direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D6.Dic5, C30⋊6M4(2), C60.187C23, C5⋊2C8⋊29D6, C10⋊5(C8⋊S3), (S3×C20).11C4, C20.111(C4×S3), C60.151(C2×C4), (C4×S3).45D10, (C2×C20).331D6, C15⋊19(C2×M4(2)), C6⋊1(C4.Dic5), (C4×S3).3Dic5, D6.3(C2×Dic5), C4.23(S3×Dic5), C15⋊3C8⋊45C22, (C2×C12).335D10, C12.28(C2×Dic5), C6.5(C22×Dic5), (S3×C20).55C22, C20.184(C22×S3), (C2×C60).233C22, C30.109(C22×C4), Dic3.6(C2×Dic5), (C2×Dic3).6Dic5, (C10×Dic3).18C4, C12.184(C22×D5), (C22×S3).4Dic5, C22.14(S3×Dic5), C5⋊7(C2×C8⋊S3), (S3×C2×C4).8D5, (S3×C2×C20).9C2, (C6×C5⋊2C8)⋊13C2, C4.157(C2×S3×D5), (C2×C5⋊2C8)⋊11S3, C3⋊1(C2×C4.Dic5), C2.7(C2×S3×Dic5), (S3×C2×C10).11C4, C10.113(S3×C2×C4), (C2×C15⋊3C8)⋊26C2, (C2×C10).76(C4×S3), (C2×C4).236(S3×D5), (S3×C10).35(C2×C4), (C2×C30).106(C2×C4), (C3×C5⋊2C8)⋊36C22, (C2×C6).16(C2×Dic5), (C5×Dic3).43(C2×C4), SmallGroup(480,370)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D6.Dic5
G = < a,b,c,d,e | a2=b6=c2=1, d10=b3, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d9 >
Subgroups: 412 in 136 conjugacy classes, 68 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, C23, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C30, C2×M4(2), C5⋊2C8, C5⋊2C8, C2×C20, C2×C20, C22×C10, C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C5⋊2C8, C2×C5⋊2C8, C4.Dic5, C22×C20, C2×C8⋊S3, C3×C5⋊2C8, C15⋊3C8, S3×C20, C10×Dic3, C2×C60, S3×C2×C10, C2×C4.Dic5, D6.Dic5, C6×C5⋊2C8, C2×C15⋊3C8, S3×C2×C20, C2×D6.Dic5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, M4(2), C22×C4, Dic5, D10, C4×S3, C22×S3, C2×M4(2), C2×Dic5, C22×D5, C8⋊S3, S3×C2×C4, S3×D5, C4.Dic5, C22×Dic5, C2×C8⋊S3, S3×Dic5, C2×S3×D5, C2×C4.Dic5, D6.Dic5, C2×S3×Dic5, C2×D6.Dic5
►On 240 points
Generators in S240
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 21)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(41 173)(42 174)(43 175)(44 176)(45 177)(46 178)(47 179)(48 180)(49 161)(50 162)(51 163)(52 164)(53 165)(54 166)(55 167)(56 168)(57 169)(58 170)(59 171)(60 172)(61 228)(62 229)(63 230)(64 231)(65 232)(66 233)(67 234)(68 235)(69 236)(70 237)(71 238)(72 239)(73 240)(74 221)(75 222)(76 223)(77 224)(78 225)(79 226)(80 227)(81 189)(82 190)(83 191)(84 192)(85 193)(86 194)(87 195)(88 196)(89 197)(90 198)(91 199)(92 200)(93 181)(94 182)(95 183)(96 184)(97 185)(98 186)(99 187)(100 188)(101 135)(102 136)(103 137)(104 138)(105 139)(106 140)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(141 213)(142 214)(143 215)(144 216)(145 217)(146 218)(147 219)(148 220)(149 201)(150 202)(151 203)(152 204)(153 205)(154 206)(155 207)(156 208)(157 209)(158 210)(159 211)(160 212)
