direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C20.48D4, C60.219D4, C4⋊Dic5⋊8C6, (C2×C30)⋊11Q8, C10.7(C6×Q8), (C2×C6)⋊7Dic10, C20.48(C3×D4), C10.43(C6×D4), C30.80(C2×Q8), (C2×Dic10)⋊6C6, C10.D4⋊2C6, C30.397(C2×D4), C15⋊35(C22⋊Q8), C2.9(C6×Dic10), C23.25(C6×D5), C23.D5.4C6, (C6×Dic10)⋊22C2, (C2×C12).434D10, (C22×C60).17C2, (C22×C20).10C6, C6.48(C2×Dic10), C22⋊3(C3×Dic10), (C22×C12).15D5, C30.192(C4○D4), C6.122(C4○D20), C12.116(C5⋊D4), (C2×C30).359C23, (C2×C60).511C22, (C22×C6).105D10, (C22×C30).155C22, (C6×Dic5).161C22, C5⋊4(C3×C22⋊Q8), (C2×C10)⋊5(C3×Q8), C2.5(C6×C5⋊D4), (C2×C4).84(C6×D5), C4.23(C3×C5⋊D4), C22.54(D5×C2×C6), (C2×C20).94(C2×C6), (C3×C4⋊Dic5)⋊26C2, C10.13(C3×C4○D4), C2.17(C3×C4○D20), C6.124(C2×C5⋊D4), (C22×C4).7(C3×D5), (C3×C10.D4)⋊2C2, (C22×C10).42(C2×C6), (C2×C10).42(C22×C6), (C2×Dic5).13(C2×C6), (C2×C6).355(C22×D5), (C3×C23.D5).10C2, SmallGroup(480,717)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C20.48D4
G = < a,b,c,d | a3=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b10c-1 >
Subgroups: 384 in 148 conjugacy classes, 74 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C23, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C3×Q8, C22×C6, C30, C30, C22⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×Q8, C3×Dic5, C60, C60, C2×C30, C2×C30, C2×C30, C10.D4, C4⋊Dic5, C23.D5, C2×Dic10, C22×C20, C3×C22⋊Q8, C3×Dic10, C6×Dic5, C2×C60, C2×C60, C22×C30, C20.48D4, C3×C10.D4, C3×C4⋊Dic5, C3×C23.D5, C6×Dic10, C22×C60, C3×C20.48D4
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, D5, C2×C6, C2×D4, C2×Q8, C4○D4, D10, C3×D4, C3×Q8, C22×C6, C3×D5, C22⋊Q8, Dic10, C5⋊D4, C22×D5, C6×D4, C6×Q8, C3×C4○D4, C6×D5, C2×Dic10, C4○D20, C2×C5⋊D4, C3×C22⋊Q8, C3×Dic10, C3×C5⋊D4, D5×C2×C6, C20.48D4, C6×Dic10, C3×C4○D20, C6×C5⋊D4, C3×C20.48D4
►On 240 points
Generators in S240
(1 162 207)(2 163 208)(3 164 209)(4 165 210)(5 166 211)(6 167 212)(7 168 213)(8 169 214)(9 170 215)(10 171 216)(11 172 217)(12 173 218)(13 174 219)(14 175 220)(15 176 201)(16 177 202)(17 178 203)(18 179 204)(19 180 205)(20 161 206)(21 110 94)(22 111 95)(23 112 96)(24 113 97)(25 114 98)(26 115 99)(27 116 100)(28 117 81)(29 118 82)(30 119 83)(31 120 84)(32 101 85)(33 102 86)(34 103 87)(35 104 88)(36 105 89)(37 106 90)(38 107 91)(39 108 92)(40 109 93)(41 230 159)(42 231 160)(43 232 141)(44 233 142)(45 234 143)(46 235 144)(47 236 145)(48 237 146)(49 238 147)(50 239 148)(51 240 149)(52 221 150)(53 222 151)(54 223 152)(55 224 153)(56 225 154)(57 226 155)(58 227 156)(59 228 157)(60 229 158)(61 140 193)(62 121 194)(63 122 195)(64 123 196)(65 124 197)(66 125 198)(67 126 199)(68 127 200)(69 128 181)(70 129 182)(71 130 183)(72 131 184)(73 132 185)(74 133 186)(75 134 187)(76 135 188)(77 136 189)(78 137 190)(79 138 191)(80 139 192)
