direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10×D6⋊C4, D6⋊6(C2×C20), (C2×C20)⋊31D6, (C22×C60)⋊3C2, (C22×C20)⋊5S3, C6.40(D4×C10), C2.3(C10×D12), C30⋊9(C22⋊C4), (C22×S3)⋊3C20, (C2×C60)⋊41C22, (C22×C12)⋊1C10, C10.87(C2×D12), (C2×C10).57D12, (C2×C30).179D4, C30.423(C2×D4), (S3×C23).2C10, C6.18(C22×C20), C22.17(S3×C20), C23.36(S3×C10), C22.16(C5×D12), C30.209(C22×C4), (C2×C30).424C23, (C22×Dic3)⋊3C10, (C22×C10).151D6, (C10×Dic3)⋊33C22, (C22×C30).175C22, (S3×C2×C10)⋊11C4, (C2×C4)⋊8(S3×C10), C6⋊1(C5×C22⋊C4), C3⋊2(C10×C22⋊C4), C2.19(S3×C2×C20), C15⋊19(C2×C22⋊C4), C10.146(S3×C2×C4), (C22×C4)⋊3(C5×S3), (C2×C6).36(C5×D4), C2.2(C10×C3⋊D4), (S3×C10)⋊31(C2×C4), (C2×C12)⋊10(C2×C10), (C2×C6).18(C2×C20), (C2×C10).89(C4×S3), (Dic3×C2×C10)⋊14C2, (S3×C22×C10).5C2, C22.23(S3×C2×C10), (C2×C30).163(C2×C4), (C2×Dic3)⋊6(C2×C10), C10.125(C2×C3⋊D4), C22.20(C5×C3⋊D4), (C2×C10).92(C3⋊D4), (S3×C2×C10).115C22, (C22×C6).37(C2×C10), (C2×C6).45(C22×C10), (C22×S3).24(C2×C10), (C2×C10).358(C22×S3), SmallGroup(480,806)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×D6⋊C4
G = < a,b,c,d | a10=b6=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >
Subgroups: 644 in 264 conjugacy classes, 114 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C23, C23, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C22×C4, C24, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, C30, C30, C2×C22⋊C4, C2×C20, C2×C20, C22×C10, C22×C10, D6⋊C4, C22×Dic3, C22×C12, S3×C23, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C30, C5×C22⋊C4, C22×C20, C22×C20, C23×C10, C2×D6⋊C4, C10×Dic3, C10×Dic3, C2×C60, C2×C60, S3×C2×C10, S3×C2×C10, C22×C30, C10×C22⋊C4, C5×D6⋊C4, Dic3×C2×C10, C22×C60, S3×C22×C10, C10×D6⋊C4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C23, C10, D6, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C4×S3, D12, C3⋊D4, C22×S3, C5×S3, C2×C22⋊C4, C2×C20, C5×D4, C22×C10, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, S3×C10, C5×C22⋊C4, C22×C20, D4×C10, C2×D6⋊C4, S3×C20, C5×D12, C5×C3⋊D4, S3×C2×C10, C10×C22⋊C4, C5×D6⋊C4, S3×C2×C20, C10×D12, C10×C3⋊D4, C10×D6⋊C4
►On 240 points
Generators in S240
(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)
(1 195 190 103 16 72)(2 196 181 104 17 73)(3 197 182 105 18 74)(4 198 183 106 19 75)(5 199 184 107 20 76)(6 200 185 108 11 77)(7 191 186 109 12 78)(8 192 187 110 13 79)(9 193 188 101 14 80)(10 194 189 102 15 71)(21 209 61 226 158 35)(22 210 62 227 159 36)(23 201 63 228 160 37)(24 202 64 229 151 38)(25 203 65 230 152 39)(26 204 66 221 153 40)(27 205 67 222 154 31)(28 206 68 223 155 32)(29 207 69 224 156 33)(30 208 70 225 157 34)(41 148 170 88 220 114)(42 149 161 89 211 115)(43 150 162 90 212 116)(44 141 163 81 213 117)(45 142 164 82 214 118)(46 143 165 83 215 119)(47 144 166 84 216 120)(48 145 167 85 217 111)(49 146 168 86 218 112)(50 147 169 87 219 113)(51 235 129 175 93 136)(52 236 130 176 94 137)(53 237 121 177 95 138)(54 238 122 178 96 139)(55 239 123 179 97 140)(56 240 124 180 98 131)(57 231 125 171 99 132)(58 232 126 172 100 133)(59 233 127 173 91 134)(60 234 128 174 92 135)
(1 185)(2 186)(3 187)(4 188)(5 189)(6 190)(7 181)(8 182)(9 183)(10 184)(11 16)(12 17)(13 18)(14 19)(15 20)(21 40)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 81)(50 82)(51 124)(52 125)(53 126)(54 127)(55 128)(56 129)(57 130)(58 121)(59 122)(60 123)(61 221)(62 222)(63 223)(64 224)(65 225)(66 226)(67 227)(68 228)(69 229)(70 230)(71 107)(72 108)(73 109)(74 110)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(91 96)(92 97)(93 98)(94 99)(95 100)(111 212)(112 213)(113 214)(114 215)(115 216)(116 217)(117 218)(118 219)(119 220)(120 211)(131 175)(132 176)(133 177)(134 178)(135 179)(136 180)(137 171)(138 172)(139 173)(140 174)(141 168)(142 169)(143 170)(144 161)(145 162)(146 163)(147 164)(148 165)(149 166)(150 167)(151 207)(152 208)(153 209)(154 210)(155 201)(156 202)(157 203)(158 204)(159 205)(160 206)(191 196)(192 197)(193 198)(194 199)(195 200)(231 236)(232 237)(233 238)(234 239)(235 240)
(1 167 179 30)(2 168 180 21)(3 169 171 22)(4 170 172 23)(5 161 173 24)(6 162 174 25)(7 163 175 26)(8 164 176 27)(9 165 177 28)(10 166 178 29)(11 43 234 152)(12 44 235 153)(13 45 236 154)(14 46 237 155)(15 47 238 156)(16 48 239 157)(17 49 240 158)(18 50 231 159)(19 41 232 160)(20 42 233 151)(31 79 142 130)(32 80 143 121)(33 71 144 122)(34 72 145 123)(35 73 146 124)(36 74 147 125)(37 75 148 126)(38 76 149 127)(39 77 150 128)(40 78 141 129)(51 221 109 117)(52 222 110 118)(53 223 101 119)(54 224 102 120)(55 225 103 111)(56 226 104 112)(57 227 105 113)(58 228 106 114)(59 229 107 115)(60 230 108 116)(61 181 218 131)(62 182 219 132)(63 183 220 133)(64 184 211 134)(65 185 212 135)(66 186 213 136)(67 187 214 137)(68 188 215 138)(69 189 216 139)(70 190 217 140)(81 93 204 191)(82 94 205 192)(83 95 206 193)(84 96 207 194)(85 97 208 195)(86 98 209 196)(87 99 210 197)(88 100 201 198)(89 91 202 199)(90 92 203 200)
G:=sub<Sym(240)| (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), (1,195,190,103,16,72)(2,196,181,104,17,73)(3,197,182,105,18,74)(4,198,183,106,19,75)(5,199,184,107,20,76)(6,200,185,108,11,77)(7,191,186,109,12,78)(8,192,187,110,13,79)(9,193,188,101,14,80)(10,194,189,102,15,71)(21,209,61,226,158,35)(22,210,62,227,159,36)(23,201,63,228,160,37)(24,202,64,229,151,38)(25,203,65,230,152,39)(26,204,66,221,153,40)(27,205,67,222,154,31)(28,206,68,223,155,32)(29,207,69,224,156,33)(30,208,70,225,157,34)(41,148,170,88,220,114)(42,149,161,89,211,115)(43,150,162,90,212,116)(44,141,163,81,213,117)(45,142,164,82,214,118)(46,143,165,83,215,119)(47,144,166,84,216,120)(48,145,167,85,217,111)(49,146,168,86,218,112)(50,147,169,87,219,113)(51,235,129,175,93,136)(52,236,130,176,94,137)(53,237,121,177,95,138)(54,238,122,178,96,139)(55,239,123,179,97,140)(56,240,124,180,98,131)(57,231,125,171,99,132)(58,232,126,172,100,133)(59,233,127,173,91,134)(60,234,128,174,92,135), (1,185)(2,186)(3,187)(4,188)(5,189)(6,190)(7,181)(8,182)(9,183)(10,184)(11,16)(12,17)(13,18)(14,19)(15,20)(21,40)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,81)(50,82)