direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10×C4.Dic3, C30⋊15M4(2), C60.289C23, (C2×C60).46C4, (C2×C12).9C20, C6⋊2(C5×M4(2)), C3⋊3(C10×M4(2)), C60.247(C2×C4), C12.37(C2×C20), (C2×C20).436D6, C15⋊31(C2×M4(2)), (C22×C6).7C20, C4.9(C10×Dic3), (C22×C60).25C2, (C22×C12).9C10, (C22×C20).18S3, (C22×C30).23C4, C6.21(C22×C20), (C2×C20).26Dic3, C20.69(C2×Dic3), C23.4(C5×Dic3), C20.247(C22×S3), C12.41(C22×C10), C30.228(C22×C4), (C2×C60).566C22, C10.44(C22×Dic3), C22.12(C10×Dic3), (C22×C10).11Dic3, (C2×C3⋊C8)⋊12C10, (C10×C3⋊C8)⋊26C2, C3⋊C8⋊12(C2×C10), C4.41(S3×C2×C10), (C5×C3⋊C8)⋊45C22, C2.3(Dic3×C2×C10), (C2×C6).32(C2×C20), (C2×C4).84(S3×C10), (C22×C4).6(C5×S3), (C2×C4).6(C5×Dic3), (C2×C30).200(C2×C4), (C2×C12).105(C2×C10), (C2×C10).43(C2×Dic3), SmallGroup(480,800)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C4.Dic3
G = < a,b,c,d | a10=b4=1, c6=b2, d2=b2c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >
Subgroups: 196 in 136 conjugacy classes, 98 normal (38 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C6, C8, C2×C4, C2×C4, C23, C10, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C3⋊C8, C2×C12, C2×C12, C22×C6, C30, C30, C30, C2×M4(2), C40, C2×C20, C2×C20, C22×C10, C2×C3⋊C8, C4.Dic3, C22×C12, C60, C60, C2×C30, C2×C30, C2×C30, C2×C40, C5×M4(2), C22×C20, C2×C4.Dic3, C5×C3⋊C8, C2×C60, C2×C60, C22×C30, C10×M4(2), C10×C3⋊C8, C5×C4.Dic3, C22×C60, C10×C4.Dic3
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C23, C10, Dic3, D6, M4(2), C22×C4, C20, C2×C10, C2×Dic3, C22×S3, C5×S3, C2×M4(2), C2×C20, C22×C10, C4.Dic3, C22×Dic3, C5×Dic3, S3×C10, C5×M4(2), C22×C20, C2×C4.Dic3, C10×Dic3, S3×C2×C10, C10×M4(2), C5×C4.Dic3, Dic3×C2×C10, C10×C4.Dic3
►On 240 points
Generators in S240
(1 91 55 79 43 67 31 115 19 103)(2 92 56 80 44 68 32 116 20 104)(3 93 57 81 45 69 33 117 21 105)(4 94 58 82 46 70 34 118 22 106)(5 95 59 83 47 71 35 119 23 107)(6 96 60 84 48 72 36 120 24 108)(7 85 49 73 37 61 25 109 13 97)(8 86 50 74 38 62 26 110 14 98)(9 87 51 75 39 63 27 111 15 99)(10 88 52 76 40 64 28 112 16 100)(11 89 53 77 41 65 29 113 17 101)(12 90 54 78 42 66 30 114 18 102)(121 205 169 193 157 181 145 229 133 217)(122 206 170 194 158 182 146 230 134 218)(123 207 171 195 159 183 147 231 135 219)(124 208 172 196 160 184 148 232 136 220)(125 209 173 197 161 185 149 233 137 221)(126 210 174 198 162 186 150 234 138 222)(127 211 175 199 163 187 151 235 139 223)(128 212 176 200 164 188 152 236 140 224)(129 213 177 201 165 189 153 237 141 225)(130 214 178 202 166 190 154 238 142 226)(131 215 179 203 167 191 155 239 143 227)(132 216 180 204 168 192 156 240 144 228)