(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 67 225 30)(2 66 226 29)(3 65 227 28)(4 64 228 27)(5 63 229 26)(6 62 230 25)(7 61 231 24)(8 80 232 23)(9 79 233 22)(10 78 234 21)(11 77 235 40)(12 76 236 39)(13 75 237 38)(14 74 238 37)(15 73 239 36)(16 72 240 35)(17 71 221 34)(18 70 222 33)(19 69 223 32)(20 68 224 31)(41 98 212 194)(42 97 213 193)(43 96 214 192)(44 95 215 191)(45 94 216 190)(46 93 217 189)(47 92 218 188)(48 91 219 187)(49 90 220 186)(50 89 201 185)(51 88 202 184)(52 87 203 183)(53 86 204 182)(54 85 205 181)(55 84 206 200)(56 83 207 199)(57 82 208 198)(58 81 209 197)(59 100 210 196)(60 99 211 195)(101 180 128 152)(102 179 129 151)(103 178 130 150)(104 177 131 149)(105 176 132 148)(106 175 133 147)(107 174 134 146)(108 173 135 145)(109 172 136 144)(110 171 137 143)(111 170 138 142)(112 169 139 141)(113 168 140 160)(114 167 121 159)(115 166 122 158)(116 165 123 157)(117 164 124 156)(118 163 125 155)(119 162 126 154)(120 161 127 153)
(1 40 11 30)(2 39 12 29)(3 38 13 28)(4 37 14 27)(5 36 15 26)(6 35 16 25)(7 34 17 24)(8 33 18 23)(9 32 19 22)(10 31 20 21)(41 184 51 194)(42 183 52 193)(43 182 53 192)(44 181 54 191)(45 200 55 190)(46 199 56 189)(47 198 57 188)(48 197 58 187)(49 196 59 186)(50 195 60 185)(61 231 71 221)(62 230 72 240)(63 229 73 239)(64 228 74 238)(65 227 75 237)(66 226 76 236)(67 225 77 235)(68 224 78 234)(69 223 79 233)(70 222 80 232)(81 209 91 219)(82 208 92 218)(83 207 93 217)(84 206 94 216)(85 205 95 215)(86 204 96 214)(87 203 97 213)(88 202 98 212)(89 201 99 211)(90 220 100 210)(101 180 111 170)(102 179 112 169)(103 178 113 168)(104 177 114 167)(105 176 115 166)(106 175 116 165)(107 174 117 164)(108 173 118 163)(109 172 119 162)(110 171 120 161)(121 159 131 149)(122 158 132 148)(123 157 133 147)(124 156 134 146)(125 155 135 145)(126 154 136 144)(127 153 137 143)(128 152 138 142)(129 151 139 141)(130 150 140 160)
G:=sub<Sym(240)| (1,162,207)(2,163,208)(3,164,209)(4,165,210)(5,166,211)(6,167,212)(7,168,213)(8,169,214)(9,170,215)(10,171,216)(11,172,217)(12,173,218)(13,174,219)(14,175,220)(15,176,201)(16,177,202)(17,178,203)(18,179,204)(19,180,205)(20,161,206)(21,110,94)(22,111,95)(23,112,96)(24,113,97)(25,114,98)(26,115,99)(27,116,100)(28,117,81)(29,118,82)(30,119,83)(31,120,84)(32,101,85)(33,102,86)(34,103,87)(35,104,88)(36,105,89)(37,106,90)(38,107,91)(39,108,92)(40,109,93)(41,230,159)(42,231,160)(43,232,141)(44,233,142)(45,234,143)(46,235,144)(47,236,145)(48,237,146)(49,238,147)(50,239,148)(51,240,149)(52,221,150)(53,222,151)(54,223,152)(55,224,153)(56,225,154)(57,226,155)(58,227,156)(59,228,157)(60,229,158)(61,140,193)(62,121,194)(63,122,195)(64,123,196)(65,124,197)(66,125,198)(67,126,199)(68,127,200)(69,128,181)(70,129,182)(71,130,183)(72,131,184)(73,132,185)(74,133,186)(75,134,187)(76,135,188)(77,136,189)(78,137,190)(79,138,191)(80,139,192), (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), (1,67,225,30)(2,66,226,29)(3,65,227,28)(4,64,228,27)(5,63,229,26)(6,62,230,25)(7,61,231,24)(8,80,232,23)(9,79,233,22)(10,78,234,21)(11,77,235,40)(12,76,236,39)(13,75,237,38)(14,74,238,37)(15,73,239,36)(16,72,240,35)(17,71,221,34)(18,70,222,33)(19,69,223,32)(20,68,224,31)(41,98,212,194)(42,97,213,193)(43,96,214,192)(44,95,215,191)(45,94,216,190)(46,93,217,189)(47,92,218,188)(48,91,219,187)(49,90,220,186)(50,89,201,185)(51,88,202,184)(52,87,203,183)(53,86,204,182)(54,85,205,181)(55,84,206,200)(56,83,207,199)(57,82,208,198)(58,81,209,197)(59,100,210,196)(60,99,211,195)(101,180,128,152)(102,179,129,151)(103,178,130,150)(104,177,131,149)(105,176,132,148)(106,175,133,147)(107,174,134,146)(108,173,135,145)(109,172,136,144)(110,171,137,143)(111,170,138,142)(112,169,139,141)(113,168,140,160)(114,167,121,159)(115,166,122,158)(116,165,123,157)(117,164,124,156)(118,163,125,155)(119,162,126,154)(120,161,127,153), (1,40,11,30)(2,39,12,29)(3,38,13,28)(4,37,14,27)(5,36,15,26)(6,35,16,25)(7,34,17,24)(8,33,18,23)(9,32,19,22)(10,31,20,21)(41,184,51,194)(42,183,52,193)(43,182,53,192)(44,181,54,191)(45,200,55,190)(46,199,56,189)(47,198,57,188)(48,197,58,187)(49,196,59,186)(50,195,60,185)(61,231,71,221)(62,230,72,240)(63,229,73,239)(64,228,74,238)(65,227,75,237)(66,226,76,236)(67,225,77,235)(68,224,78,234)(69,223,79,233)(70,222,80,232)(81,209,91,219)(82,208,92,218)(83,207,93,217)(84,206,94,216)(85,205,95,215)(86,204,96,214)(87,203,97,213)(88,202,98,212)(89,201,99,211)(90,220,100,210)(101,180,111,170)(102,179,112,169)(103,178,113,168)(104,177,114,167)(105,176,115,166)(106,175,116,165)(107,174,117,164)(108,173,118,163)(109,172,119,162)(110,171,120,161)(121,159,131,149)(122,158,132,148)(123,157,133,147)(124,156,134,146)(125,155,135,145)(126,154,136,144)(127,153,137,143)(128,152,138,142)(129,151,139,141)(130,150,140,160)>;
G:=Group( (1,162,207)(2,163,208)(3,164,209)(4,165,210)(5,166,211)(6,167,212)(7,168,213)(8,169,214)(9,170,215)(10,171,216)(11,172,217)(12,173,218)(13,174,219)(14,175,220)(15,176,201)(16,177,202)(17,178,203)(18,179,204)(19,180,205)(20,161,206)(21,110,94)(22,111,95)(23,112,96)(24,113,97)(25,114,98)(26,115,99)(27,116,100)(28,117,81)(29,118,82)(30,119,83)(31,120,84)(32,101,85)(33,102,86)(34,103,87)(35,104,88)(36,105,89)(37,106,90)(38,107,91)(39,108,92)(40,109,93)(41,230,159)(42,231,160)(43,232,141)(44,233,142)(45,234,143)(46,235,144)(47,236,145)(48,237,146)(49,238,147)(50,239,148)(51,240,149)(52,221,150)(53,222,151)(54,223,152)(55,224,153)(56,225,154)(57,226,155)(58,227,156)(59,228,157)(60,229,158)(61,140,193)(62,121,194)(63,122,195)(64,123,196)(65,124,197)(66,125,198)(67,126,199)(68,127,200)(69,128,181)(70,129,182)(71,130,183)(72,131,184)(73,132,185)(74,133,186)(75,134,187)(76,135,188)(77,136,189)(78,137,190)(79,138,191)(80,139,192), (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), (1,67,225,30)(2,66,226,29)(3,65,227,28)(4,64,228,27)(5,63,229,26)(6,62,230,25)(7,61,231,24)(8,80,232,23)(9,79,233,22)(10,78,234,21)(11,77,235,40)(12,76,236,39)(13,75,237,38)(14,74,238,37)(15,73,239,36)(16,72,240,35)(17,71,221,34)(18,70,222,33)(19,69,223,32)(20,68,224,31)(41,98,212,194)(42,97,213,193)(43,96,214,192)(44,95,215,191)(45,94,216,190)(46,93,217,189)(47,92,218,188)(48,91,219,187)(49,90,220,186)(50,89,201,185)(51,88,202,184)(52,87,203,183)(53,86,204,182)(54,85,205,181)(55,84,206,200)(56,83,207,199)(57,82,208,198)(58,81,209,197)(59,100,210,196)(60,99,211,195)(101,180,128,152)(102,179,129,151)(103,178,130,150)(104,177,131,149)(105,176,132,148)(106,175,133,147)(107,174,134,146)(108,173,135,145)(109,172,136,144)(110,171,137,143)(111,170,138,142)(112,169,139,141)(113,168,140,160)(114,167,121,159)(115,166,122,158)(116,165,123,157)(117,164,124,156)(118,163,125,155)(119,162,126,154)(120,161,127,153), (1,40,11,30)(2,39,12,29)(3,38,13,28)(4,37,14,27)(5,36,15,26)(6,35,16,25)(7,34,17,24)(8,33,18,23)(9,32,19,22)(10,31,20,21)(41,184,51,194)(42,183,52,193)(43,182,53,192)(44,181,54,191)(45,200,55,190)(46,199,56,189)(47,198,57,188)(48,197,58,187)(49,196,59,186)(50,195,60,185)(61,231,71,221)(62,230,72,240)(63,229,73,239)(64,228,74,238)(65,227,75,237)(66,226,76,236)(67,225,77,235)(68,224,78,234)(69,223,79,233)(70,222,80,232)(81,209,91,219)(82,208,92,218)(83,207,93,217)(84,206,94,216)(85,205,95,215)(86,204,96,214)(87,203,97,213)(88,202,98,212)(89,201,99,211)(90,220,100,210)(101,180,111,170)(102,179,112,169)(103,178,113,168)(104,177,114,167)(105,176,115,166)(106,175,116,165)(107,174,117,164)(108,173,118,163)(109,172,119,162)(110,171,120,161)(121,159,131,149)(122,158,132,148)(123,157,133,147)(124,156,134,146)(125,155,135,145)(126,154,136,144)(127,153,137,143)(128,152,138,142)(129,151,139,141)(130,150,140,160) );
G=PermutationGroup([[(1,162,207),(2,163,208),(3,164,209),(4,165,210),(5,166,211),(6,167,212),(7,168,213),(8,169,214),(9,170,215),(10,171,216),(11,172,217),(12,173,218),(13,174,219),(14,175,220),(15,176,201),(16,177,202),(17,178,203),(18,179,204),(19,180,205),(20,161,206),(21,110,94),(22,111,95),(23,112,96),(24,113,97),(25,114,98),(26,115,99),(27,116,100),(28,117,81),(29,118,82),(30,119,83),(31,120,84),(32,101,85),(33,102,86),(34,103,87),(35,104,88),(36,105,89),(37,106,90),(38,107,91),(39,108,92),(40,109,93),(41,230,159),(42,231,160),(43,232,141),(44,233,142),(45,234,143),(46,235,144),(47,236,145),(48,237,146),(49,238,147),(50,239,148),(51,240,149),(52,221,150),(53,222,151),(54,223,152),(55,224,153),(56,225,154),(57,226,155),(58,227,156),(59,228,157),(60,229,158),(61,140,193),(62,121,194),(63,122,195),(64,123,196),(65,124,197),(66,125,198),(67,126,199),(68,127,200),(69,128,181),(70,129,182),(71,130,183),(72,131,184),(73,132,185),(74,133,186),(75,134,187),(76,135,188),(77,136,189),(78,137,190),(79,138,191),(80,139,192)], [(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)], [(1,67,225,30),(2,66,226,29),(3,65,227,28),(4,64,228,27),(5,63,229,26),(6,62,230,25),(7,61,231,24),(8,80,232,23),(9,79,233,22),(10,78,234,21),(11,77,235,40),(12,76,236,39),(13,75,237,38),(14,74,238,37),(15,73,239,36),(16,72,240,35),(17,71,221,34),(18,70,222,33),(19,69,223,32),(20,68,224,31),(41,98,212,194),(42,97,213,193),(43,96,214,192),(44,95,215,191),(45,94,216,190),(46,93,217,189),(47,92,218,188),(48,91,219,187),(49,90,220,186),(50,89,201,185),(51,88,202,184),(52,87,203,183),(53,86,204,182),(54,85,205,181),(55,84,206,200),(56,83,207,199),(57,82,208,198),(58,81,209,197),(59,100,210,196),(60,99,211,195),(101,180,128,152),(102,179,129,151),(103,178,130,150),(104,177,131,149),(105,176,132,148),(106,175,133,147),(107,174,134,146),(108,173,135,145),(109,172,136,144),(110,171,137,143),(111,170,138,142),(112,169,139,141),(113,168,140,160),(114,167,121,159),(115,166,122,158),(116,165,123,157),(117,164,124,156),(118,163,125,155),(119,162,126,154),(120,161,127,153)], [(1,40,11,30),(2,39,12,29),(3,38,13,28),(4,37,14,27),(5,36,15,26),(6,35,16,25),(7,34,17,24),(8,33,18,23),(9,32,19,22),(10,31,20,21),(41,184,51,194),(42,183,52,193),(43,182,53,192),(44,181,54,191),(45,200,55,190),(46,199,56,189),(47,198,57,188),(48,197,58,187),(49,196,59,186),(50,195,60,185),(61,231,71,221),(62,230,72,240),(63,229,73,239),(64,228,74,238),(65,227,75,237),(66,226,76,236),(67,225,77,235),(68,224,78,234),(69,223,79,233),(70,222,80,232),(81,209,91,219),(82,208,92,218),(83,207,93,217),(84,206,94,216),(85,205,95,215),(86,204,96,214),(87,203,97,213),(88,202,98,212),(89,201,99,211),(90,220,100,210),(101,180,111,170),(102,179,112,169),(103,178,113,168),(104,177,114,167),(105,176,115,166),(106,175,116,165),(107,174,117,164),(108,173,118,163),(109,172,119,162),(110,171,120,161),(121,159,131,149),(122,158,132,148),(123,157,133,147),(124,156,134,146),(125,155,135,145),(126,154,136,144),(127,153,137,143),(128,152,138,142),(129,151,139,141),(130,150,140,160)]])
138 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 10A | ··· | 10N | 12A | ··· | 12H | 12I | ··· | 12P | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
138 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 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | Q8 | D5 | C4○D4 | D10 | D10 | C3×D4 | C3×Q8 | C3×D5 | C5⋊D4 | Dic10 | C3×C4○D4 | C6×D5 | C6×D5 | C4○D20 | C3×C5⋊D4 | C3×Dic10 | C3×C4○D20 |
| kernel | C3×C20.48D4 | C3×C10.D4 | C3×C4⋊Dic5 | C3×C23.D5 | C6×Dic10 | C22×C60 | C20.48D4 | C10.D4 | C4⋊Dic5 | C23.D5 | C2×Dic10 | C22×C20 | C60 | C2×C30 | C22×C12 | C30 | C2×C12 | C22×C6 | C20 | C2×C10 | C22×C4 | C12 | C2×C6 | C10 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 8 | 4 | 8 | 16 | 16 | 16 |
Matrix representation of C3×C20.48D4 ►in GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 37 | 58 | 0 | 0 |
| 0 | 33 | 0 | 0 |
| 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 58 |
| 57 | 7 | 0 | 0 |
| 15 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 57 | 7 | 0 | 0 |
| 15 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[37,0,0,0,58,33,0,0,0,0,20,0,0,0,0,58],[57,15,0,0,7,4,0,0,0,0,0,60,0,0,1,0],[57,15,0,0,7,4,0,0,0,0,0,1,0,0,1,0] >;
C3×C20.48D4 in GAP, Magma, Sage, TeX
C_3\times C_{20}._{48}D_4 % in TeX
G:=Group("C3xC20.48D4"); // GroupNames label
G:=SmallGroup(480,717);
// by ID
G=gap.SmallGroup(480,717);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,701,344,590,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^20=c^4=1,d^2=b^10,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^10*c^-1>;
// generators/relations