direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D10⋊1C8, D10⋊1C24, C60.230D4, C12.71D20, C30.30M4(2), (C2×C40)⋊1C6, (C6×D5)⋊3C8, (C2×C24)⋊1D5, (C2×C120)⋊1C2, C2.5(D5×C24), C6.20(C8×D5), C30.52(C2×C8), C4.19(C3×D20), C20.52(C3×D4), C15⋊11(C22⋊C8), C10.14(C2×C24), (C2×C12).444D10, C6.12(C8⋊D5), (C6×Dic5).14C4, (C2×Dic5).5C12, (C22×D5).3C12, C22.11(D5×C12), C10.8(C3×M4(2)), C12.120(C5⋊D4), C30.85(C22⋊C4), (C2×C60).544C22, C6.38(D10⋊C4), (C2×C8)⋊1(C3×D5), C5⋊3(C3×C22⋊C8), (D5×C2×C6).9C4, (C2×C4×D5).6C6, (C2×C5⋊2C8)⋊9C6, (C6×C5⋊2C8)⋊23C2, (D5×C2×C12).21C2, (C2×C4).94(C6×D5), (C2×C6).60(C4×D5), C2.3(C3×C8⋊D5), C4.27(C3×C5⋊D4), (C2×C10).32(C2×C12), (C2×C20).110(C2×C6), (C2×C30).146(C2×C4), C2.1(C3×D10⋊C4), C10.17(C3×C22⋊C4), SmallGroup(480,98)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D10⋊1C8
G = < a,b,c,d | a3=b10=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b5c >
Subgroups: 320 in 100 conjugacy classes, 50 normal (46 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, C23, D5, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, D10, C2×C10, C24, C2×C12, C2×C12, C22×C6, C3×D5, C30, C22⋊C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C24, C2×C24, C22×C12, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C2×C5⋊2C8, C2×C40, C2×C4×D5, C3×C22⋊C8, C3×C5⋊2C8, C120, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, D10⋊1C8, C6×C5⋊2C8, C2×C120, D5×C2×C12, C3×D10⋊1C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, D4, D5, C12, C2×C6, C22⋊C4, C2×C8, M4(2), D10, C24, C2×C12, C3×D4, C3×D5, C22⋊C8, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C2×C24, C3×M4(2), C6×D5, C8×D5, C8⋊D5, D10⋊C4, C3×C22⋊C8, D5×C12, C3×D20, C3×C5⋊D4, D10⋊1C8, D5×C24, C3×C8⋊D5, C3×D10⋊C4, C3×D10⋊1C8
►On 240 points
Generators in S240
(1 96 56)(2 97 57)(3 98 58)(4 99 59)(5 100 60)(6 91 51)(7 92 52)(8 93 53)(9 94 54)(10 95 55)(11 205 165)(12 206 166)(13 207 167)(14 208 168)(15 209 169)(16 210 170)(17 201 161)(18 202 162)(19 203 163)(20 204 164)(21 101 61)(22 102 62)(23 103 63)(24 104 64)(25 105 65)(26 106 66)(27 107 67)(28 108 68)(29 109 69)(30 110 70)(31 111 71)(32 112 72)(33 113 73)(34 114 74)(35 115 75)(36 116 76)(37 117 77)(38 118 78)(39 119 79)(40 120 80)(41 121 81)(42 122 82)(43 123 83)(44 124 84)(45 125 85)(46 126 86)(47 127 87)(48 128 88)(49 129 89)(50 130 90)(131 211 171)(132 212 172)(133 213 173)(134 214 174)(135 215 175)(136 216 176)(137 217 177)(138 218 178)(139 219 179)(140 220 180)(141 221 181)(142 222 182)(143 223 183)(144 224 184)(145 225 185)(146 226 186)(147 227 187)(148 228 188)(149 229 189)(150 230 190)(151 231 191)(152 232 192)(153 233 193)(154 234 194)(155 235 195)(156 236 196)(157 237 197)(158 238 198)(159 239 199)(160 240 200)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 15)(12 14)(16 20)(17 19)(21 28)(22 27)(23 26)(24 25)(29 30)(31 40)(32 39)(33 38)(34 37)(35 36)(41 48)(42 47)(43 46)(44 45)(49 50)(51 60)(52 59)(53 58)(54 57)(55 56)(61 68)(62 67)(63 66)(64 65)(69 70)(71 80)(72 79)(73 78)(74 77)(75 76)(81 88)(82 87)(83 86)(84 85)(89 90)(91 100)(92 99)(93 98)(94 97)(95 96)(101 108)(102 107)(103 106)(104 105)(109 110)(111 120)(112 119)(113 118)(114 117)(115 116)(121 128)(122 127)(123 126)(124 125)(129 130)(131 135)(132 134)(136 140)(137 139)(141 143)(144 150)(145 149)(146 148)(151 155)(152 154)(156 160)(157 159)(161 163)(164 170)(165 169)(166 168)(171 175)(172 174)(176 180)(177 179)(181 183)(184 190)(185 189)(186 188)(191 195)(192 194)(196 200)(197 199)(201 203)(204 210)(205 209)(206 208)(211 215)(212 214)(216 220)(217 219)(221 223)(224 230)(225 229)(226 228)(231 235)(232 234)(236 240)(237 239)
(1 165 45 145 25 156 36 136)(2 166 46 146 26 157 37 137)(3 167 47 147 27 158 38 138)(4 168 48 148 28 159 39 139)(5 169 49 149 29 160 40 140)(6 170 50 150 30 151 31 131)(7 161 41 141 21 152 32 132)(8 162 42 142 22 153 33 133)(9 163 43 143 23 154 34 134)(10 164 44 144 24 155 35 135)(11 125 225 105 236 116 216 96)(12 126 226 106 237 117 217 97)(13 127 227 107 238 118 218 98)(14 128 228 108 239 119 219 99)(15 129 229 109 240 120 220 100)(16 130 230 110 231 111 211 91)(17 121 221 101 232 112 212 92)(18 122 222 102 233 113 213 93)(19 123 223 103 234 114 214 94)(20 124 224 104 235 115 215 95)(51 210 90 190 70 191 71 171)(52 201 81 181 61 192 72 172)(53 202 82 182 62 193 73 173)(54 203 83 183 63 194 74 174)(55 204 84 184 64 195 75 175)(56 205 85 185 65 196 76 176)(57 206 86 186 66 197 77 177)(58 207 87 187 67 198 78 178)(59 208 88 188 68 199 79 179)(60 209 89 189 69 200 80 180)
G:=sub<Sym(240)| (1,96,56)(2,97,57)(3,98,58)(4,99,59)(5,100,60)(6,91,51)(7,92,52)(8,93,53)(9,94,54)(10,95,55)(11,205,165)(12,206,166)(13,207,167)(14,208,168)(15,209,169)(16,210,170)(17,201,161)(18,202,162)(19,203,163)(20,204,164)(21,101,61)(22,102,62)(23,103,63)(24,104,64)(25,105,65)(26,106,66)(27,107,67)(28,108,68)(29,109,69)(30,110,70)(31,111,71)(32,112,72)(33,113,73)(34,114,74)(35,115,75)(36,116,76)(37,117,77)(38,118,78)(39,119,79)(40,120,80)(41,121,81)(42,122,82)(43,123,83)(44,124,84)(45,125,85)(46,126,86)(47,127,87)(48,128,88)(49,129,89)(50,130,90)(131,211,171)(132,212,172)(133,213,173)(134,214,174)(135,215,175)(136,216,176)(137,217,177)(138,218,178)(139,219,179)(140,220,180)(141,221,181)(142,222,182)(143,223,183)(144,224,184)(145,225,185)(146,226,186)(147,227,187)(148,228,188)(149,229,189)(150,230,190)(151,231,191)(152,232,192)(153,233,193)(154,234,194)(155,235,195)(156,236,196)(157,237,197)(158,238,198)(159,239,199)(160,240,200), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,15)(12,14)(16,20)(17,19)(21,28)(22,27)(23,26)(24,25)(29,30)(31,40)(32,39)(33,38)(34,37)(35,36)(41,48)(42,47)(43,46)(44,45)(49,50)(51,60)(52,59)(53,58)(54,57)(55,56)(61,68)(62,67)(63,66)(64,65)(69,70)(71,80)(72,79)(73,78)(74,77)(75,76)(81,88)(82,87)(83,86)(84,85)(89,90)(91,100)(92,99)(93,98)(94,97)(95,96)(101,108)(102,107)(103,106)(104,105)(109,110)(111,120)(112,119)(113,118)(114,117)(115,116)(121,128)(122,127)(123,126)(124,125)(129,130)(131,135)(132,134)(136,140)(137,139)(141,143)(144,150)(145,149)(146,148)(151,155)(152,154)(156,160)(157,159)(161,163)(164,170)(165,169)(166,168)(171,175)(172,174)(176,180)(177,179)(181,183)(184,190)(185,189)(186,188)(191,195)(192,194)(196,200)(197,199)(201,203)(204,210)(205,209)(206,208)(211,215)(212,214)(216,220)(217,219)(221,223)(224,230)(225,229)(226,228)(231,235)(232,234)(236,240)(237,239), (1,165,45,145,25,156,36,136)(2,166,46,146,26,157,37,137)(3,167,47,147,27,158,38,138)(4,168,48,148,28,159,39,139)(5,169,49,149,29,160,40,140)(6,170,50,150,30,151,31,131)(7,161,41,141,21,152,32,132)(8,162,42,142,22,153,33,133)(9,163,43,143,23,154,34,134)(10,164,44,144,24,155,35,135)(11,125,225,105,236,116,216,96)(12,126,226,106,237,117,217,97)(13,127,227,107,238,118,218,98)(14,128,228,108,239,119,219,99)(15,129,229,109,240,120,220,100)(16,130,230,110,231,111,211,91)(17,121,221,101,232,112,212,92)(18,122,222,102,233,113,213,93)(19,123,223,103,234,114,214,94)(20,124,224,104,235,115,215,95)(51,210,90,190,70,191,71,171)(52,201,81,181,61,192,72,172)(53,202,82,182,62,193,73,173)(54,203,83,183,63,194,74,174)(55,204,84,184,64,195,75,175)(56,205,85,185,65,196,76,176)(57,206,86,186,66,197,77,177)(58,207,87,187,67,198,78,178)(59,208,88,188,68,199,79,179)(60,209,89,189,69,200,80,180)>;
G:=Group( (1,96,56)(2,97,57)(3,98,58)(4,99,59)(5,100,60)(6,91,51)(7,92,52)(8,93,53)(9,94,54)(10,95,55)(11,205,165)(12,206,166)(13,207,167)(14,208,168)(15,209,169)(16,210,170)(17,201,161)(18,202,162)(19,203,163)(20,204,164)(21,101,61)(22,102,62)(23,103,63)(24,104,64)(25,105,65)(26,106,66)(27,107,67)(28,108,68)(29,109,69)(30,110,70)(31,111,71)(32,112,72)(33,113,73)(34,114,74)(35,115,75)(36,116,76)(37,117,77)(38,118,78)(39,119,79)(40,120,80)(41,121,81)(42,122,82)(43,123,83)(44,124,84)(45,125,85)(46,126,86)(47,127,87)(48,128,88)(49,129,89)(50,130,90)(131,211,171)(132,212,172)(133,213,173)(134,214,174)(135,215,175)(136,216,176)(137,217,177)(138,218,178)(139,219,179)(140,220,180)(141,221,181)(142,222,182)(143,223,183)(144,224,184)(145,225,185)(146,226,186)(147,227,187)(148,228,188)(149,229,189)(150,230,190)(151,231,191)(152,232,192)(153,233,193)(154,234,194)(155,235,195)(156,236,196)(157,237,197)(158,238,198)(159,239,199)(160,240,200), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,15)(12,14)(16,20)(17,19)(21,28)(22,27)(23,26)(24,25)(29,30)(31,40)(32,39)(33,38)(34,37)(35,36)(41,48)(42,47)(43,46)(44,45)(49,50)(51,60)(52,59)(53,58)(54,57)(55,56)(61,68)(62,67)(63,66)(64,65)(69,70)(71,80)(72,79)(73,78)(74,77)(75,76)(81,88)(82,87)(83,86)(84,85)(89,90)(91,100)(92,99)(93,98)(94,97)(95,96)(101,108)(102,107)(103,106)(104,105)(109,110)(111,120)(112,119)(113,118)(114,117)(115,116)(121,128)(122,127)(123,126)(124,125)(129,130)(131,135)(132,134)(136,140)(137,139)(141,143)(144,150)(145