(1 64 7 70)(2 65 8 71)(3 66 9 72)(4 67 10 61)(5 68 11 62)(6 69 12 63)(13 82 19 76)(14 83 20 77)(15 84 21 78)(16 73 22 79)(17 74 23 80)(18 75 24 81)(25 94 31 88)(26 95 32 89)(27 96 33 90)(28 85 34 91)(29 86 35 92)(30 87 36 93)(37 106 43 100)(38 107 44 101)(39 108 45 102)(40 97 46 103)(41 98 47 104)(42 99 48 105)(49 118 55 112)(50 119 56 113)(51 120 57 114)(52 109 58 115)(53 110 59 116)(54 111 60 117)(121 184 127 190)(122 185 128 191)(123 186 129 192)(124 187 130 181)(125 188 131 182)(126 189 132 183)(133 196 139 202)(134 197 140 203)(135 198 141 204)(136 199 142 193)(137 200 143 194)(138 201 144 195)(145 208 151 214)(146 209 152 215)(147 210 153 216)(148 211 154 205)(149 212 155 206)(150 213 156 207)(157 220 163 226)(158 221 164 227)(159 222 165 228)(160 223 166 217)(161 224 167 218)(162 225 168 219)(169 232 175 238)(170 233 176 239)(171 234 177 240)(172 235 178 229)(173 236 179 230)(174 237 180 231)
(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)
(1 128 10 125 7 122 4 131)(2 121 11 130 8 127 5 124)(3 126 12 123 9 132 6 129)(13 134 22 143 19 140 16 137)(14 139 23 136 20 133 17 142)(15 144 24 141 21 138 18 135)(25 146 34 155 31 152 28 149)(26 151 35 148 32 145 29 154)(27 156 36 153 33 150 30 147)(37 158 46 167 43 164 40 161)(38 163 47 160 44 157 41 166)(39 168 48 165 45 162 42 159)(49 170 58 179 55 176 52 173)(50 175 59 172 56 169 53 178)(51 180 60 177 57 174 54 171)(61 182 70 191 67 188 64 185)(62 187 71 184 68 181 65 190)(63 192 72 189 69 186 66 183)(73 194 82 203 79 200 76 197)(74 199 83 196 80 193 77 202)(75 204 84 201 81 198 78 195)(85 206 94 215 91 212 88 209)(86 211 95 208 92 205 89 214)(87 216 96 213 93 210 90 207)(97 218 106 227 103 224 100 221)(98 223 107 220 104 217 101 226)(99 228 108 225 105 222 102 219)(109 230 118 239 115 236 112 233)(110 235 119 232 116 229 113 238)(111 240 120 237 117 234 114 231)
G:=sub<Sym(240)| (1,91,55,79,43,67,31,115,19,103)(2,92,56,80,44,68,32,116,20,104)(3,93,57,81,45,69,33,117,21,105)(4,94,58,82,46,70,34,118,22,106)(5,95,59,83,47,71,35,119,23,107)(6,96,60,84,48,72,36,120,24,108)(7,85,49,73,37,61,25,109,13,97)(8,86,50,74,38,62,26,110,14,98)(9,87,51,75,39,63,27,111,15,99)(10,88,52,76,40,64,28,112,16,100)(11,89,53,77,41,65,29,113,17,101)(12,90,54,78,42,66,30,114,18,102)(121,205,169,193,157,181,145,229,133,217)(122,206,170,194,158,182,146,230,134,218)(123,207,171,195,159,183,147,231,135,219)(124,208,172,196,160,184,148,232,136,220)(125,209,173,197,161,185,149,233,137,221)(126,210,174,198,162,186,150,234,138,222)(127,211,175,199,163,187,151,235,139,223)(128,212,176,200,164,188,152,236,140,224)(129,213,177,201,165,189,153,237,141,225)(130,214,178,202,166,190,154,238,142,226)(131,215,179,203,167,191,155,239,143,227)(132,216,180,204,168,192,156,240,144,228), (1,64,7,70)(2,65,8,71)(3,66,9,72)(4,67,10,61)(5,68,11,62)(6,69,12,63)(13,82,19,76)(14,83,20,77)(15,84,21,78)(16,73,22,79)(17,74,23,80)(18,75,24,81)(25,94,31,88)(26,95,32,89)(27,96,33,90)(28,85,34,91)(29,86,35,92)(30,87,36,93)(37,106,43,100)(38,107,44,101)(39,108,45,102)(40,97,46,103)(41,98,47,104)(42,99,48,105)(49,118,55,112)(50,119,56,113)(51,120,57,114)(52,109,58,115)(53,110,59,116)(54,111,60,117)(121,184,127,190)(122,185,128,191)(123,186,129,192)(124,187,130,181)(125,188