direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C24.C4, C24.1C20, C120.15C4, C60.191D4, C20.70D12, C40.10Dic3, (C2×C24).7C10, (C2×C40).15S3, C4.18(C5×D12), C12.34(C5×D4), C30.59(C4⋊C4), (C2×C30).20Q8, C8.1(C5×Dic3), C12.36(C2×C20), (C2×C120).22C2, C60.246(C2×C4), (C2×C20).423D6, C4.8(C10×Dic3), C15⋊12(C8.C4), C20.68(C2×Dic3), (C2×C10).10Dic6, C4.Dic3.1C10, C22.2(C5×Dic6), (C2×C60).535C22, C10.21(C4⋊Dic3), C6.7(C5×C4⋊C4), C3⋊1(C5×C8.C4), (C2×C8).5(C5×S3), (C2×C6).3(C5×Q8), C2.5(C5×C4⋊Dic3), (C2×C4).73(S3×C10), (C2×C12).102(C2×C10), (C5×C4.Dic3).5C2, SmallGroup(480,138)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C24.C4
G = < a,b,c,d | a5=b8=1, c6=b4, d2=b4c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >
Subgroups: 100 in 60 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C8, C2×C4, C10, C10, C12, C2×C6, C15, C2×C8, M4(2), C20, C2×C10, C3⋊C8, C24, C2×C12, C30, C30, C8.C4, C40, C40, C2×C20, C4.Dic3, C2×C24, C60, C2×C30, C2×C40, C5×M4(2), C24.C4, C5×C3⋊C8, C120, C2×C60, C5×C8.C4, C5×C4.Dic3, C2×C120, C5×C24.C4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, Q8, C10, Dic3, D6, C4⋊C4, C20, C2×C10, Dic6, D12, C2×Dic3, C5×S3, C8.C4, C2×C20, C5×D4, C5×Q8, C4⋊Dic3, C5×Dic3, S3×C10, C5×C4⋊C4, C24.C4, C5×Dic6, C5×D12, C10×Dic3, C5×C8.C4, C5×C4⋊Dic3, C5×C24.C4
►On 240 points
Generators in S240
(1 55 43 31 19)(2 56 44 32 20)(3 57 45 33 21)(4 58 46 34 22)(5 59 47 35 23)(6 60 48 36 24)(7 49 37 25 13)(8 50 38 26 14)(9 51 39 27 15)(10 52 40 28 16)(11 53 41 29 17)(12 54 42 30 18)(61 115 97 85 76)(62 116 98 86 77)(63 117 99 87 78)(64 118 100 88 79)(65 119 101 89 80)(66 120 102 90 81)(67 109 103 91 82)(68 110 104 92 83)(69 111 105 93 84)(70 112 106 94 73)(71 113 107 95 74)(72 114 108 96 75)(121 174 167 155 143)(122 175 168 156 144)(123 176 157 145 133)(124 177 158 146 134)(125 178 159 147 135)(126 179 160 148 136)(127 180 161 149 137)(128 169 162 150 138)(129 170 163 151 139)(130 171 164 152 140)(131 172 165 153 141)(132 173 166 154 142)(181 230 225 211 199)(182 231 226 212 200)(183 232 227 213 201)(184 233 228 214 202)(185 234 217 215 203)(186 235 218 216 204)(187 236 219 205 193)(188 237 220 206 194)(189 238 221 207 195)(190 239 222 208 196)(191 240 223 209 197)(192 229 224 210 198)
(1 67 10 64 7 61 4 70)(2 68 11 65 8 62 5 71)(3 69 12 66 9 63 6 72)(13 76 22 73 19 82 16 79)(14 77 23 74 20 83 17 80)(15 78 24 75 21 84 18 81)(25 85 34 94 31 91 28 88)(26 86 35 95 32 92 29 89)(27 87 36 96 33 93 30 90)(37 97 46 106 43 103 40 100)(38 98 47 107 44 104 41 101)(39 99 48 108 45 105 42 102)(49 115 58 112 55 109 52 118)(50 116 59 113 56 110 53 119)(51 117 60 114 57 111 54 120)(121 181 124 184 127 187 130 190)(122 182 125 185 128 188 131 191)(123 183 126 186 129 189 132 192)(133 201 136 204 139 195 142 198)(134 202 137 193 140 196 143 199)(135 203 138 194 141 197 144 200)(145 213 148 216 151 207 154 210)(146 214 149 205 152 208 155 211)(147 215 150 206 153 209 156 212)(157 227 160 218 163 221 166 224)(158 228 161 219 164 222 167 225)(159 217 162 220 165 223 168 226)(169 237 172 240 175 231 178 234)(170 238 173 229 176 232 179 235)(171 239 174 230 177 233 180 236)