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,121)(59,122)(60,123)(61,221)(62,222)(63,223)(64,224)(65,225)(66,226)(67,227)(68,228)(69,229)(70,230)(71,107)(72,108)(73,109)(74,110)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(91,96)(92,97)(93,98)(94,99)(95,100)(111,212)(112,213)(113,214)(114,215)(115,216)(116,217)(117,218)(118,219)(119,220)(120,211)(131,175)(132,176)(133,177)(134,178)(135,179)(136,180)(137,171)(138,172)(139,173)(140,174)(141,168)(142,169)(143,170)(144,161)(145,162)(146,163)(147,164)(148,165)(149,166)(150,167)(151,207)(152,208)(153,209)(154,210)(155,201)(156,202)(157,203)(158,204)(159,205)(160,206)(191,196)(192,197)(193,198)(194,199)(195,200)(231,236)(232,237)(233,238)(234,239)(235,240), (1,167,179,30)(2,168,180,21)(3,169,171,22)(4,170,172,23)(5,161,173,24)(6,162,174,25)(7,163,175,26)(8,164,176,27)(9,165,177,28)(10,166,178,29)(11,43,234,152)(12,44,235,153)(13,45,236,154)(14,46,237,155)(15,47,238,156)(16,48,239,157)(17,49,240,158)(18,50,231,159)(19,41,232,160)(20,42,233,151)(31,79,142,130)(32,80,143,121)(33,71,144,122)(34,72,145,123)(35,73,146,124)(36,74,147,125)(37,75,148,126)(38,76,149,127)(39,77,150,128)(40,78,141,129)(51,221,109,117)(52,222,110,118)(53,223,101,119)(54,224,102,120)(55,225,103,111)(56,226,104,112)(57,227,105,113)(58,228,106,114)(59,229,107,115)(60,230,108,116)(61,181,218,131)(62,182,219,132)(63,183,220,133)(64,184,211,134)(65,185,212,135)(66,186,213,136)(67,187,214,137)(68,188,215,138)(69,189,216,139)(70,190,217,140)(81,93,204,191)(82,94,205,192)(83,95,206,193)(84,96,207,194)(85,97,208,195)(86,98,209,196)(87,99,210,197)(88,100,201,198)(89,91,202,199)(90,92,203,200)>;
G:=Group( (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), (1,195,190,103,16,72)(2,196,181,104,17,73)(3,197,182,105,18,74)(4,198,183,106,19,75)(5,199,184,107,20,76)(6,200,185,108,11,77)(7,191,186,109,12,78)(8,192,187,110,13,79)(9,193,188,101,14,80)(10,194,189,102,15,71)(21,209,61,226,158,35)(22,210,62,227,159,36)(23,201,63,228,160,37)(24,202,64,229,151,38)(25,203,65,230,152,39)(26,204,66,221,153,40)(27,205,67,222,154,31)(28,206,68,223,155,32)(29,207,69,224,156,33)(30,208,70,225,157,34)(41,148,170,88,220,114)(42,149,161,89,211,115)(43,150,162,90,212,116)(44,141,163,81,213,117)(45,142,164,82,214,118)(46,143,165,83,215,119)(47,144,166,84,216,120)(48,145,167,85,217,111)(49,146,168,86,218,112)(50,147,169,87,219,113)(51,235,129,175,93,136)(52,236,130,176,94,137)(53,237,121,177,95,138)(54,238,122,178,96,139)(55,239,123,179,97,140)(56,240,124,180,98,131)(57,231,125,171,99,132)(58,232,126,172,100,133)(59,233,127,173,91,134)(60,234,128,174,92,135), (1,185)(2,186)(3,187)(4,188)(5,189)(6,190)(7,181)(8,182)(9,183)(10,184)(11,16)(12,17)(13,18)(14,19)(15,20)(21,40)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,81)(50,82)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,121)(59,122)(60,123)(61,221)(62,222)(63,223)(64,224)(65,225)(66,226)(67,227)(68,228)(69,229)(70,230)(71,107)(72,108)(73,109)(74,110)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(91,96)(92,97)(93,98)(94,99)(95,100)(111,212)(112,213)(113,214)(114,215)(115,216)(116,217)(117,218)(118,219)(119,220)(120,211)(131,175)(132,176)(133,177)(134,178)(135,179)(136,180)(137,171)(138,172)(139,173)(140,174)(141,168)(142,169)(143,170)(144,161)(145,162)(146,163)(147,164)(148,165)(149,166)(150,167)(151,207)(152,208)(153,209)(154,210)(155,201)(156,202)(157,203)(158,204)(159,205)(160,206)(191,196)(192,197)(193,198)(194,199)(195,200)(231,236)(232,237)(233,238)(234,239)(235,240), (1,167,179,30)(2,168,180,21)(3,169,171,22)(4,170,172,23)(5,161,173,24)(6,162,174,25)(7,163,175,26)(8,164,176,27)(9,165,177,28)(10,166,178,29)(11,43,234,152)(12,44,235,153)(13,45,236,154)(14,46,237,155)(15,47,238,156)(16,48,239,157)(17,49,240,158)(18,50,231,159)(19,41,232,160)(20,42,233,151)(31,79,142,130)(32,80,143,121)(33,71,144,122)(34,72,145,123)(35,73,146,124)(36,74,147,125)(37,75,148,126)(38,76,149,127)(39,77,150,128)(40,78,141,129)(51,221,109,117)(52,222,110,118)(53,223,101,119)(54,224,102,120)(55,225,103,111)(56,226,104,112)(57,227,105,113)(58,228,106,114)(59,229,107,115)(60,230,108,116)(61,181,218,131)(62,182,219,132)(63,183,220,133)(64,184,211,134)(65,185,212,135)(66,186,213,136)(67,187,214,137)(68,188,215,138)(69,189,216,139)(70,190,217,140)(81,93,204,191)(82,94,205,192)(83,95,206,193)(84,96,207,194)(85,97,208,195)(86,98,209,196)(87,99,210,197)(88,100,201,198)(89,91,202,199)(90,92,203,200) );
G=PermutationGroup([[(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)], [(1,195,190,103,16,72),(2,196,181,104,17,73),(3,197,182,105,18,74),(4,198,183,106,19,75),(5,199,184,107,20,76),(6,200,185,108,11,77),(7,191,186,109,12,78),(8,192,187,110,13,79),(9,193,188,101,14,80),(10,194,189,102,15,71),(21,209,61,226,158,35),(22,210,62,227,159,36),(23,201,63,228,160,37),(24,202,64,229,151,38),(25,203,65,230,152,39),(26,204,66,221,153,40),(27,205,67,222,154,31),(28,206,68,223,155,32),(29,207,69,224,156,33),(30,208,70,225,157,34),(41,148,170,88,220,114),(42,149,161,89,211,115),(43,150,162,90,212,116),(44,141,163,81,213,117),(45,142,164,82,214,118),(46,143,165,83,215,119),(47,144,166,84,216,120),(48,145,167,85,217,111),(49,146,168,86,218,112),(50,147,169,87,219,113),(51,235,129,175,93,136),(52,236,130,176,94,137),(53,237,121,177,95,138),(54,238,122,178,96,139),(55,239,123,179,97,140),(56,240,124,180,98,131),(57,231,125,171,99,132),(58,232,126,172,100,133),(59,233,127,173,91,134),(60,234,128,174,92,135)], [(1,185),(2,186),(3,187),(4,188),(5,189),(6,190),(7,181),(8,182),(9,183),(10,184),(11,16),(12,17),(13,18),(14,19),(15,20),(21,40),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39),(41,83),(42,84),(43,85),(44,86),(45,87),(46,88),(47,89),(48,90),(49,81),(50,82),(51,124),(52,125),(53,126),(54,127),(55,128),(56,129),(57,130),(58,121),(59,122),(60,123),(61,221),(62,222),(63,223),(64,224),(65,225),(66,226),(67,227),(68,228),(69,229),(70,230),(71,107),(72,108),(73,109),(74,110),(75,101),(76,102),(77,103),(78,104),(79,105),(80,106),(91,96),(92,97),(93,98),(94,99),(95,100),(111,212),(112,213),(113,214),(114,215),(115,216),(116,217),(117,218),(118,219),(119,220),(120,211),(131,175),(132,176),(133,177),(134,178),(135,179),(136,180),(137,171),(138,172),(139,173),(140,174),(141,168),(142,169),(143,170),(144,161),(145,162),(146,163),(147,164),(148,165),(149,166),(150,167),(151,207),(152,208),(153,209),(154,210),(155,201),(156,202),(157,203),(158,204),(159,205),(160,206),(191,196),(192,197),(193,198),(194,199),(195,200),(231,236),(232,237),(233,238),(234,239),(235,240)], [(1,167,179,30),(2,168,180,21),(3,169,171,22),(4,170,172,23),(5,161,173,24),(6,162,174,25),(7,163,175,26),(8,164,176,27),(9,165,177,28),(10,166,178,29),(11,43,234,152),(12,44,235,153),(13,45,236,154),(14,46,237,155),(15,47,238,156),(16,48,239,157),(17,49,240,158),(18,50,231,159),(19,41,232,160),(20,42,233,151),(31,79,142,130),(32,80,143,121),(33,71,144,122),(34,72,145,123),(35,73,146,124),(36,74,147,125),(37,75,148,126),(38,76,149,127),(39,77,150,128),(40,78,141,129),(51,221,109,117),(52,222,110,118),(53,223,101,119),(54,224,102,120),(55,225,103,111),(56,226,104,112),(57,227,105,113),(58,228,106,114),(59,229,107,115),(60,230,108,116),(61,181,218,131),(62,182,219,132),(63,183,220,133),(64,184,211,134),(65,185,212,135),(66,186,213,136),(67,187,214,137),(68,188,215,138),(69,189,216,139),(70,190,217,140),(81,93,204,191),(82,94,205,192),(83,95,206,193),(84,96,207,194),(85,97,208,195),(86,98,209,196),(87,99,210,197),(88,100,201,198),(89,91,202,199),(90,92,203,200)]])
180 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 6A | ··· | 6G | 10A | ··· | 10AB | 10AC | ··· | 10AR | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20AF | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 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 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | S3 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 | C5×S3 | C5×D4 | S3×C10 | S3×C10 | S3×C20 | C5×D12 | C5×C3⋊D4 |
| kernel | C10×D6⋊C4 | C5×D6⋊C4 | Dic3×C2×C10 | C22×C60 | S3×C22×C10 | S3×C2×C10 | C2×D6⋊C4 | D6⋊C4 | C22×Dic3 | C22×C12 | S3×C23 | C22×S3 | C22×C20 | C2×C30 | C2×C20 | C22×C10 | C2×C10 | C2×C10 | C2×C10 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 16 | 4 | 4 | 4 | 32 | 1 | 4 | 2 | 1 | 4 | 4 | 4 | 4 | 16 | 8 | 4 | 16 | 16 | 16 |
Matrix representation of C10×D6⋊C4 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 60 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 9 | 18 |
| 0 | 0 | 43 | 52 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,1,0,0,0,0,27,0,0,0,0,27],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,1],[1,0,0,0,0,60,0,0,0,0,0,60,0,0,60,0],[1,0,0,0,0,50,0,0,0,0,9,43,0,0,18,52] >;
C10×D6⋊C4 in GAP, Magma, Sage, TeX
C_{10}\times D_6\rtimes C_4 % in TeX
G:=Group("C10xD6:C4"); // GroupNames label
G:=SmallGroup(480,806);
// by ID
G=gap.SmallGroup(480,806);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1766,226,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^6=c^2=d^4=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^3*c>;
// generators/relations