,131,182)(126,189,132,183)(133,196,139,202)(134,197,140,203)(135,198,141,204)(136,199,142,193)(137,200,143,194)(138,201,144,195)(145,208,151,214)(146,209,152,215)(147,210,153,216)(148,211,154,205)(149,212,155,206)(150,213,156,207)(157,220,163,226)(158,221,164,227)(159,222,165,228)(160,223,166,217)(161,224,167,218)(162,225,168,219)(169,232,175,238)(170,233,176,239)(171,234,177,240)(172,235,178,229)(173,236,179,230)(174,237,180,231), (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), (1,128,10,125,7,122,4,131)(2,121,11,130,8,127,5,124)(3,126,12,123,9,132,6,129)(13,134,22,143,19,140,16,137)(14,139,23,136,20,133,17,142)(15,144,24,141,21,138,18,135)(25,146,34,155,31,152,28,149)(26,151,35,148,32,145,29,154)(27,156,36,153,33,150,30,147)(37,158,46,167,43,164,40,161)(38,163,47,160,44,157,41,166)(39,168,48,165,45,162,42,159)(49,170,58,179,55,176,52,173)(50,175,59,172,56,169,53,178)(51,180,60,177,57,174,54,171)(61,182,70,191,67,188,64,185)(62,187,71,184,68,181,65,190)(63,192,72,189,69,186,66,183)(73,194,82,203,79,200,76,197)(74,199,83,196,80,193,77,202)(75,204,84,201,81,198,78,195)(85,206,94,215,91,212,88,209)(86,211,95,208,92,205,89,214)(87,216,96,213,93,210,90,207)(97,218,106,227,103,224,100,221)(98,223,107,220,104,217,101,226)(99,228,108,225,105,222,102,219)(109,230,118,239,115,236,112,233)(110,235,119,232,116,229,113,238)(111,240,120,237,117,234,114,231)>;
G:=Group( (1,91,55,79,43,67,31,115,19,103)(2,92,56,80,44,68,32,116,20,104)(3,93,57,81,45,69,33,117,21,105)(4,94,58,82,46,70,34,118,22,106)(5,95,59,83,47,71,35,119,23,107)(6,96,60,84,48,72,36,120,24,108)(7,85,49,73,37,61,25,109,13,97)(8,86,50,74,38,62,26,110,14,98)(9,87,51,75,39,63,27,111,15,99)(10,88,52,76,40,64,28,112,16,100)(11,89,53,77,41,65,29,113,17,101)(12,90,54,78,42,66,30,114,18,102)(121,205,169,193,157,181,145,229,133,217)(122,206,170,194,158,182,146,230,134,218)(123,207,171,195,159,183,147,231,135,219)(124,208,172,196,160,184,148,232,136,220)(125,209,173,197,161,185,149,233,137,221)(126,210,174,198,162,186,150,234,138,222)(127,211,175,199,163,187,151,235,139,223)(128,212,176,200,164,188,152,236,140,224)(129,213,177,201,165,189,153,237,141,225)(130,214,178,202,166,190,154,238,142,226)(131,215,179,203,167,191,155,239,143,227)(132,216,180,204,168,192,156,240,144,228), (1,64,7,70)(2,65,8,71)(3,66,9,72)(4,67,10,61)(5,68,11,62)(6,69,12,63)(13,82,19,76)(14,83,20,77)(15,84,21,78)(16,73,22,79)(17,74,23,80)(18,75,24,81)(25,94,31,88)(26,95,32,89)(27,96,33,90)(28,85,34,91)(29,86,35,92)(30,87,36,93)(37,106,43,100)(38,107,44,101)(39,108,45,102)(40,97,46,103)(41,98,47,104)(42,99,48,105)(49,118,55,112)(50,119,56,113)(51,120,57,114)(52,109,58,115)(53,110,59,116)(54,111,60,117)(121,184,127,190)(122,185,128,191)(123,186,129,192)(124,187,130,181)(125,188,131,182)(126,189,132,183)(133,196,139,202)(134,197,140,203)(135,198,141,204)(136,199,142,193)(137,200,143,194)(138,201,144,195)(145,208,151,214)(146,209,152,215)(147,210,153,216)(148,211,154,205)(149,212,155,206)(150,213,156,207)(157,220,163,226)(158,221,164,227)(159,222,165,228)(160,223,166,217)(161,224,167,218)(162,225,168,219)(169,232,175,238)(170,233,176,239)(171,234,177,240)(172,235,178,229)(173,236,179,230)(174,237,180,231), (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), (1,128,10,125,7,122,4,131)(2,121,11,130,8,127,5,124)(3,126,12,123,9,132,6,129)(13,134,22,143,19,140,16,137)(14,139,23,136,20,133,17,142)(15,144,24,141,21,138,18,135)(25,146,34,155,31,152,28,149)(26,151,35,148,32,145,29,154)(27,156,36,153,33,150,30,147)(37,158,46,167,43,164,40,161)(38,163,47,160,44,157,41,166)(39,168,48,165,45,162,42,159)(49,170,58,179,55,176,52,173)(50,175,59,172,56,169,53,178)(51,180,60,177,57,174,54,171)(61,182,70,191,67,188,64,185)(62,187,71,184,68,181,65,190)(63,192,72,189,69,186,66,183)(73,194,82,203,79,200,76,197)(74,199,83,196,80,193,77,202)(75,204,84,201,81,198,78,195)(85,206,94,215,91,212,88,209)(86,211,95,208,92,205,89,214)(87,216,96,213,93,210,90,207)(97,218,106,227,103,224,100,221)(98,223,107,220,104,217,101,226)(99,228,108,225,105,222,102,219)(109,230,118,239,115,236,112,233)(110,235,119,232,116,229,113,238)(111,240,120,237,117,234,114,231) );
G=PermutationGroup([[(1,91,55,79,43,67,31,115,19,103),(2,92,56,80,44,68,32,116,20,104),(3,93,57,81,45,69,33,117,21,105),(4,94,58,82,46,70,34,118,22,106),(5,95,59,83,47,71,35,119,23,107),(6,96,60,84,48,72,36,120,24,108),(7,85,49,73,37,61,25,109,13,97),(8,86,50,74,38,62,26,110,14,98),(9,87,51,75,39,63,27,111,15,99),(10,88,52,76,40,64,28,112,16,100),(11,89,53,77,41,65,29,113,17,101),(12,90,54,78,42,66,30,114,18,102),(121,205,169,193,157,181,145,229,133,217),(122,206,170,194,158,182,146,230,134,218),(123,207,171,195,159,183,147,231,135,219),(124,208,172,196,160,184,148,232,136,220),(125,209,173,197,161,185,149,233,137,221),(126,210,174,198,162,186,150,234,138,222),(127,211,175,199,163,187,151,235,139,223),(128,212,176,200,164,188,152,236,140,224),(129,213,177,201,165,189,153,237,141,225),(130,214,178,202,166,190,154,238,142,226),(131,215,179,203,167,191,155,239,143,227),(132,216,180,204,168,192,156,240,144,228)], [(1,64,7,70),(2,65,8,71),(3,66,9,72),(4,67,10,61),(5,68,11,62),(6,69,12,63),(13,82,19,76),(14,83,20,77),(15,84,21,78),(16,73,22,79),(17,74,23,80),(18,75,24,81),(25,94,31,88),(26,95,32,89),(27,96,33,90),(28,85,34,91),(29,86,35,92),(30,87,36,93),(37,106,43,100),(38,107,44,101),(39,108,45,102),(40,97,46,103),(41,98,47,104),(42,99,48,105),(49,118,55,112),(50,119,56,113),(51,120,57,114),(52,109,58,115),(53,110,59,116),(54,111,60,117),(121,184,127,190),(122,185,128,191),(123,186,129,192),(124,187,130,181),(125,188,131,182),(126,189,132,183),(133,196,139,202),(134,197,140,203),(135,198,141,204),(136,199,142,193),(137,200,143,194),(138,201,144,195),(145,208,151,214),(146,209,152,215),(147,210,153,216),(148,211,154,205),(149,212,155,206),(150,213,156,207),(157,220,163,226),(158,221,164,227),(159,222,165,228),(160,223,166,217),(161,224,167,218),(162,225,168,219),(169,232,175,238),(170,233,176,239),(171,234,177,240),(172,235,178,229),(173,236,179,230),(174,237,180,231)], [(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)], [(1,128,10,125,7,122,4,131),(2,121,11,130,8,127,5,124),(3,126,12,123,9,132,6,129),(13,134,22,143,19,140,16,137),(14,139,23,136,20,133,17,142),(15,144,24,141,21,138,18,135),(25,146,34,155,31,152,28,149),(26,151,35,148,32,145,29,154),(27,156,36,153,33,150,30,147),(37,158,46,167,43,164,40,161),(38,163,47,160,44,157,41,166),(39,168,48,165,45,162,42,159),(49,170,58,179,55,176,52,173),(50,175,59,172,56,169,53,178),(51,180,60,177,57,174,54,171),(61,182,70,191,67,188,64,185),(62,187,71,184,68,181,65,190),(63,192,72,189,69,186,66,183),(73,194,82,203,79,200,76,197),(74,199,83,196,80,193,77,202),(75,204,84,201,81,198,78,195),(85,206,94,215,91,212,88,209),(86,211,95,208,92,205,89,214),(87,216,96,213,93,210,90,207),(97,218,106,227,103,224,100,221),(98,223,107,220,104,217,101,226),(99,228,108,225,105,222,102,219),(109,230,118,239,115,236,112,233),(110,235,119,232,116,229,113,238),(111,240,120,237,117,234,114,231)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | ··· | 6G | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20X | 30A | ··· | 30AB | 40A | ··· | 40AF | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
180 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 |
| type | + | + | + | + | + | - | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | S3 | Dic3 | D6 | Dic3 | M4(2) | C5×S3 | C4.Dic3 | C5×Dic3 | S3×C10 | C5×Dic3 | C5×M4(2) | C5×C4.Dic3 |
| kernel | C10×C4.Dic3 | C10×C3⋊C8 | C5×C4.Dic3 | C22×C60 | C2×C60 | C22×C30 | C2×C4.Dic3 | C2×C3⋊C8 | C4.Dic3 | C22×C12 | C2×C12 | C22×C6 | C22×C20 | C2×C20 | C2×C20 | C22×C10 | C30 | C22×C4 | C10 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 4 | 8 | 16 | 4 | 24 | 8 | 1 | 3 | 3 | 1 | 4 | 4 | 8 | 12 | 12 | 4 | 16 | 32 |
Matrix representation of C10×C4.Dic3 ►in GL4(𝔽241) generated by
| 150 | 0 | 0 | 0 |
| 0 | 150 | 0 | 0 |
| 0 | 0 | 87 | 0 |
| 0 | 0 | 0 | 87 |
| 64 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 177 |
| 177 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 |
| 0 | 0 | 181 | 0 |
| 0 | 0 | 0 | 237 |
| 0 | 1 | 0 | 0 |
| 177 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 64 | 0 |
G:=sub<GL(4,GF(241))| [150,0,0,0,0,150,0,0,0,0,87,0,0,0,0,87],[64,0,0,0,0,177,0,0,0,0,64,0,0,0,0,177],[177,0,0,0,0,177,0,0,0,0,181,0,0,0,0,237],[0,177,0,0,1,0,0,0,0,0,0,64,0,0,1,0] >;
C10×C4.Dic3 in GAP, Magma, Sage, TeX
C_{10}\times C_4.{\rm Dic}_3 % in TeX
G:=Group("C10xC4.Dic3"); // GroupNames label
G:=SmallGroup(480,800);
// by ID
G=gap.SmallGroup(480,800);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,280,1766,102,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=1,c^6=b^2,d^2=b^2*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