,149)(146,148)(151,155)(152,154)(156,160)(157,159)(161,163)(164,170)(165,169)(166,168)(171,175)(172,174)(176,180)(177,179)(181,183)(184,190)(185,189)(186,188)(191,195)(192,194)(196,200)(197,199)(201,203)(204,210)(205,209)(206,208)(211,215)(212,214)(216,220)(217,219)(221,223)(224,230)(225,229)(226,228)(231,235)(232,234)(236,240)(237,239), (1,165,45,145,25,156,36,136)(2,166,46,146,26,157,37,137)(3,167,47,147,27,158,38,138)(4,168,48,148,28,159,39,139)(5,169,49,149,29,160,40,140)(6,170,50,150,30,151,31,131)(7,161,41,141,21,152,32,132)(8,162,42,142,22,153,33,133)(9,163,43,143,23,154,34,134)(10,164,44,144,24,155,35,135)(11,125,225,105,236,116,216,96)(12,126,226,106,237,117,217,97)(13,127,227,107,238,118,218,98)(14,128,228,108,239,119,219,99)(15,129,229,109,240,120,220,100)(16,130,230,110,231,111,211,91)(17,121,221,101,232,112,212,92)(18,122,222,102,233,113,213,93)(19,123,223,103,234,114,214,94)(20,124,224,104,235,115,215,95)(51,210,90,190,70,191,71,171)(52,201,81,181,61,192,72,172)(53,202,82,182,62,193,73,173)(54,203,83,183,63,194,74,174)(55,204,84,184,64,195,75,175)(56,205,85,185,65,196,76,176)(57,206,86,186,66,197,77,177)(58,207,87,187,67,198,78,178)(59,208,88,188,68,199,79,179)(60,209,89,189,69,200,80,180) );
G=PermutationGroup([[(1,96,56),(2,97,57),(3,98,58),(4,99,59),(5,100,60),(6,91,51),(7,92,52),(8,93,53),(9,94,54),(10,95,55),(11,205,165),(12,206,166),(13,207,167),(14,208,168),(15,209,169),(16,210,170),(17,201,161),(18,202,162),(19,203,163),(20,204,164),(21,101,61),(22,102,62),(23,103,63),(24,104,64),(25,105,65),(26,106,66),(27,107,67),(28,108,68),(29,109,69),(30,110,70),(31,111,71),(32,112,72),(33,113,73),(34,114,74),(35,115,75),(36,116,76),(37,117,77),(38,118,78),(39,119,79),(40,120,80),(41,121,81),(42,122,82),(43,123,83),(44,124,84),(45,125,85),(46,126,86),(47,127,87),(48,128,88),(49,129,89),(50,130,90),(131,211,171),(132,212,172),(133,213,173),(134,214,174),(135,215,175),(136,216,176),(137,217,177),(138,218,178),(139,219,179),(140,220,180),(141,221,181),(142,222,182),(143,223,183),(144,224,184),(145,225,185),(146,226,186),(147,227,187),(148,228,188),(149,229,189),(150,230,190),(151,231,191),(152,232,192),(153,233,193),(154,234,194),(155,235,195),(156,236,196),(157,237,197),(158,238,198),(159,239,199),(160,240,200)], [(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,15),(12,14),(16,20),(17,19),(21,28),(22,27),(23,26),(24,25),(29,30),(31,40),(32,39),(33,38),(34,37),(35,36),(41,48),(42,47),(43,46),(44,45),(49,50),(51,60),(52,59),(53,58),(54,57),(55,56),(61,68),(62,67),(63,66),(64,65),(69,70),(71,80),(72,79),(73,78),(74,77),(75,76),(81,88),(82,87),(83,86),(84,85),(89,90),(91,100),(92,99),(93,98),(94,97),(95,96),(101,108),(102,107),(103,106),(104,105),(109,110),(111,120),(112,119),(113,118),(114,117),(115,116),(121,128),(122,127),(123,126),(124,125),(129,130),(131,135),(132,134),(136,140),(137,139),(141,143),(144,150),(145,149),(146,148),(151,155),(152,154),(156,160),(157,159),(161,163),(164,170),(165,169),(166,168),(171,175),(172,174),(176,180),(177,179),(181,183),(184,190),(185,189),(186,188),(191,195),(192,194),(196,200),(197,199),(201,203),(204,210),(205,209),(206,208),(211,215),(212,214),(216,220),(217,219),(221,223),(224,230),(225,229),(226,228),(231,235),(232,234),(236,240),(237,239)], [(1,165,45,145,25,156,36,136),(2,166,46,146,26,157,37,137),(3,167,47,147,27,158,38,138),(4,168,48,148,28,159,39,139),(5,169,49,149,29,160,40,140),(6,170,50,150,30,151,31,131),(7,161,41,141,21,152,32,132),(8,162,42,142,22,153,33,133),(9,163,43,143,23,154,34,134),(10,164,44,144,24,155,35,135),(11,125,225,105,236,116,216,96),(12,126,226,106,237,117,217,97),(13,127,227,107,238,118,218,98),(14,128,228,108,239,119,219,99),(15,129,229,109,240,120,220,100),(16,130,230,110,231,111,211,91),(17,121,221,101,232,112,212,92),(18,122,222,102,233,113,213,93),(19,123,223,103,234,114,214,94),(20,124,224,104,235,115,215,95),(51,210,90,190,70,191,71,171),(52,201,81,181,61,192,72,172),(53,202,82,182,62,193,73,173),(54,203,83,183,63,194,74,174),(55,204,84,184,64,195,75,175),(56,205,85,185,65,196,76,176),(57,206,86,186,66,197,77,177),(58,207,87,187,67,198,78,178),(59,208,88,188,68,199,79,179),(60,209,89,189,69,200,80,180)]])
156 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24H | 24I | ··· | 24P | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
156 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C12 | C12 | C24 | D4 | D5 | M4(2) | D10 | C3×D4 | C3×D5 | D20 | C5⋊D4 | C4×D5 | C3×M4(2) | C6×D5 | C8×D5 | C8⋊D5 | C3×D20 | C3×C5⋊D4 | D5×C12 | D5×C24 | C3×C8⋊D5 |
| kernel | C3×D10⋊1C8 | C6×C5⋊2C8 | C2×C120 | D5×C2×C12 | D10⋊1C8 | C6×Dic5 | D5×C2×C6 | C2×C5⋊2C8 | C2×C40 | C2×C4×D5 | C6×D5 | C2×Dic5 | C22×D5 | D10 | C60 | C2×C24 | C30 | C2×C12 | C20 | C2×C8 | C12 | C12 | C2×C6 | C10 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 16 | 16 |
Matrix representation of C3×D10⋊1C8 ►in GL4(𝔽241) generated by
| 225 | 0 | 0 | 0 |
| 0 | 225 | 0 | 0 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 0 | 225 |
| 240 | 51 | 0 | 0 |
| 190 | 190 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 240 | 51 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 71 | 1 |
| 211 | 0 | 0 | 0 |
| 0 | 211 | 0 | 0 |
| 0 | 0 | 68 | 192 |
| 0 | 0 | 76 | 173 |
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,225,0,0,0,0,225],[240,190,0,0,51,190,0,0,0,0,240,0,0,0,0,240],[240,0,0,0,51,1,0,0,0,0,240,71,0,0,0,1],[211,0,0,0,0,211,0,0,0,0,68,76,0,0,192,173] >;
C3×D10⋊1C8 in GAP, Magma, Sage, TeX
C_3\times D_{10}\rtimes_1C_8 % in TeX
G:=Group("C3xD10:1C8"); // GroupNames label
G:=SmallGroup(480,98);
// by ID
G=gap.SmallGroup(480,98);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,365,92,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^10=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^5*c>;
// generators/relations