(1 60 104 11 50 114)(2 41 105 12 51 115)(3 42 106 13 52 116)(4 43 107 14 53 117)(5 44 108 15 54 118)(6 45 109 16 55 119)(7 46 110 17 56 120)(8 47 111 18 57 101)(9 48 112 19 58 102)(10 49 113 20 59 103)(21 165 131 31 175 121)(22 166 132 32 176 122)(23 167 133 33 177 123)(24 168 134 34 178 124)(25 169 135 35 179 125)(26 170 136 36 180 126)(27 171 137 37 161 127)(28 172 138 38 162 128)(29 173 139 39 163 129)(30 174 140 40 164 130)(61 219 185 71 209 195)(62 220 186 72 210 196)(63 201 187 73 211 197)(64 202 188 74 212 198)(65 203 189 75 213 199)(66 204 190 76 214 200)(67 205 191 77 215 181)(68 206 192 78 216 182)(69 207 193 79 217 183)(70 208 194 80 218 184)(81 222 141 91 232 151)(82 223 142 92 233 152)(83 224 143 93 234 153)(84 225 144 94 235 154)(85 226 145 95 236 155)(86 227 146 96 237 156)(87 228 147 97 238 157)(88 229 148 98 239 158)(89 230 149 99 240 159)(90 231 150 100 221 160)
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 101)(9 102)(10 103)(11 104)(12 105)(13 106)(14 107)(15 108)(16 109)(17 110)(18 111)(19 112)(20 113)(21 121)(22 122)(23 123)(24 124)(25 125)(26 126)(27 127)(28 128)(29 129)(30 130)(31 131)(32 132)(33 133)(34 134)(35 135)(36 136)(37 137)(38 138)(39 139)(40 140)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 185)(62 186)(63 187)(64 188)(65 189)(66 190)(67 191)(68 192)(69 193)(70 194)(71 195)(72 196)(73 197)(74 198)(75 199)(76 200)(77 181)(78 182)(79 183)(80 184)(81 232)(82 233)(83 234)(84 235)(85 236)(86 237)(87 238)(88 239)(89 240)(90 221)(91 222)(92 223)(93 224)(94 225)(95 226)(96 227)(97 228)(98 229)(99 230)(100 231)(161 171)(162 172)(163 173)(164 174)(165 175)(166 176)(167 177)(168 178)(169 179)(170 180)
(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 64 6 69 11 74 16 79)(2 73 7 78 12 63 17 68)(3 62 8 67 13 72 18 77)(4 71 9 76 14 61 19 66)(5 80 10 65 15 70 20 75)(21 228 26 233 31 238 36 223)(22 237 27 222 32 227 37 232)(23 226 28 231 33 236 38 221)(24 235 29 240 34 225 39 230)(25 224 30 229 35 234 40 239)(41 211 46 216 51 201 56 206)(42 220 47 205 52 210 57 215)(43 209 48 214 53 219 58 204)(44 218 49 203 54 208 59 213)(45 207 50 212 55 217 60 202)(81 132 86 137 91 122 96 127)(82 121 87 126 92 131 97 136)(83 130 88 135 93 140 98 125)(84 139 89 124 94 129 99 134)(85 128 90 133 95 138 100 123)(101 181 106 186 111 191 116 196)(102 190 107 195 112 200 117 185)(103 199 108 184 113 189 118 194)(104 188 109 193 114 198 119 183)(105 197 110 182 115 187 120 192)(141 176 146 161 151 166 156 171)(142 165 147 170 152 175 157 180)(143 174 148 179 153 164 158 169)(144 163 149 168 154 173 159 178)(145 172 150 177 155 162 160 167)
G:=sub<Sym(240)| (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,21)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(41,173)(42,174)(43,175)(44,176)(45,177)(46,178)(47,179)(48,180)(49,161)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,228)(62,229)(63,230)(64,231)(65,232)(66,233)(67,234)(68,235)(69,236)(70,237)(71,238)(72,239)(73,240)(74,221)(75,222)(76,223)(77,224)(78,225)(79,226)(80,227)(81,189)(82,190)(83,191)(84,192)(85,193)(86,194)(87,195)(88,196)(89,197)(90,198)(91,199)(92,200)(93,181)(94,182)(95,183)(96,184)(97,185)(98,186)(99,187)(100,188)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(141,213)(142,214)(143,215)(144,216)(145,217)(146,218)(147,219)(148,220)(149,201)(150,202)(151,203)(152,204)(153,205)(154,206)(155,207)(156,208)(157,209)(158,210)(159,211)(160,212), (1,60,104,11,50,114)(2,41,105,12,51,115)(3,42,106,13,52,116)(4,43,107,14,53,117)(5,44,108,15,54,118)(6,45,109,16,55,119)(7,46,110,17,56,120)(8,47,111,18,57,101)(9,48,112,19,58,102)(10,49,113,20,59,103)(21,165,131,31,175,121)(22,166,132,32,176,122)(23,167,133,33,177,123)(24,168,134,34,178,124)(25,169,135,35,179,125)(26,170,136,36,180,126)(27,171,137,37,161,127)(28,172,138,38,162,128)(29,173,139,39,163,129)(30,174,140,40,164,130)(61,219,185,71,209,195)(62,220,186,72,210,196)(63,201,187,73,211,197)(64,202,188,74,212,198)(65,203,189,75,213,199)(66,204,190,76,214,200)(67,205,191,77,215,181)(68,206,192,78,216,182)(69,207,193,79,217,183)(70,208,194,80,218,184)(81,222,141,91,232,151)(82,223,142,92,233,152)(83,224,143,93,234,153)(84,225,144,94,235,154)(85,226,145,95,236,155)(86,227,146,96,237,156)(87,228,147,97,238,157)(88,229,148,98,239,158)(89,230,149,99,240,159)(90,231,150,100,221,160), (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,101)(9,102)(10,103)(11,104)(12,105)(13,106)(14,107)(15,108)(16,109)(17,110)(18,111)(19,112)(20,113)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,127)(28,128)(29,129)(30,130)(31,131)(32,132)(33,133)(34,134)(35,135)(36,136)(37,137)(38,138)(39,139)(40,140)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,185)(62,186)(63,187)(64,188)(65,189)(66,190)(67,191)(68,192)(69,193)(70,194)(71,195)(72,196)(73,197)(74,198)(75,199)(76,200)(77,181)(78,182)(79,183)(80,184)(81,232)(82,233)(83,234)(84,235)(85,236)(86,237)(87,238)(88,239)(89,240)(90,221)(91,222)(92,223)(93,224)(94,225)(95,226)(96,227)(97,228)(98,229)(99,230)(100,231)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178)(169,179)(170,180), (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), (1,64,6,69,11,74,16,79)(2,73,7,78,12,63,17,68)(3,62,8,67,13,72,18,77)(4,71,9,76,14,61,19,66)(5,80,10,65,15,70,20,75)(21,228,26,233,31,238,36,223)(22,237,27,222,32,227,37,232)(23,226,28,231,33,236,38,221)(24,235,29,240,34,225,39,230)(25,224,30,229,35,234,40,239)(41,211,46,216,51,201,56,206)(42,220,47,205,52,210,57,215)(43,209,48,214,53,219,58,204)(44,218,49,203,54,208,59,213)(45,207,50,212,55,217,60,202)(81,132,86,137,91,122,96,127)(82,121,87,126,92,131,97,136)(83,130,88,135,93,140,98,125)(84,139,89,124,94,129,99,134)(85,128,90,133,95,138,100,123)(101,181,106,186,111,191,116,196)(102,190,107,195,112,200,117,185)(103,199,108,184,113,189,118,194)(104,188,109,193,114,198,119,183)(105,197,110,182,115,187,120,192)(141,176,146,161,151,166,156,171)(142,165,147,170,152,175,157,180)(143,174,148,179,153,164,158,169)(144,163,149,168,154,173,159,178)(145,172,150,177,155,162,160,167)>;
G:=Group( (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,21)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(41,173)(42,174)(43,175)(44,176)(45,177)(46,178)(47,179)(48,180)(49,161)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,228)(62,229)(63,230)(64,231)(65,232)(66,233)(67,234)(68,235)(69,236)(70,237)(71,238)(72,239)(73,240)(74,221)(75,222)(76,223)(77,224)(78,225)(79,226)(80,227)(81,189)(82,190)(83,191)(84,192)(85,193)(86,194)(87,195)(88,196)(89,197)(90,198)(91,199)(92,200)(93,181)(94,182)(95,183)(96,184)(97,185)(98,186)(99,187)(100,188)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(141,213)(142,214)(143,215)(144,216)(145,217)(146,218)(147,219)(148,220)(149,201)(150,202)(151,203)(152,204)(153,205)(154,206)(155,207)(156,208)(157,209)(158,210)(159,211)(160,212), (1,60,104,11,50,114)(2,41,105,12,51,115)(3,42,106,13,52,116)(4,43,107,14,53,117)(5,44,108,15,54,118)(6,45,109,16,55,119)(7,46,110,17,56,120)(8,47,111,18,57,101)(9,48,112,19,58,102)(10,49,113,20,59,103)(21,165,131,31,175,121)(22,166,132,32,176,122)(23,167,133,33,177,123)(24,168,134,34,178,124)(25,169,135,35,179,125)(26,170,136,36,180,126)(27,171,137,37,161,127)(28,172,138,38,162,128)(29,173,139,39,163,129)(30,174,140,40,164,130)(61,219,185,71,209,195)(62,220,186,72,210,196)(63,201,187,73,211,197)(64,202,188,74,212,198)(65,203,189,75,213,199)(66,204,190,76,214,200)(67,205,191,77,215,181)(68,206,192,78,216,182)(69,207,193,79,217,183)(70,208,194,80,218,184)(81,222,141,91,232,151)(82,223,142,92,233,152)(83,224,143,93,234,153)(84,225,144,94,235,154)(85,226,145,95,236,155)(86,227,146,96,237,156)(87,228,147,97,238,157)(88,229,148,98,239,158)(89,230,149,99,240,159)(90,231,150,100,221,160), (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,101)(9,102)(10,103)(11,104)(12,105)(13,106)(14,107)(15,108)(16,109)(17,110)(18,111)(19,112)(20,113)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,127)(28,128)(29,129)(30,130)(31,131)(32,132)(33,133)(34,134)(35,135)(36,136)(37,137)(38,138)(39,139)(40,140)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,185)(62,186)(63,187)(64,188)(65,189)(66,190)(67,191)(68,192)(69,193)(70,194)(71,195)(72,196)(73,197)(74,198)(75,199)(76,200)(77,181)(78,182)(79,183)(80,184)(81,232)(82,233)(83,234)(84,235)(85,236)(86,237)(87,238)(88,239)(89,240)(90,221)(91,222)(92,223)(93,224)(94,225)(95,226)(96,227)(97,228)(98,229)(99,230)(100,231)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178)(169,179)(170,180), (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), (1,64,6,69,11,74,16,79)(2,73,7,78,12,63,17,68)(3,62,8,67,13,72,18,77)(4,71,9,76,14,61,19,66)(5,80,10,65,15,70,20,75)(21,228,26,233,31,238,36,223)(22,237,27,222,32,227,37,232)(23,226,28,231,33,236,38,221)(24,235,29,240,34,225,39,230)(25,224,30,229,35,234,40,239)(41,211,46,216,51,201,56,206)(42,220,47,205,52,210,57,215)(43,209,48,214,53,219,58,204)(44,218,49,203,54,208,59,213)(45,207,50,212,55,217,60,202)(81,132,86,137,91,122,96,127)(82,121,87,126,92,131,97,136)(83,130,88,135,93,140,98,125)(84,139,89,124,94,129,99,134)(85,128,90,133,95,138,100,123)(101,181,106,186,111,191,116,196)(102,190,107,195,112,200,117,185)(103,199,108,184,113,189,118,194)(104,188,109,193,114,198,119,183)(105,197,110,182,115,187,120,192)(141,176,146,161,151,166,156,171)(142,165,147,170,152,175,157,180)(143,174,148,179,153,164,158,169)(144,163,149,168,154,173,159,178)(145,172,150,177,155,162,160,167) );
G=PermutationGroup([[(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,21),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(41,173),(42,174),(43,175),(44,176),(45,177),(46,178),(47,179),(48,180),(49,161),(50,162),(51,163),(52,164),(53,165),(54,166),(55,167),(56,168),(57,169),(58,170),(59,171),(60,172),(61,228),(62,229),(63,230),(64,231),(65,232),(66,233),(67,234),(68,235),(69,236),(70,237),(71,238),(72,239),(73,240),(74,221),(75,222),(76,223),(77,224),(78,225),(79,226),(80,227),(81,189),(82,190),(83,191),(84,192),(85,193),(86,194),(87,195),(88,196),(89,197),(90,198),(91,199),(92,200),(93,181),(94,182),(95,183),(96,184),(97,185),(98,186),(99,187),(100,188),(101,135),(102,136),(103,137),(104,138),(105,139),(106,140),(107,121),(108,122),(109,123),(110,124),(111,125),(112,126),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(141,213),(142,214),(143,215),(144,216),(145,217),(146,218),(147,219),(148,220),(149,201),(150,202),(151,203),(152,204),(153,205),(154,206),(155,207),(156,208),(157,209),(158,210),(159,211),(160,212)], [(1,60,104,11,50,114),(2,41,105,12,51,115),(3,42,106,13,52,116),(4,43,107,14,53,117),(5,44,108,15,54,118),(6,45,109,16,55,119),(7,46,110,17,56,120),(8,47,111,18,57,101),(9,48,112,19,58,102),(10,49,113,20,59,103),(21,165,131,31,175,121),(22,166,132,32,176,122),(23,167,133,33,177,123),(24,168,134,34,178,124),(25,169,135,35,179,125),(26,170,136,36,180,126),(27,171,137,37,161,127),(28,172,138,38,162,128),(29,173,139,39,163,129),(30,174,140,40,164,130),(61,219,185,71,209,195),(62,220,186,72,210,196),(63,201,187,73,211,197),(64,202,188,74,212,198),(65,203,189,75,213,199),(66,204,190,76,214,200),(67,205,191,77,215,181),(68,206,192,78,216,182),(69,207,193,79,217,183),(70,208,194,80,218,184),(81,222,141,91,232,151),(82,223,142,92,233,152),(83,224,143,93,234,153),(84,225,144,94,235,154),(85,226,145,95,236,155),(86,227,146,96,237,156),(87,228,147,97,238,157),(88,229,148,98,239,158),(89,230,149,99,240,159),(90,231,150,100,221,160)], [(1,114),(2,115),(3,116),(4,117),(5,118),(6,119),(7,120),(8,101),(9,102),(10,103),(11,104),(12,105),(13,106),(14,107),(15,108),(16,109),(17,110),(18,111),(19,112),(20,113),(21,121),(22,122),(23,123),(24,124),(25,125),(26,126),(27,127),(28,128),(29,129),(30,130),(31,131),(32,132),(33,133),(34,134),(35,135),(36,136),(37,137),(38,138),(39,139),(40,140),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,185),(62,186),(63,187),(64,188),(65,189),(66,190),(67,191),(68,192),(69,193),(70,194),(71,195),(72,196),(73,197),(74,198),(75,199),(76,200),(77,181),(78,182),(79,183),(80,184),(81,232),(82,233),(83,234),(84,235),(85,236),(86,237),(87,238),(88,239),(89,240),(90,221),(91,222),(92,223),(93,224),(94,225),(95,226),(96,227),(97,228),(98,229),(99,230),(100,231),(161,171),(162,172),(163,173),(164,174),(165,175),(166,176),(167,177),(168,178),(169,179),(170,180)], [(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)], [(1,64,6,69,11,74,16,79),(2,73,7,78,12,63,17,68),(3,62,8,67,13,72,18,77),(4,71,9,76,14,61,19,66),(5,80,10,65,15,70,20,75),(21,228,26,233,31,238,36,223),(22,237,27,222,32,227,37,232),(23,226,28,231,33,236,38,221),(24,235,29,240,34,225,39,230),(25,224,30,229,35,234,40,239),(41,211,46,216,51,201,56,206),(42,220,47,205,52,210,57,215),(43,209,48,214,53,219,58,204),(44,218,49,203,54,208,59,213),(45,207,50,212,55,217,60,202),(81,132,86,137,91,122,96,127),(82,121,87,126,92,131,97,136),(83,130,88,135,93,140,98,125),(84,139,89,124,94,129,99,134),(85,128,90,133,95,138,100,123),(101,181,106,186,111,191,116,196),(102,190,107,195,112,200,117,185),(103,199,108,184,113,189,118,194),(104,188,109,193,114,198,119,183),(105,197,110,182,115,187,120,192),(141,176,146,161,151,166,156,171),(142,165,147,170,152,175,157,180),(143,174,148,179,153,164,158,169),(144,163,149,168,154,173,159,178),(145,172,150,177,155,162,160,167)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 24A | ··· | 24H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
84 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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | + | - | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D5 | D6 | D6 | M4(2) | Dic5 | D10 | Dic5 | D10 | Dic5 | C4×S3 | C4×S3 | C8⋊S3 | C4.Dic5 | S3×D5 | S3×Dic5 | C2×S3×D5 | S3×Dic5 | D6.Dic5 |
| kernel | C2×D6.Dic5 | D6.Dic5 | C6×C5⋊2C8 | C2×C15⋊3C8 | S3×C2×C20 | S3×C20 | C10×Dic3 | S3×C2×C10 | C2×C5⋊2C8 | S3×C2×C4 | C5⋊2C8 | C2×C20 | C30 | C4×S3 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | C20 | C2×C10 | C10 | C6 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 1 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 8 | 16 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C2×D6.Dic5 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 233 | 0 | 0 | 0 | 0 |
| 151 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 90 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 221 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 177 | 0 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 51 |
| 207 | 101 | 0 | 0 | 0 | 0 |
| 112 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 211 | 3 | 0 | 0 |
| 0 | 0 | 0 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 92 | 40 |
| 0 | 0 | 0 | 0 | 168 | 149 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,151,0,0,0,0,233,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,90,0,0,0,0,0,1,0,0,0,0,0,0,240,221,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[177,0,0,0,0,0,0,177,0,0,0,0,0,0,177,0,0,0,0,0,0,177,0,0,0,0,0,0,0,1,0,0,0,0,240,51],[207,112,0,0,0,0,101,34,0,0,0,0,0,0,211,0,0,0,0,0,3,30,0,0,0,0,0,0,92,168,0,0,0,0,40,149] >;
C2×D6.Dic5 in GAP, Magma, Sage, TeX
C_2\times D_6.{\rm Dic}_5 % in TeX
G:=Group("C2xD6.Dic5"); // GroupNames label
G:=SmallGroup(480,370);
// by ID
G=gap.SmallGroup(480,370);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^2=1,d^10=b^3,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=d^9>;
// generators/relations