(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)
(1 127 10 124 7 121 4 130)(2 132 11 129 8 126 5 123)(3 125 12 122 9 131 6 128)(13 143 22 140 19 137 16 134)(14 136 23 133 20 142 17 139)(15 141 24 138 21 135 18 144)(25 155 34 152 31 149 28 146)(26 148 35 145 32 154 29 151)(27 153 36 150 33 147 30 156)(37 167 46 164 43 161 40 158)(38 160 47 157 44 166 41 163)(39 165 48 162 45 159 42 168)(49 174 58 171 55 180 52 177)(50 179 59 176 56 173 53 170)(51 172 60 169 57 178 54 175)(61 190 70 187 67 184 64 181)(62 183 71 192 68 189 65 186)(63 188 72 185 69 182 66 191)(73 193 82 202 79 199 76 196)(74 198 83 195 80 204 77 201)(75 203 84 200 81 197 78 194)(85 208 94 205 91 214 88 211)(86 213 95 210 92 207 89 216)(87 206 96 215 93 212 90 209)(97 222 106 219 103 228 100 225)(98 227 107 224 104 221 101 218)(99 220 108 217 105 226 102 223)(109 233 118 230 115 239 112 236)(110 238 119 235 116 232 113 229)(111 231 120 240 117 237 114 234)
G:=sub<Sym(240)| (1,55,43,31,19)(2,56,44,32,20)(3,57,45,33,21)(4,58,46,34,22)(5,59,47,35,23)(6,60,48,36,24)(7,49,37,25,13)(8,50,38,26,14)(9,51,39,27,15)(10,52,40,28,16)(11,53,41,29,17)(12,54,42,30,18)(61,115,97,85,76)(62,116,98,86,77)(63,117,99,87,78)(64,118,100,88,79)(65,119,101,89,80)(66,120,102,90,81)(67,109,103,91,82)(68,110,104,92,83)(69,111,105,93,84)(70,112,106,94,73)(71,113,107,95,74)(72,114,108,96,75)(121,174,167,155,143)(122,175,168,156,144)(123,176,157,145,133)(124,177,158,146,134)(125,178,159,147,135)(126,179,160,148,136)(127,180,161,149,137)(128,169,162,150,138)(129,170,163,151,139)(130,171,164,152,140)(131,172,165,153,141)(132,173,166,154,142)(181,230,225,211,199)(182,231,226,212,200)(183,232,227,213,201)(184,233,228,214,202)(185,234,217,215,203)(186,235,218,216,204)(187,236,219,205,193)(188,237,220,206,194)(189,238,221,207,195)(190,239,222,208,196)(191,240,223,209,197)(192,229,224,210,198), (1,67,10,64,7,61,4,70)(2,68,11,65,8,62,5,71)(3,69,12,66,9,63,6,72)(13,76,22,73,19,82,16,79)(14,77,23,74,20,83,17,80)(15,78,24,75,21,84,18,81)(25,85,34,94,31,91,28,88)(26,86,35,95,32,92,29,89)(27,87,36,96,33,93,30,90)(37,97,46,106,43,103,40,100)(38,98,47,107,44,104,41,101)(39,99,48,108,45,105,42,102)(49,115,58,112,55,109,52,118)(50,116,59,113,56,110,53,119)(51,117,60,114,57,111,54,120)(121,181,124,184,127,187,130,190)(122,182,125,185,128,188,131,191)(123,183,126,186,129,189,132,192)(133,201,136,204,139,195,142,198)(134,202,137,193,140,196,143,199)(135,203,138,194,141,197,144,200)(145,213,148,216,151,207,154,210)(146,214,149,205,152,208,155,211)(147,215,150,206,153,209,156,212)(157,227,160,218,163,221,166,224)(158,228,161,219,164,222,167,225)(159,217,162,220,165,223,168,226)(169,237,172,240,175,231,178,234)(170,238,173,229,176,232,179,235)(171,239,174,230,177,233,180,236), (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), (1,127,10,124,7,121,4,130)(2,132,11,129,8,126,5,123)(3,125,12,122,9,131,6,128)(13,143,22,140,19,137,16,134)(14,136,23,133,20,142,17,139)(15,141,24,138,21,135,18,144)(25,155,34,152,31,149,28,146)(26,148,35,145,32,154,29,151)(27,153,36,150,33,147,30,156)(37,167,46,164,43,161,40,158)(38,160,47,157,44,166,41,163)(39,165,48,162,45,159,42,168)(49,174,58,171,55,180,52,177)(50,179,59,176,56,173,53,170)(51,172,60,169,57,178,54,175)(61,190,70,187,67,184,64,181)(62,183,71,192,68,189,65,186)(63,188,72,185,69,182,66,191)(73,193,82,202,79,199,76,196)(74,198,83,195,80,204,77,201)(75,203,84,200,81,197,78,194)(85,208,94,205,91,214,88,211)(86,213,95,210,92,207,89,216)(87,206,96,215,93,212,90,209)(97,222,106,219,103,228,100,225)(98,227,107,224,104,221,101,218)(99,220,108,217,105,226,102,223)(109,233,118,230,115,239,112,236)(110,238,119,235,116,232,113,229)(111,231,120,240,117,237,114,234)>;
G:=Group( (1,55,43,31,19)(2,56,44,32,20)(3,57,45,33,21)(4,58,46,34,22)(5,59,47,35,23)(6,60,48,36,24)(7,49,37,25,13)(8,50,38,26,14)(9,51,39,27,15)(10,52,40,28,16)(11,53,41,29,17)(12,54,42,30,18)(61,115,97,85,76)(62,116,98,86,77)(63,117,99,87,78)(64,118,100,88,79)(65,119,101,89,80)(66,120,102,90,81)(67,109,103,91,82)(68,110,104,92,83)(69,111,105,93,84)(70,112,106,94,73)(71,113,107,95,74)(72,114,108,96,75)(121,174,167,155,143)(122,175,168,156,144)(123,176,157,145,133)(124,177,158,146,134)(125,178,159,147,135)(126,179,160,148,136)(127,180,161,149,137)(128,169,162,150,138)(129,170,163,151,139)(130,171,164,152,140)(131,172,165,153,141)(132,173,166,154,142)(181,230,225,211,199)(182,231,226,212,200)(183,232,227,213,201)(184,233,228,214,202)(185,234,217,215,203)(186,235,218,216,204)(187,236,219,205,193)(188,237,220,206,194)(189,238,221,207,195)(190,239,222,208,196)(191,240,223,209,197)(192,229,224,210,198), (1,67,10,64,7,61,4,70)(2,68,11,65,8,62,5,71)(3,69,12,66,9,63,6,72)(13,76,22,73,19,82,16,79)(14,77,23,74,20,83,17,80)(15,78,24,75,21,84,18,81)(25,85,34,94,31,91,28,88)(26,86,35,95,32,92,29,89)(27,87,36,96,33,93,30,90)(37,97,46,106,43,103,40,100)(38,98,47,107,44,104,41,101)(39,99,48,108,45,105,42,102)(49,115,58,112,55,109,52,118)(50,116,59,113,56,110,53,119)(51,117,60,114,57,111,54,120)(121,181,124,184,127,187,130,190)(122,182,125,185,128,188,131,191)(123,183,126,186,129,189,132,192)(133,201,136,204,139,195,142,198)(134,202,137,193,140,196,143,199)(135,203,138,194,141,197,144,200)(145,213,148,216,151,207,154,210)(146,214,149,205,152,208,155,211)(147,215,150,206,153,209,156,212)(157,227,160,218,163,221,166,224)(158,228,161,219,164,222,167,225)(159,217,162,220,165,223,168,226)(169,237,172,240,175,231,178,234)(170,238,173,229,176,232,179,235)(171,239,174,230,177,233,180,236), (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), (1,127,10,124,7,121,4,130)(2,132,11,129,8,126,5,123)(3,125,12,122,9,131,6,128)(13,143,22,140,19,137,16,134)(14,136,23,133,20,142,17,139)(15,141,24,138,21,135,18,144)(25,155,34,152,31,149,28,146)(26,148,35,145,32,154,29,151)(27,153,36,150,33,147,30,156)(37,167,46,164,43,161,40,158)(38,160,47,157,44,166,41,163)(39,165,48,162,45,159,42,168)(49,174,58,171,55,180,52,177)(50,179,59,176,56,173,53,170)(51,172,60,169,57,178,54,175)(61,190,70,187,67,184,64,181)(62,183,71,192,68,189,65,186)(63,188,72,185,69,182,66,191)(73,193,82,202,79,199,76,196)(74,198,83,195,80,204,77,201)(75,203,84,200,81,197,78,194)(85,208,94,205,91,214,88,211)(86,213,95,210,92,207,89,216)(87,206,96,215,93,212,90,209)(97,222,106,219,103,228,100,225)(98,227,107,224,104,221,101,218)(99,220,108,217,105,226,102,223)(109,233,118,230,115,239,112,236)(110,238,119,235,116,232,113,229)(111,231,120,240,117,237,114,234) );
G=PermutationGroup([[(1,55,43,31,19),(2,56,44,32,20),(3,57,45,33,21),(4,58,46,34,22),(5,59,47,35,23),(6,60,48,36,24),(7,49,37,25,13),(8,50,38,26,14),(9,51,39,27,15),(10,52,40,28,16),(11,53,41,29,17),(12,54,42,30,18),(61,115,97,85,76),(62,116,98,86,77),(63,117,99,87,78),(64,118,100,88,79),(65,119,101,89,80),(66,120,102,90,81),(67,109,103,91,82),(68,110,104,92,83),(69,111,105,93,84),(70,112,106,94,73),(71,113,107,95,74),(72,114,108,96,75),(121,174,167,155,143),(122,175,168,156,144),(123,176,157,145,133),(124,177,158,146,134),(125,178,159,147,135),(126,179,160,148,136),(127,180,161,149,137),(128,169,162,150,138),(129,170,163,151,139),(130,171,164,152,140),(131,172,165,153,141),(132,173,166,154,142),(181,230,225,211,199),(182,231,226,212,200),(183,232,227,213,201),(184,233,228,214,202),(185,234,217,215,203),(186,235,218,216,204),(187,236,219,205,193),(188,237,220,206,194),(189,238,221,207,195),(190,239,222,208,196),(191,240,223,209,197),(192,229,224,210,198)], [(1,67,10,64,7,61,4,70),(2,68,11,65,8,62,5,71),(3,69,12,66,9,63,6,72),(13,76,22,73,19,82,16,79),(14,77,23,74,20,83,17,80),(15,78,24,75,21,84,18,81),(25,85,34,94,31,91,28,88),(26,86,35,95,32,92,29,89),(27,87,36,96,33,93,30,90),(37,97,46,106,43,103,40,100),(38,98,47,107,44,104,41,101),(39,99,48,108,45,105,42,102),(49,115,58,112,55,109,52,118),(50,116,59,113,56,110,53