direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×C8⋊D5, C24⋊35D10, C120⋊45C22, C30⋊11M4(2), C60.275C23, C8⋊9(C6×D5), (C2×C40)⋊9C6, C40⋊11(C2×C6), (C2×C24)⋊14D5, (C2×C120)⋊22C2, C5⋊4(C6×M4(2)), (D5×C12).9C4, (C4×D5).3C12, C4.24(D5×C12), C12.94(C4×D5), C20.49(C2×C12), C60.215(C2×C4), C10⋊3(C3×M4(2)), C15⋊24(C2×M4(2)), D10.11(C2×C12), (C2×C12).450D10, C20.35(C22×C6), (C2×Dic5).6C12, (C6×Dic5).15C4, (C22×D5).4C12, C22.14(D5×C12), (C2×C60).550C22, C10.26(C22×C12), C30.184(C22×C4), Dic5.12(C2×C12), C12.242(C22×D5), (D5×C12).117C22, (C2×C8)⋊6(C3×D5), C4.36(D5×C2×C6), (D5×C2×C6).10C4, (C2×C4×D5).12C6, C6.109(C2×C4×D5), (C6×C5⋊2C8)⋊25C2, (C2×C5⋊2C8)⋊11C6, C2.14(D5×C2×C12), C5⋊2C8⋊10(C2×C6), (D5×C2×C12).31C2, (C2×C4).99(C6×D5), (C2×C6).63(C4×D5), (C6×D5).55(C2×C4), (C4×D5).22(C2×C6), (C2×C10).34(C2×C12), (C2×C20).114(C2×C6), (C2×C30).149(C2×C4), (C3×C5⋊2C8)⋊43C22, (C3×Dic5).63(C2×C4), SmallGroup(480,693)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C8⋊D5
G = < a,b,c,d | a6=b8=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Subgroups: 368 in 136 conjugacy classes, 82 normal (38 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C24, C24, C2×C12, C2×C12, C22×C6, C3×D5, C30, C30, C2×M4(2), C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C24, C2×C24, C3×M4(2), C22×C12, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C8⋊D5, C2×C5⋊2C8, C2×C40, C2×C4×D5, C6×M4(2), C3×C5⋊2C8, C120, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, C2×C8⋊D5, C3×C8⋊D5, C6×C5⋊2C8, C2×C120, D5×C2×C12, C6×C8⋊D5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, D5, C12, C2×C6, M4(2), C22×C4, D10, C2×C12, C22×C6, C3×D5, C2×M4(2), C4×D5, C22×D5, C3×M4(2), C22×C12, C6×D5, C8⋊D5, C2×C4×D5, C6×M4(2), D5×C12, D5×C2×C6, C2×C8⋊D5, C3×C8⋊D5, D5×C2×C12, C6×C8⋊D5
►On 240 points
Generators in S240
(1 115 83 19 185 56)(2 116 84 20 186 49)(3 117 85 21 187 50)(4 118 86 22 188 51)(5 119 87 23 189 52)(6 120 88 24 190 53)(7 113 81 17 191 54)(8 114 82 18 192 55)(9 96 71 112 231 33)(10 89 72 105 232 34)(11 90 65 106 225 35)(12 91 66 107 226 36)(13 92 67 108 227 37)(14 93 68 109 228 38)(15 94 69 110 229 39)(16 95 70 111 230 40)(25 240 221 78 101 141)(26 233 222 79 102 142)(27 234 223 80 103 143)(28 235 224 73 104 144)(29 236 217 74 97 137)(30 237 218 75 98 138)(31 238 219 76 99 139)(32 239 220 77 100 140)(41 204 177 149 124 165)(42 205 178 150 125 166)(43 206 179 151 126 167)(44 207 180 152 127 168)(45 208 181 145 128 161)(46 201 182 146 121 162)(47 202 183 147 122 163)(48 203 184 148 123 164)(57 212 193 157 132 173)(58 213 194 158 133 174)(59 214 195 159 134 175)(60 215 196 160 135 176)(61 216 197 153 136 169)(62 209 198 154 129 170)(63 210 199 155 130 171)(64 211 200 156 131 172)
(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)
(1 228 75 46 59)(2 229 76 47 60)(3 230 77 48 61)(4 231 78 41 62)(5 232 79 42 63)(6 225 80 43 64)(7 226 73 44 57)(8 227 74 45 58)(9 141 177 198 86)(10 142 178 199 87)(11 143 179 200 88)(12 144 180 193 81)(13 137 181 194 82)(14 138 182 195 83)(15 139 183 196 84)(16 140 184 197 85)(17 91 28 152 157)(18 92 29 145 158)(19 93 30 146 159)(20 94 31 147 160)(21 95 32 148 153)(22 96 25 149 154)(23 89 26 150 155)(24 90 27 151 156)(33 101 204 209 118)(34 102 205 210 119)(35 103 206 211 120)(36 104 207 212 113)(37 97 208 213 114)(38 98 201 214 115)(39 99 202 215 116)(40 100 203 216 117)(49 110 219 163 176)(50 111 220 164 169)(51 112 221 165 170)(52 105 222 166 171)(53 106 223 167 172)(54 107 224 168 173)(55 108 217 161 174)(56 109 218 162 175)(65 234 126 131 190)(66 235 127 132 191)(67 236 128 133 192)(68 237 121 134 185)(69 238 122 135 186)(70 239 123 136 187)(71 240 124 129 188)(72 233 125 130 189)
(1 59)(2 64)(3 61)(4 58)(5 63)(6 60)(7 57)(8 62)(9 181)(10 178)(11 183)(12 180)(13 177)(14 182)(15 179)(16 184)(17 157)(18 154)(19 159)(20 156)(21 153)(22 158)(23 155)(24 160)(25 29)(27 31)(33 208)(34 205)(35 202)(36 207)(37 204)(38 201)(39 206)(40 203)(41 227)(42 232)(43 229)(44 226)(45 231)(46 228)(47 225)(48 230)(49 172)(50 169)(51 174)(52 171)(53 176)(54 173)(55 170)(56 175)(65 122)(66 127)(67 124)(68 121)(69 126)(70 123)(71 128)(72 125)(74 78)(76 80)(81 193)(82 198)(83 195)(84 200)(85 197)(86 194)(87 199)(88 196)(89 150)(90 147)(91 152)(92 149)(93 146)(94 151)(95 148)(96 145)(97 101)(99 103)(105 166)(106 163)(107 168)(108 165)(109 162)(110 167)(111 164)(112 161)(113 212)(114 209)(115 214)(116 211)(117 216)(118 213)(119 210)(120 215)(129 192)(130 189)(131 186)(132 191)(133 188)(134 185)(135 190)(136 187)(137 141)(139 143)(217 221)(219 223)(234 238)(236 240)
G:=sub<Sym(240)| (1,115,83,19,185,56)(2,116,84,20,186,49)(3,117,85,21,187,50)(4,118,86,22,188,51)(5,119,87,23,189,52)(6,120,88,24,190,53)(7,113,81,17,191,54)(8,114,82,18,192,55)(9,96,71,112,231,33)(10,89,72,105,232,34)(11,90,65,106,225,35)(12,91,66,107,226,36)(13,92,67,108,227,37)(14,93,68,109,228,38)(15,94,69,110,229,39)(16,95,70,111,230,40)(25,240,221,78,101,141)(26,233,222,79,102,142)(27,234,223,80,103,143)(28,235,224,73,104,144)(29,236,217,74,97,137)(30,237,218,75,98,138)(31,238,219,76,99,139)(32,239,220,77,100,140)(41,204,177,149,124,165)(42,205,178,150,125,166)(43,206,179,151,126,167)(44,207,180,152,127,168)(45,208,181,145,128,161)(46,201,182,146,121,162)(47,202,183,147,122,163)(48,203,184,148,123,164)(57,212,193,157,132,173)(58,213,194,158,133,174)(59,214,195,159,134,175)(60,215,196,160,135,176)(61,216,197,153,136,169)(62,209,198,154,129,170)(63,210,199,155,130,171)(64,211,200,156,131,172), (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), (1,228,75,46,59)(2,229,76,47,60)(3,230,77,48,61)(4,231,78,41,62)(5,232,79,42,63)(6,225,80,43,64)(7,226,73,44,57)(8,227,74,45,58)(9,141,177,198,86)(10,142,178,199,87)(11,143,179,200,88)(12,144,180,193,81)(13,137,181,194,82)(14,138,182,195,83)(15,139,183,196,84)(16,140,184,197,85)(17,91,28,152,157)(18,92,29,145,158)(19,93,30,146,159)(20,94,31,147,160)(21,95,32,148,153)(22,96,25,149,154)(23,89,26,150,155)(24,90,27,151,156)(33,101,204,209,118)(34,102,205,210,119)(35,103,206,211,120)(36,104,207,212,113)(37,97,208,213,114)(38,98,201,214,115)(39,99,202,215,116)(40,100,203,216,117)(49,110,219,163,176)(50,111,220,164,169)(51,112,221,165,170)(52,105,222,166,171)(53,106,223,167,172)(54,107,224,168,173)(55,108,217,161,174)(56,109,218,162,175)(65,234,126,131,190)(66,235,127,132,191)(67,236,128,133,192)(68,237,121,134,185)(69,238,122,135,186)(70,239,123,136,187)(71,240,124,129,188)(72,233,125,130,189), (1,59)(2,64)(3,61)(4,58)(5,63)(6,60)(7,57)(8,62)(9,181)(10,178)(11,183)(12,180)(13,177)(14,182)(15,179)(16,184)(17,157)(18,154)(19,159)(20,156)(21,153)(22,158)(23,155)(24,160)(25,29)(27,31)(33,208)(34,205)(35,202)(36,207)(37,204)(38,201)(39,206)(40,203)(41,227)(42,232)(43,229)(44,226)(45,231)(46,228)(47,225)(48,230)(49,172)(50,169)(51,174)(52,171)(53,176)(54,173)(55,170)(56,175)(65,122)(66,127)(67,124)(68,121)(69,126)(70,123)(71,128)(72,125)(74,78)(76,80)(81,193)(82,198)(83,195)(84,200)(85,197)(86,194)(87,199)(88,196)(89,150)(90,147)(91,152)(92,149)(93,146)(94,151)(95,148)(96,145)(97,101)(99,103)(105,166)(106,163)(107,168)(108,165)(109,162)(110,167)(111,164)(112,161)(113,212)(114,209)(115,214)(116,211)(117,216)(118,213)(119,210)(120,215)(129,192)(130,189)(131,186)(132,191)(133,188)(134,185)(135,190)(136,187)(137,141)(139,143)(217,221)(219,223)(234,238)(236,240)>;
G:=Group( (1,115,83,19,185,56)(2,116,84,20,186,49)(3,117,85,21,187,50)(4,118,86,22,188,51)(5,119,87,23,189,52)(6,120,88,24,190,53)(7,113,81,17,191,54)(8,114,82,18,192,55)(9,96,71,112,231,33)(10,89,72,105,232,34)(11,90,65,106,225,35)(12,91,66,107,226,36)(13,92,67,108,227,37)(14,93,68,109,228,38)(15,94,69,110,229,39)(16,95,70,111,230,40)(25,240,221,78,101,141)(26,233,222,79,102,142)(27,234,223,80,103,143)(28,235,224,73,104,144)(29,236,217,74,97,137)(30,237,218,75,98,138)(31,238,219,76,99,139)(32,239,220,77,100,140)(41,204,177,149,124,165)(42,205,178,150,125,166)(43,206,179,151,126,167)(44,207,180,152,127,168)(45,208,181,145,128,161)(46,201,182,146,121,162)(47,202,183,147,122,163)(48,203,184,148,123,164)(57,212,193,157,132,173)(58,213,194,158,133,174)(59,214,195,159,134,175)(60,215,196,160,135,176)(61,216,197,153,136,169)(62,209,198,154,129,170)(63,210,199,155,130,171)(64,211,200,156,131,172), (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), (1,228,75,46,59)(2,229,76,47,60)(3,230,77,48,61)(4,231,78,41,62)(5,232,79,42,63)(6,225,80,43,64)(7,226,73,44,57)(8,227,74,45,58)(9,141,177,198,86)(10,142,178,199,87)(11,143,179,200,88)(12,144,180,193,81)(13,137,181,194,82)(14,138,182,195,83)(15,139,183,196,84)(16,140,184,197,85)(17,91,28,152,157)(18,92,29,145,158)(19,93,30,146,159)(20,94,31,147,160)(21,95,32,148,153)(22,96,25,149,154)(23,89,26,150,155)(24,90,27,151,156)(33,101,204,209,118)(34,102,205,210,119)(35,103,206,211,120)(36,104,207,212,113)(37,97,208,213,114)(38,98,201,214,115)(39,99,202,215,116)(40,100,203,216,117)(49,110,219,163,176)(50,111,220,164,169)(51,112,221,165,170)(52,105,222,166,171)(53,106,223,167,172)(54,107,224,168,173)(55,108,217,161,174)(56,109,218,162,175)(65,234,126,131,190)(66,235,127,132,191)(67,236,128,133,192)(68,237,121,134,185)(69,238,122,135,186)(70,239,123,136,187)(71,240,124,129,188)(72,233,125,130,189), (1,59)(2,64)(3,61)(4,58)(5,63)(6,60)(7,57)(8,62)(9,181)(10,178)(11,183)(12,180)(13,177)(14,182)(15,179)(16,184)(17,157)(18,154)(19,159)(20,156)(21,153)(22,158)(23,155)(24,160)(25,29)(27,31)(33,208)(34,205)(35,202)(36,207)(37,204)(38,201)(39,206)(40,203)(41,227)(42,232)(43,229)(44,226)(45,231)(46,228)(47,225)(48,230)(49,172)(50,169)(51,174)(52,171)(53,176)(54,173)(55,170)(56,175)(65,122)(66,127)(67,124)(68,121)(69,126)(70,123)(71,128)(72,125)(74,78)(76,80)(81,193)(82,198)(83,195)(84,200)(85,197)(86,194)(87,199)(88,196)(89,150)(90,147)(91,152)(92,149)(93,146)(94,151)(95,148)(96,145)(97,101)(99,103)(105,166)(106,163)(107,168)(108,165)(109,162)(110,167)(111,164)(112,161)(113,212)(114,209)(115,214)(116,211)(117,216)(118,213)(119,210)(120,215)(129,192)(130,189)(131,186)(132,191)(133,188)(134,185)(135,190)(136,187)(137,141)(139,143)(217,221)(219,223)(234,238)(236,240) );
G=PermutationGroup([[(1,115,83,19,185,56),(2,116,84,20,186,49),(3,117,85,21,187,50),(4,118,86,22,188,51),(5,119,87,23,189,52),(6,120,88,24,190,53),(7,113,81,17,191,54),(8,114,82,18,192,55),(9,96,71,112,231,33),(10,89,72,105,232,34),(11,90,65,106,225,35),(12,91,66,107,226,36),(13,92,67,108,227,37),(14,93,68,109,228,38),(15,94,69,110,229,39),(16,95,70,111,230,40),(25,240,221,78,101,141),(26,233,222,79,102,142),(27,234,223,80,103,143),(28,235,224,73,104,144),(29,236,217,74,97,137),(30,237,218,75,98,138),(31,238,219,76,99,139),(32,239,220,77,100,140),(41,204,177,149,124,165),(42,205,178,150,125,166),(43,206,179,151,126,167),(44,207,180,152,127,168),(45,208,181,145,128,161),(46,201,182,146,121,162),(47,202,183,147,122,163),(48,203,184,148,123,164),(57,212,193,157,132,173),(58,213,194,158,133,174),(59,214,195,159,134,175),(60,215,196,160,135,176),(61,216,197,153,136,169),(62,209,198,154,129,170),(63,210,199,155,130,171),(64,211,200,156,131,172)], [(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)], [(1,228,75,46,59),(2,229,76,47,60),(3,230,77,48,61),(4,231,78,41,62),(5,232,79,42,63),(6,225,80,43,64),(7,226,73,44,57),(8,227,74,45,58),(9,141,177,198,86),(10,142,178,199,87),(11,143,179,200,88),(12,144,180,193,81),(13,137,181,194,82),(14,138,182,195,83),(15,139,183,196,84),(16,140,184,197,85),(17,91,28,152,157),(18,92,29,145,158),(19,93,30,146,159),(20,94,31,147,160),(21,95,32,148,153),(22,96,25,149,154),(23,89,26,150,155),(24,90,27,151,156),(33,101,204,209,118),(34,102,205,210,119),(35,103,206,211,120),(36,104,207,212,113),(37,97,208,213,114),(38,98,201,214,115),(39,99,202,215,116),(40,100,203,216,117),(49,110,219,163,176),(50,111,220,164,169),(51,112,221,165,170),(52,105,222,166,171),(53,106,223,167,172),(54,107,224,168,173),(55,108,217,161,174),(56,109,218,162,175),(65,234,126,131,190),(66,235,127,132,191),(67,236,128,133,192),(68,237,121,134,185),(69,238,122,135,186),(70,239,123,136,187),(71,240,124,129,188),(72,233,125,130,189)], [(1,59),(2,64),(3,61),(4,58),(5,63),(6,60),(7,57),(8,62),(9,181),(10,178),(11,183),(12,180),(13,177),(14,182),(15,179),(16,184),(17,157),(18,154),(19,159),(20,156),(21,153),(22,158),(23,155),(24,160),(25,29),(27,31),(33,208),(34,205),(35,202),(36,207),(37,204),(38,201),(39,206),(40,203),(41,227),(42,232),(43,229),(44,226),(45,231),(46,228),(47,225),(48,230),(49,172),(50,169),(51,174),(52,171),(53,176),(54,173),(55,170),(56,175),(65,122),(66,127),(67,124),(68,121),(69,126),(70,123),(71,128),(72,125),(74,78),(76,80),(81,193),(82,198),(83,195),(84,200),(85,197),(86,194),(87,199),(88,196),(89,150),(90,147),(91,152),(92,149),(93,146),(94,151),(95,148),(96,145),(97,101),(99,103),(105,166),(106,163),(107,168),(108,165),(109,162),(110,167),(111,164),(112,161),(113,212),(114,209),(115,214),(116,211),(117,216),(118,213),(119,210),(120,215),(129,192),(130,189),(131,186),(132,191),(133,188),(134,185),(135,190),(136,187),(137,141),(139,143),(217,221),(219,223),(234,238),(236,240)]])
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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C12 | C12 | C12 | D5 | M4(2) | D10 | D10 | C3×D5 | C4×D5 | C4×D5 | C3×M4(2) | C6×D5 | C6×D5 | C8⋊D5 | D5×C12 | D5×C12 | C3×C8⋊D5 |
| kernel | C6×C8⋊D5 | C3×C8⋊D5 | C6×C5⋊2C8 | C2×C120 | D5×C2×C12 | C2×C8⋊D5 | D5×C12 | C6×Dic5 | D5×C2×C6 | C8⋊D5 | C2×C5⋊2C8 | C2×C40 | C2×C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | C2×C24 | C30 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C10 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 8 | 2 | 2 | 2 | 8 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 16 | 8 | 8 | 32 |
Matrix representation of C6×C8⋊D5 ►in GL5(𝔽241)
| 240 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 |
| 0 | 0 | 0 | 105 | 212 |
| 0 | 0 | 0 | 158 | 136 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 52 | 188 | 0 | 0 |
| 0 | 1 | 240 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 0 | 0 |
| 0 | 51 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 139 |
| 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,15,0,0,0,0,0,15,0,0,0,0,0,1,0,0,0,0,0,1],[240,0,0,0,0,0,177,0,0,0,0,0,177,0,0,0,0,0,105,158,0,0,0,212,136],[1,0,0,0,0,0,52,1,0,0,0,188,240,0,0,0,0,0,1,0,0,0,0,0,1],[240,0,0,0,0,0,0,51,0,0,0,52,0,0,0,0,0,0,1,0,0,0,0,139,240] >;
C6×C8⋊D5 in GAP, Magma, Sage, TeX
C_6\times C_8\rtimes D_5
% in TeX
G:=Group("C6xC8:D5"); // GroupNames label
G:=SmallGroup(480,693);
// by ID
G=gap.SmallGroup(480,693);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,1094,142,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^8=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations