63,108,85)(64,109,86)(65,110,87)(66,111,88)(67,112,89)(68,113,90)(69,114,91)(70,115,92)(71,116,93)(72,117,94)(73,118,95)(74,119,96)(75,120,97)(76,101,98)(77,102,99)(78,103,100)(79,104,81)(80,105,82)(121,161,156)(122,162,157)(123,163,158)(124,164,159)(125,165,160)(126,166,141)(127,167,142)(128,168,143)(129,169,144)(130,170,145)(131,171,146)(132,172,147)(133,173,148)(134,174,149)(135,175,150)(136,176,151)(137,177,152)(138,178,153)(139,179,154)(140,180,155)(181,221,211)(182,222,212)(183,223,213)(184,224,214)(185,225,215)(186,226,216)(187,227,217)(188,228,218)(189,229,219)(190,230,220)(191,231,201)(192,232,202)(193,233,203)(194,234,204)(195,235,205)(196,236,206)(197,237,207)(198,238,208)(199,239,209)(200,240,210), (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), (1,136,16,131,11,126,6,121)(2,125,17,140,12,135,7,130)(3,134,18,129,13,124,8,139)(4,123,19,138,14,133,9,128)(5,132,20,127,15,122,10,137)(21,158,36,153,31,148,26,143)(22,147,37,142,32,157,27,152)(23,156,38,151,33,146,28,141)(24,145,39,160,34,155,29,150)(25,154,40,149,35,144,30,159)(41,163,56,178,51,173,46,168)(42,172,57,167,52,162,47,177)(43,161,58,176,53,171,48,166)(44,170,59,165,54,180,49,175)(45,179,60,174,55,169,50,164)(61,183,66,188,71,193,76,198)(62,192,67,197,72,182,77,187)(63,181,68,186,73,191,78,196)(64,190,69,195,74,200,79,185)(65,199,70,184,75,189,80,194)(81,215,86,220,91,205,96,210)(82,204,87,209,92,214,97,219)(83,213,88,218,93,203,98,208)(84,202,89,207,94,212,99,217)(85,211,90,216,95,201,100,206)(101,238,106,223,111,228,116,233)(102,227,107,232,112,237,117,222)(103,236,108,221,113,226,118,231)(104,225,109,230,114,235,119,240)(105,234,110,239,115,224,120,229), (1,191,6,196,11,181,16,186)(2,200,7,185,12,190,17,195)(3,189,8,194,13,199,18,184)(4,198,9,183,14,188,19,193)(5,187,10,192,15,197,20,182)(21,208,26,213,31,218,36,203)(22,217,27,202,32,207,37,212)(23,206,28,211,33,216,38,201)(24,215,29,220,34,205,39,210)(25,204,30,209,35,214,40,219)(41,238,46,223,51,228,56,233)(42,227,47,232,52,237,57,222)(43,236,48,221,53,226,58,231)(44,225,49,230,54,235,59,240)(45,234,50,239,55,224,60,229)(61,133,66,138,71,123,76,128)(62,122,67,127,72,132,77,137)(63,131,68,136,73,121,78,126)(64,140,69,125,74,130,79,135)(65,129,70,134,75,139,80,124)(81,150,86,155,91,160,96,145)(82,159,87,144,92,149,97,154)(83,148,88,153,93,158,98,143)(84,157,89,142,94,147,99,152)(85,146,90,151,95,156,100,141)(101,168,106,173,111,178,116,163)(102,177,107,162,112,167,117,172)(103,166,108,171,113,176,118,161)(104,175,109,180,114,165,119,170)(105,164,110,169,115,174,120,179)>;
G:=Group( (1,58,38)(2,59,39)(3,60,40)(4,41,21)(5,42,22)(6,43,23)(7,44,24)(8,45,25)(9,46,26)(10,47,27)(11,48,28)(12,49,29)(13,50,30)(14,51,31)(15,52,32)(16,53,33)(17,54,34)(18,55,35)(19,56,36)(20,57,37)(61,106,83)(62,107,84)(63,108,85)(64,109,86)(65,110,87)(66,111,88)(67,112,89)(68,113,90)(69,114,91)(70,115,92)(71,116,93)(72,117,94)(73,118,95)(74,119,96)(75,120,97)(76,101,98)(77,102,99)(78,103,100)(79,104,81)(80,105,82)(121,161,156)(122,162,157)(123,163,158)(124,164,159)(125,165,160)(126,166,141)(127,167,142)(128,168,143)(129,169,144)(130,170,145)(131,171,146)(132,172,147)(133,173,148)(134,174,149)(135,175,150)(136,176,151)(137,177,152)(138,178,153)(139,179,154)(140,180,155)(181,221,211)(182,222,212)(183,223,213)(184,224,214)(185,225,215)(186,226,216)(187,227,217)(188,228,218)(189,229,219)(190,230,220)(191,231,201)(192,232,202)(193,233,203)(194,234,204)(195,235,205)(196,236,206)(197,237,207)(198,238,208)(199,239,209)(200,240,210), (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), (1,136,16,131,11,126,6,121)(2,125,17,140,12,135,7,130)(3,134,18,129,13,124,8,139)(4,123,19,138,14,133,9,128)(5,132,20,127,15,122,10,137)(21,158,36,153,31,148,26,143)(22,147,37,142,32,157,27,152)(23,156,38,151,33,146,28,141)(24,145,39,160,34,155,29,150)(25,154,40,149,35,144,30,159)(41,163,56,178,51,173,46,168)(42,172,57,167,52,162,47,177)(43,161,58,176,53,171,48,166)(44,170,59,165,54,180,49,175)(45,179,60,174,55,169,50,164)(61,183,66,188,71,193,76,198)(62,192,67,197,72,182,77,187)(63,181,68,186,73,191,78,196)(64,190,69,195,74,200,79,185)(65,199,70,184,75,189,80,194)(81,215,86,220,91,205,96,210)(82,204,87,209,92,214,97,219)(83,213,88,218,93,203,98,208)(84,202,89,207,94,212,99,217)(85,211,90,216,95,201,100,206)(101,238,106,223,111,228,116,233)(102,227,107,232,112,237,117,222)(103,236,108,221,113,226,118,231)(104,225,109,230,114,235,119,240)(105,234,110,239,115,224,120,229), (1,191,6,196,11,181,16,186)(2,200,7,185,12,190,17,195)(3,189,8,194,13,199,18,184)(4,198,9,183,14,188,19,193)(5,187,10,192,15,197,20,182)(21,208,26,213,31,218,36,203)(22,217,27,202,32,207,37,212)(23,206,28,211,33,216,38,201)(24,215,29,220,34,205,39,210)(25,204,30,209,35,214,40,219)(41,238,46,223,51,228,56,233)(42,227,47,232,52,237,57,222)(43,236,48,221,53,226,58,231)(44,225,49,230,54,235,59,240)(45,234,50,239,55,224,60,229)(61,133,66,138,71,123,76,128)(62,122,67,127,72,132,77,137)(63,131,68,136,73,121,78,126)(64,140,69,125,74,130,79,135)(65,129,70,134,75,139,80,124)(81,150,86,155,91,160,96,145)(82,159,87,144,92,149,97,154)(83,148,88,153,93,158,98,143)(84,157,89,142,94,147,99,152)(85,146,90,151,95,156,100,141)(101,168,106,173,111,178,116,163)(102,177,107,162,112,167,117,172)(103,166,108,171,113,176,118,161)(104,175,109,180,114,165,119,170)(105,164,110,169,115,174,120,179) );
G=PermutationGroup([[(1,58,38),(2,59,39),(3,60,40),(4,41,21),(5,42,22),(6,43,23),(7,44,24),(8,45,25),(9,46,26),(10,47,27),(11,48,28),(12,49,29),(13,50,30),(14,51,31),(15,52,32),(16,53,33),(17,54,34),(18,55,35),(19,56,36),(20,57,37),(61,106,83),(62,107,84),(63,108,85),(64,109,86),(65,110,87),(66,111,88),(67,112,89),(68,113,90),(69,114,91),(70,115,92),(71,116,93),(72,117,94),(73,118,95),(74,119,96),(75,120,97),(76,101,98),(77,102,99),(78,103,100),(79,104,81),(80,105,82),(121,161,156),(122,162,157),(123,163,158),(124,164,159),(125,165,160),(126,166,141),(127,167,142),(128,168,143),(129,169,144),(130,170,145),(131,171,146),(132,172,147),(133,173,148),(134,174,149),(135,175,150),(136,176,151),(137,177,152),(138,178,153),(139,179,154),(140,180,155),(181,221,211),(182,222,212),(183,223,213),(184,224,214),(185,225,215),(186,226,216),(187,227,217),(188,228,218),(189,229,219),(190,230,220),(191,231,201),(192,232,202),(193,233,203),(194,234,204),(195,235,205),(196,236,206),(197,237,207),(198,238,208),(199,239,209),(200,240,210)], [(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)], [(1,136,16,131,11,126,6,121),(2,125,17,140,12,135,7,130),(3,134,18,129,13,124,8,139),(4,123,19,138,14,133,9,128),(5,132,20,127,15,122,10,137),(21,158,36,153,31,148,26,143),(22,147,37,142,32,157,27,152),(23,156,38,151,33,146,28,141),(24,145,39,160,34,155,29,150),(25,154,40,149,35,144,30,159),(41,163,56,178,51,173,46,168),(42,172,57,167,52,162,47,177),(43,161,58,176,53,171,48,166),(44,170,59,165,54,180,49,175),(45,179,60,174,55,169,50,164),(61,183,66,188,71,193,76,198),(62,192,67,197,72,182,77,187),(63,181,68,186,73,191,78,196),(64,190,69,195,74,200,79,185),(65,199,70,184,75,189,80,194),(81,215,86,220,91,205,96,210),(82,204,87,209,92,214,97,219),(83,213,88,218,93,203,98,208),(84,202,89,207,94,212,99,217),(85,211,90,216,95,201,100,206),(101,238,106,223,111,228,116,233),(102,227,107,232,112,237,117,222),(103,236,108,221,113,226,118,231),(104,225,109,230,114,235,119,240),(105,234,110,239,115,224,120,229)], [(1,191,6,196,11,181,16,186),(2,200,7,185,12,190,17,195),(3,189,8,194,13,199,18,184),(4,198,9,183,14,188,19,193),(5,187,10,192,15,197,20,182),(21,208,26,213,31,218,36,203),(22,217,27,202,32,207,37,212),(23,206,28,211,33,216,38,201),(24,215,29,220,34,205,39,210),(25,204,30,209,35,214,40,219),(41,238,46,223,51,228,56,233),(42,227,47,232,52,237,57,222),(43,236,48,221,53,226,58,231),(44,225,49,230,54,235,59,240),(45,234,50,239,55,224,60,229),(61,133,66,138,71,123,76,128),(62,122,67,127,72,132,77,137),(63,131,68,136,73,121,78,126),(64,140,69,125,74,130,79,135),(65,129,70,134,75,139,80,124),(81,150,86,155,91,160,96,145),(82,159,87,144,92,149,97,154),(83,148,88,153,93,158,98,143),(84,157,89,142,94,147,99,152),(85,146,90,151,95,156,100,141),(101,168,106,173,111,178,116,163),(102,177,107,162,112,167,117,172),(103,166,108,171,113,176,118,161),(104,175,109,180,114,165,119,170),(105,164,110,169,115,174,120,179)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 24E | ··· | 24L | 24M | 24N | 24O | 24P | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
102 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 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | Q8 | D5 | D10 | C3×D4 | C3×Q8 | C3×D5 | C8.C4 | C4×D5 | C5⋊D4 | Dic10 | C6×D5 | C3×C8.C4 | D5×C12 | C3×C5⋊D4 | C3×Dic10 | C20.53D4 | C3×C20.53D4 |
| kernel | C3×C20.53D4 | C6×C5⋊2C8 | C3×C4.Dic5 | C15×M4(2) | C20.53D4 | C3×C5⋊2C8 | C2×C5⋊2C8 | C4.Dic5 | C5×M4(2) | C5⋊2C8 | C60 | C2×C30 | C3×M4(2) | C2×C12 | C20 | C2×C10 | M4(2) | C15 | C12 | C12 | C2×C6 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 8 |
Matrix representation of C3×C20.53D4 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 0 | 225 |
| 51 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 177 |
| 192 | 98 | 0 | 0 |
| 187 | 49 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 136 | 211 |
| 192 | 98 | 0 | 0 |
| 187 | 49 | 0 | 0 |
| 0 | 0 | 90 | 67 |
| 0 | 0 | 58 | 151 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,225,0,0,0,0,225],[51,240,0,0,1,0,0,0,0,0,177,0,0,0,0,177],[192,187,0,0,98,49,0,0,0,0,8,136,0,0,0,211],[192,187,0,0,98,49,0,0,0,0,90,58,0,0,67,151] >;
C3×C20.53D4 in GAP, Magma, Sage, TeX
C_3\times C_{20}._{53}D_4 % in TeX
G:=Group("C3xC20.53D4"); // GroupNames label
G:=SmallGroup(480,100);
// by ID
G=gap.SmallGroup(480,100);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,168,365,92,136,1271,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^20=1,c^4=b^10,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^9,d*c*d^-1=b^10*c^3>;
// generators/relations