,119),(51,117,60,114,57,111,54,120),(121,181,124,184,127,187,130,190),(122,182,125,185,128,188,131,191),(123,183,126,186,129,189,132,192),(133,201,136,204,139,195,142,198),(134,202,137,193,140,196,143,199),(135,203,138,194,141,197,144,200),(145,213,148,216,151,207,154,210),(146,214,149,205,152,208,155,211),(147,215,150,206,153,209,156,212),(157,227,160,218,163,221,166,224),(158,228,161,219,164,222,167,225),(159,217,162,220,165,223,168,226),(169,237,172,240,175,231,178,234),(170,238,173,229,176,232,179,235),(171,239,174,230,177,233,180,236)], [(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)], [(1,127,10,124,7,121,4,130),(2,132,11,129,8,126,5,123),(3,125,12,122,9,131,6,128),(13,143,22,140,19,137,16,134),(14,136,23,133,20,142,17,139),(15,141,24,138,21,135,18,144),(25,155,34,152,31,149,28,146),(26,148,35,145,32,154,29,151),(27,153,36,150,33,147,30,156),(37,167,46,164,43,161,40,158),(38,160,47,157,44,166,41,163),(39,165,48,162,45,159,42,168),(49,174,58,171,55,180,52,177),(50,179,59,176,56,173,53,170),(51,172,60,169,57,178,54,175),(61,190,70,187,67,184,64,181),(62,183,71,192,68,189,65,186),(63,188,72,185,69,182,66,191),(73,193,82,202,79,199,76,196),(74,198,83,195,80,204,77,201),(75,203,84,200,81,197,78,194),(85,208,94,205,91,214,88,211),(86,213,95,210,92,207,89,216),(87,206,96,215,93,212,90,209),(97,222,106,219,103,228,100,225),(98,227,107,224,104,221,101,218),(99,220,108,217,105,226,102,223),(109,233,118,230,115,239,112,236),(110,238,119,235,116,232,113,229),(111,231,120,240,117,237,114,234)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 40Q | ··· | 40AF | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
150 irreducible representations
| dim | 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 | C4 | C5 | C10 | C10 | C20 | S3 | D4 | Q8 | Dic3 | D6 | D12 | Dic6 | C5×S3 | C8.C4 | C5×D4 | C5×Q8 | C5×Dic3 | S3×C10 | C24.C4 | C5×D12 | C5×Dic6 | C5×C8.C4 | C5×C24.C4 |
| kernel | C5×C24.C4 | C5×C4.Dic3 | C2×C120 | C120 | C24.C4 | C4.Dic3 | C2×C24 | C24 | C2×C40 | C60 | C2×C30 | C40 | C2×C20 | C20 | C2×C10 | C2×C8 | C15 | C12 | C2×C6 | C8 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C5×C24.C4 ►in GL2(𝔽241) generated by
| 205 | 0 |
| 0 | 205 |
| 211 | 0 |
| 103 | 8 |
| 60 | 0 |
| 178 | 4 |
| 58 | 239 |
| 27 | 183 |
G:=sub<GL(2,GF(241))| [205,0,0,205],[211,103,0,8],[60,178,0,4],[58,27,239,183] >;
C5×C24.C4 in GAP, Magma, Sage, TeX
C_5\times C_{24}.C_4 % in TeX
G:=Group("C5xC24.C4"); // GroupNames label
G:=SmallGroup(480,138);
// by ID
G=gap.SmallGroup(480,138);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,288,136,4204,102,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=1,c^6=b^4,d^2=b^4*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations