direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D6⋊Dic5, C10⋊4(D6⋊C4), D6⋊5(C2×Dic5), (C2×C30).69D4, C30⋊5(C22⋊C4), (C2×Dic5)⋊19D6, C30.218(C2×D4), (C2×C10).44D12, C10.64(C2×D12), C6⋊1(C23.D5), (S3×C23).1D5, C23.62(S3×D5), (C22×Dic5)⋊4S3, (C22×S3)⋊2Dic5, (C22×C6).83D10, (C2×C30).180C23, C30.138(C22×C4), (C6×Dic5)⋊23C22, (C22×S3).77D10, (C22×C10).101D6, C6.16(C22×Dic5), C22.16(S3×Dic5), (C22×Dic15)⋊12C2, (C2×Dic15)⋊31C22, C22.23(C5⋊D12), C22.24(C15⋊D4), (C22×C30).42C22, C5⋊5(C2×D6⋊C4), (S3×C2×C10)⋊7C4, C15⋊9(C2×C22⋊C4), (C2×C6×Dic5)⋊1C2, C3⋊2(C2×C23.D5), C10.123(S3×C2×C4), C2.2(C2×C5⋊D12), C2.3(C2×C15⋊D4), C6.91(C2×C5⋊D4), (S3×C10)⋊27(C2×C4), C2.16(C2×S3×Dic5), C22.78(C2×S3×D5), (C2×C10).78(C4×S3), C10.92(C2×C3⋊D4), (S3×C22×C10).1C2, (C2×C30).113(C2×C4), (C2×C6).35(C5⋊D4), (S3×C2×C10).93C22, (C2×C6).18(C2×Dic5), (C2×C10).58(C3⋊D4), (C2×C6).192(C22×D5), (C2×C10).192(C22×S3), SmallGroup(480,614)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D6⋊Dic5
G = < a,b,c,d,e | a2=b6=c2=d10=1, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d-1 >
Subgroups: 988 in 264 conjugacy classes, 100 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, C23, C23, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, Dic5, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, C30, C30, C2×C22⋊C4, C2×Dic5, C2×Dic5, C22×C10, C22×C10, D6⋊C4, C22×Dic3, C22×C12, S3×C23, C3×Dic5, Dic15, S3×C10, S3×C10, C2×C30, C2×C30, C23.D5, C22×Dic5, C22×Dic5, C23×C10, C2×D6⋊C4, C6×Dic5, C6×Dic5, C2×Dic15, C2×Dic15, S3×C2×C10, S3×C2×C10, C22×C30, C2×C23.D5, D6⋊Dic5, C2×C6×Dic5, C22×Dic15, S3×C22×C10, C2×D6⋊Dic5
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22⋊C4, C22×C4, C2×D4, Dic5, D10, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C2×Dic5, C5⋊D4, C22×D5, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, S3×D5, C23.D5, C22×Dic5, C2×C5⋊D4, C2×D6⋊C4, S3×Dic5, C15⋊D4, C5⋊D12, C2×S3×D5, C2×C23.D5, D6⋊Dic5, C2×S3×Dic5, C2×C15⋊D4, C2×C5⋊D12, C2×D6⋊Dic5
►On 240 points
Generators in S240
(1 116)(2 117)(3 118)(4 119)(5 120)(6 111)(7 112)(8 113)(9 114)(10 115)(11 82)(12 83)(13 84)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 81)(21 99)(22 100)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(31 170)(32 161)(33 162)(34 163)(35 164)(36 165)(37 166)(38 167)(39 168)(40 169)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)(61 104)(62 105)(63 106)(64 107)(65 108)(66 109)(67 110)(68 101)(69 102)(70 103)(131 231)(132 232)(133 233)(134 234)(135 235)(136 236)(137 237)(138 238)(139 239)(140 240)(141 218)(142 219)(143 220)(144 211)(145 212)(146 213)(147 214)(148 215)(149 216)(150 217)(151 204)(152 205)(153 206)(154 207)(155 208)(156 209)(157 210)(158 201)(159 202)(160 203)(171 191)(172 192)(173 193)(174 194)(175 195)(176 196)(177 197)(178 198)(179 199)(180 200)(181 223)(182 224)(183 225)(184 226)(185 227)(186 228)(187 229)(188 230)(189 221)(190 222)
(1 67 27 51 45 11)(2 68 28 52 46 12)(3 69 29 53 47 13)(4 70 30 54 48 14)(5 61 21 55 49 15)(6 62 22 56 50 16)(7 63 23 57 41 17)(8 64 24 58 42 18)(9 65 25 59 43 19)(10 66 26 60 44 20)(31 208 239 225 220 194)(32 209 240 226 211 195)(33 210 231 227 212 196)(34 201 232 228 213 197)(35 202 233 229 214 198)(36 203 234 230 215 199)(37 204 235 221 216 200)(38 205 236 222 217 191)(39 206 237 223 218 192)(40 207 238 224 219 193)(71 130 87 111 105 100)(72 121 88 112 106 91)(73 122 89 113 107 92)(74 123 90 114 108 93)(75 124 81 115 109 94)(76 125 82 116 110 95)(77 126 83 117 101 96)(78 127 84 118 102 97)(79 128 85 119 103 98)(80 129 86 120 104 99)(131 185 145 176 162 157)(132 186 146 177 163 158)(133 187 147 178 164 159)(134 188 148 179 165 160)(135 189 149 180 166 151)(136 190 150 171 167 152)(137 181 141 172 168 153)(138 182 142 173 169 154)(139 183 143 174 170 155)(140 184 144 175 161 156)
(1 95)(2 96)(3 97)(4 98)(5 99)(6 100)(7 91)(8 92)(9 93)(10 94)(11 76)(12 77)(13 78)(14 79)(15 80)(16 71)(17 72)(18 73)(19 74)(20 75)(21 120)(22 111)(23 112)(24 113)(25 114)(26 115)(27 116)(28 117)(29 118)(30 119)(31 183)(32 184)(33 185)(34 186)(35 187)(36 188)(37 189)(38 190)(39 181)(40 182)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 82)(52 83)(53 84)(54 85)(55 86)(56 87)(57 88)(58 89)(59 90)(60 81)(61 104)(62 105)(63 106)(64 107)(65 108)(66 109)(67 110)(68 101)(69 102)(70 103)(131 210)(132 201)(133 202)(134 203)(135 204)(136 205)(137 206)(138 207)(139 208)(140 209)(141 192)(142 193)(143 194)(144 195)(145 196)(146 197)(147 198)(148 199)(149 200)(150 191)(151 235)(152 236)(153 237)(154 238)(155 239)(156 240)(157 231)(158 232)(159 233)(160 234)(161 226)(162 227)(163 228)(164 229)(165 230)(166 221)(167 222)(168 223)(169 224)(170 225)(171 217)(172 218)(173 219)(174 220)(175 211)(176 212)(177 213)(178 214)(179 215)(180 216)
(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)
(1 199 6 194)(2 198 7 193)(3 197 8 192)(4 196 9 191)(5 195 10 200)(11 215 16 220)(12 214 17 219)(13 213 18 218)(14 212 19 217)(15 211 20 216)(21 209 26 204)(22 208 27 203)(23 207 28 202)(24 206 29 201)(25 205 30 210)(31 67 36 62)(32 66 37 61)(33 65 38 70)(34 64 39 69)(35 63 40 68)(41 224 46 229)(42 223 47 228)(43 222 48 227)(44 221 49 226)(45 230 50 225)(51 234 56 239)(52 233 57 238)(53 232 58 237)(54 231 59 236)(55 240 60 235)(71 139 76 134)(72 138 77 133)(73 137 78 132)(74 136 79 131)(75 135 80 140)(81 149 86 144)(82 148 87 143)(83 147 88 142)(84 146 89 141)(85 145 90 150)(91 154 96 159)(92 153 97 158)(93 152 98 157)(94 151 99 156)(95 160 100 155)(101 164 106 169)(102 163 107 168)(103 162 108 167)(104 161 109 166)(105 170 110 165)(111 174 116 179)(112 173 117 178)(113 172 118 177)(114 171 119 176)(115 180 120 175)(121 182 126 187)(122 181 127 186)(123 190 128 185)(124 189 129 184)(125 188 130 183)
G:=sub<Sym(240)| (1,116)(2,117)(3,118)(4,119)(5,120)(6,111)(7,112)(8,113)(9,114)(10,115)(11,82)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,81)(21,99)(22,100)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,170)(32,161)(33,162)(34,163)(35,164)(36,165)(37,166)(38,167)(39,168)(40,169)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,76)(52,77)(53,78)(54,79)(55,80)(56,71)(57,72)(58,73)(59,74)(60,75)(61,104)(62,105)(63,106)(64,107)(65,108)(66,109)(67,110)(68,101)(69,102)(70,103)(131,231)(132,232)(133,233)(134,234)(135,235)(136,236)(137,237)(138,238)(139,239)(140,240)(141,218)(142,219)(143,220)(144,211)(145,212)(146,213)(147,214)(148,215)(149,216)(150,217)(151,204)(152,205)(153,206)(154,207)(155,208)(156,209)(157,210)(158,201)(159,202)(160,203)(171,191)(172,192)(173,193)(174,194)(175,195)(176,196)(177,197)(178,198)(179,199)(180,200)(181,223)(182,224)(183,225)(184,226)(185,227)(186,228)(187,229)(188,230)(189,221)(190,222), (1,67,27,51,45,11)(2,68,28,52,46,12)(3,69,29,53,47,13)(4,70,30,54,48,14)(5,61,21,55,49,15)(6,62,22,56,50,16)(7,63,23,57,41,17)(8,64,24,58,42,18)(9,65,25,59,43,19)(10,66,26,60,44,20)(31,208,239,225,220,194)(32,209,240,226,211,195)(33,210,231,227,212,196)(34,201,232,228,213,197)(35,202,233,229,214,198)(36,203,234,230,215,199)(37,204,235,221,216,200)(38,205,236,222,217,191)(39,206,237,223,218,192)(40,207,238,224,219,193)(71,130,87,111,105,100)(72,121,88,112,106,91)(73,122,89,113,107,92)(74,123,90,114,108,93)(75,124,81,115,109,94)(76,125,82,116,110,95)(77,126,83,117,101,96)(78,127,84,118,102,97)(79,128,85,119,103,98)(80,129,86,120,104,99)(131,185,145,176,162,157)(132,186,146,177,163,158)(133,187,147,178,164,159)(134,188,148,179,165,160)(135,189,149,180,166,151)(136,190,150,171,167,152)(137,181,141,172,168,153)(138,182,142,173,169,154)(139,183,143,174,170,155)(140,184,144,175,161,156), (1,95)(2,96)(3,97)(4,98)(5,99)(6,100)(7,91)(8,92)(9,93)(10,94)(11,76)(12,77)(13,78)(14,79)(15,80)(16,71)(17,72)(18,73)(19,74)(20,75)(21,120)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,183)(32,184)(33,185)(34,186)(35,187)(36,188)(37,189)(38,190)(39,181)(40,182)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,81)(61,104)(62,105)(63,106)(64,107)(65,108)(66,109)(67,110)(68,101)(69,102)(70,103)(131,210)(132,201)(133,202)(134,203)(135,204)(136,205)(137,206)(138,207)(139,208)(140,209)(141,192)(142,193)(143,194)(144,195)(145,196)(146,197)(147,198)(148,199)(149,200)(150,191)(151,235)(152,236)(153,237)(154,238)(155,239)(156,240)(157,231)(158,232)(159,233)(160,234)(161,226)(162,227)(163,228)(164,229)(165,230)(166,221)(167,222)(168,223)(169,224)(170,225)(171,217)(172,218)(173,219)(174,220)(175,211)(176,212)(177,213)(178,214)(179,215)(180,216), (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), (1,199,6,194)(2,198,7,193)(3,197,8,192)(4,196,9,191)(5,195,10,200)(11,215,16,220)(12,214,17,219)(13,213,18,218)(14,212,19,217)(15,211,20,216)(21,209,26,204)(22,208,27,203)(23,207,28,202)(24,206,29,201)(25,205,30,210)(31,67,36,62)(32,66,37,61)(33,65,38,70)(34,64,39,69)(35,63,40,68)(41,224,46,229)(42,223,47,228)(43,222,48,227)(44,221,49,226)(45,230,50,225)(51,234,56,239)(52,233,57,238)(53,232,58,237)(54,231,59,236)(55,240,60,235)(71,139,76,134)(72,138,77,133)(73,137,78,132)(74,136,79,131)(75,135,80,140)(81,149,86,144)(82,148,87,143)(83,147,88,142)(84,146,89,141)(85,145,90,150)(91,154,96,159)(92,153,97,158)(93,152,98,157)(94,151,99,156)(95,160,100,155)(101,164,106,169)(102,163,107,168)(103,162,108,167)(104,161,109,166)(105,170,110,165)(111,174,116,179)(112,173,117,178)(113,172,118,177)(114,171,119,176)(115,180,120,175)(121,182,126,187)(122,181,127,186)(123,190,128,185)(124,189,129,184)(125,188,130,183)>;
G:=Group( (1,116)(2,117)(3,118)(4,119)(5,120)(6,111)(7,112)(8,113)(9,114)(10,115)(11,82)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,81)(21,99)(22,100)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,170)(32,161)(33,162)(34,163)(35,164)(36,165)(37,166)(38,167)(39,168)(40,169)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,76)(52,77)(53,78)(54,79)(55,80)(56,71)(57,72)(58,73)(59,74)(60,75)(61,104)(62,105)(63,106)(64,107)(65,108)(66,109)(67,110)(68,101)(69,102)(70,103)(131,231)(132,232)(133,233)(134,234)(135,235)(136,236)(137,237)(138,238)(139,239)(140,240)(141,218)(142,219)(143,220)(144,211)(145,212)(146,213)(147,214)(148,215)(149,216)(150,217)(151,204)(152,205)(153,206)(154,207)(155,208)(156,209)(157,210)(158,201)(159,202)(160,203)(171,191)(172,192)(173,193)(174,194)(175,195)(176,196)(177,197)(178,198)(179,199)(180,200)(181,223)(182,224)(183,225)(184,226)(185,227)(186,228)(187,229)(188,230)(189,221)(190,222), (1,67,27,51,45,11)(2,68,28,52,46,12)(3,69,29,53,47,13)(4,70,30,54,48,14)(5,61,21,55,49,15)(6,62,22,56,50,16)(7,63,23,57,41,17)(8,64,24,58,42,18)(9,65,25,59,43,19)(10,66,26,60,44,20)(31,208,239,225,220,194)(32,209,240,226,211,195)(33,210,231,227,212,196)(34,201,232,228,213,197)(35,202,233,229,214,198)(36,203,234,230,215,199)(37,204,235,221,216,200)(38,205,236,222,217,191)(39,206,237,223,218,192)(40,207,238,224,219,193)(71,130,87,111,105,100)(72,121,88,112,106,91)(73,122,89,113,107,92)(74,123,90,114,108,93)(75,124,81,115,109,94)(76,125,82,116,110,95)(77,126,83,117,101,96)(78,127,84,118,102,97)(79,128,85,119,103,98)(80,129,86,120,104,99)(131,185,145,176,162,157)(132,186,146,177,163,158)(133,187,147,178,164,159)(134,188,148,179,165,160)(135,189,149,180,166,151)(136,190,150,171,167,152)(137,181,141,172,168,153)(138,182,142,173,169,154)(139,183,143,174,170,155)(140,184,144,175,161,156), (1,95)(2,96)(3,97)(4,98)(5,99)(6,100)(7,91)(8,92)(9,93)(10,94)(11,76)(12,77)(13,78)(14,79)(15,80)(16,71)(17,72)(18,73)(19,74)(20,75)(21,120)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,183)(32,184)(33,185)(34,186)(35,187)(36,188)(37,189)(38,190)(39,181)(40,182)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,81)(61,104)(62,105)(63,106)(64,107)(65,108)(66,109)(67,110)(68,101)(69,102)(70,103)(131,210)(132,201)(133,202)(134,203)(135,204)(136,205)(137,206)(138,207)(139,208)(140,209)(141,192)(142,193)(143,194)(144,195)(145,196)(146,197)(147,198)(148,199)(149,200)(150,191)(151,235)(152,236)(153,237)(154,238)(155,239)(156,240)(157,231)(158,232)(159,233)(160,234)(161,226)(162,227)(163,228)(164,229)(165,230)(166,221)(167,222)(168,223)(169,224)(170,225)(171,217)(172,218)(173,219)(174,220)(175,211)(176,212)(177,213)(178,214)(179,215)(180,216), (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), (1,199,6,194)(2,198,7,193)(3,197,8,192)(4,196,9,191)(5,195,10,200)(11,215,16,220)(12,214,17,219)(13,213,18,218)(14,212,19,217)(15,211,20,216)(21,209,26,204)(22,208,27,203)(23,207,28,202)(24,206,29,201)(25,205,30,210)(31,67,36,62)(32,66,37,61)(33,65,38,70)(34,64,39,69)(35,63,40,68)(41,224,46,229)(42,223,47,228)(43,222,48,227)(44,221,49,226)(45,230,50,225)(51,234,56,239)(52,233,57,238)(53,232,58,237)(54,231,59,236)(55,240,60,235)(71,139,76,134)(72,138,77,133)(73,137,78,132)(74,136,79,131)(75,135,80,140)(81,149,86,144)(82,148,87,143)(83,147,88,142)(84,146,89,141)(85,145,90,150)(91,154,96,159)(92,153,97,158)(93,152,98,157)(94,151,99,156)(95,160,100,155)(101,164,106,169)(102,163,107,168)(103,162,108,167)(104,161,109,166)(105,170,110,165)(111,174,116,179)(112,173,117,178)(113,172,118,177)(114,171,119,176)(115,180,120,175)(121,182,126,187)(122,181,127,186)(123,190,128,185)(124,189,129,184)(125,188,130,183) );
G=PermutationGroup([[(1,116),(2,117),(3,118),(4,119),(5,120),(6,111),(7,112),(8,113),(9,114),(10,115),(11,82),(12,83),(13,84),(14,85),(15,86),(16,87),(17,88),(18,89),(19,90),(20,81),(21,99),(22,100),(23,91),(24,92),(25,93),(26,94),(27,95),(28,96),(29,97),(30,98),(31,170),(32,161),(33,162),(34,163),(35,164),(36,165),(37,166),(38,167),(39,168),(40,169),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,76),(52,77),(53,78),(54,79),(55,80),(56,71),(57,72),(58,73),(59,74),(60,75),(61,104),(62,105),(63,106),(64,107),(65,108),(66,109),(67,110),(68,101),(69,102),(70,103),(131,231),(132,232),(133,233),(134,234),(135,235),(136,236),(137,237),(138,238),(139,239),(140,240),(141,218),(142,219),(143,220),(144,211),(145,212),(146,213),(147,214),(148,215),(149,216),(150,217),(151,204),(152,205),(153,206),(154,207),(155,208),(156,209),(157,210),(158,201),(159,202),(160,203),(171,191),(172,192),(173,193),(174,194),(175,195),(176,196),(177,197),(178,198),(179,199),(180,200),(181,223),(182,224),(183,225),(184,226),(185,227),(186,228),(187,229),(188,230),(189,221),(190,222)], [(1,67,27,51,45,11),(2,68,28,52,46,12),(3,69,29,53,47,13),(4,70,30,54,48,14),(5,61,21,55,49,15),(6,62,22,56,50,16),(7,63,23,57,41,17),(8,64,24,58,42,18),(9,65,25,59,43,19),(10,66,26,60,44,20),(31,208,239,225,220,194),(32,209,240,226,211,195),(33,210,231,227,212,196),(34,201,232,228,213,197),(35,202,233,229,214,198),(36,203,234,230,215,199),(37,204,235,221,216,200),(38,205,236,222,217,191),(39,206,237,223,218,192),(40,207,238,224,219,193),(71,130,87,111,105,100),(72,121,88,112,106,91),(73,122,89,113,107,92),(74,123,90,114,108,93),(75,124,81,115,109,94),(76,125,82,116,110,95),(77,126,83,117,101,96),(78,127,84,118,102,97),(79,128,85,119,103,98),(80,129,86,120,104,99),(131,185,145,176,162,157),(132,186,146,177,163,158),(133,187,147,178,164,159),(134,188,148,179,165,160),(135,189,149,180,166,151),(136,190,150,171,167,152),(137,181,141,172,168,153),(138,182,142,173,169,154),(139,183,143,174,170,155),(140,184,144,175,161,156)], [(1,95),(2,96),(3,97),(4,98),(5,99),(6,100),(7,91),(8,92),(9,93),(10,94),(11,76),(12,77),(13,78),(14,79),(15,80),(16,71),(17,72),(18,73),(19,74),(20,75),(21,120),(22,111),(23,112),(24,113),(25,114),(26,115),(27,116),(28,117),(29,118),(30,119),(31,183),(32,184),(33,185),(34,186),(35,187),(36,188),(37,189),(38,190),(39,181),(40,182),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,82),(52,83),(53,84),(54,85),(55,86),(56,87),(57,88),(58,89),(59,90),(60,81),(61,104),(62,105),(63,106),(64,107),(65,108),(66,109),(67,110),(68,101),(69,102),(70,103),(131,210),(132,201),(133,202),(134,203),(135,204),(136,205),(137,206),(138,207),(139,208),(140,209),(141,192),(142,193),(143,194),(144,195),(145,196),(146,197),(147,198),(148,199),(149,200),(150,191),(151,235),(152,236),(153,237),(154,238),(155,239),(156,240),(157,231),(158,232),(159,233),(160,234),(161,226),(162,227),(163,228),(164,229),(165,230),(166,221),(167,222),(168,223),(169,224),(170,225),(171,217),(172,218),(173,219),(174,220),(175,211),(176,212),(177,213),(178,214),(179,215),(180,216)], [(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)], [(1,199,6,194),(2,198,7,193),(3,197,8,192),(4,196,9,191),(5,195,10,200),(11,215,16,220),(12,214,17,219),(13,213,18,218),(14,212,19,217),(15,211,20,216),(21,209,26,204),(22,208,27,203),(23,207,28,202),(24,206,29,201),(25,205,30,210),(31,67,36,62),(32,66,37,61),(33,65,38,70),(34,64,39,69),(35,63,40,68),(41,224,46,229),(42,223,47,228),(43,222,48,227),(44,221,49,226),(45,230,50,225),(51,234,56,239),(52,233,57,238),(53,232,58,237),(54,231,59,236),(55,240,60,235),(71,139,76,134),(72,138,77,133),(73,137,78,132),(74,136,79,131),(75,135,80,140),(81,149,86,144),(82,148,87,143),(83,147,88,142),(84,146,89,141),(85,145,90,150),(91,154,96,159),(92,153,97,158),(93,152,98,157),(94,151,99,156),(95,160,100,155),(101,164,106,169),(102,163,107,168),(103,162,108,167),(104,161,109,166),(105,170,110,165),(111,174,116,179),(112,173,117,178),(113,172,118,177),(114,171,119,176),(115,180,120,175),(121,182,126,187),(122,181,127,186),(123,190,128,185),(124,189,129,184),(125,188,130,183)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10N | 10O | ··· | 10AD | 12A | ··· | 12H | 15A | 15B | 30A | ··· | 30N |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 2 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 10 | ··· | 10 | 4 | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 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 | S3 | D4 | D5 | D6 | D6 | Dic5 | D10 | D10 | C4×S3 | D12 | C3⋊D4 | C5⋊D4 | S3×D5 | S3×Dic5 | C15⋊D4 | C5⋊D12 | C2×S3×D5 |
| kernel | C2×D6⋊Dic5 | D6⋊Dic5 | C2×C6×Dic5 | C22×Dic15 | S3×C22×C10 | S3×C2×C10 | C22×Dic5 | C2×C30 | S3×C23 | C2×Dic5 | C22×C10 | C22×S3 | C22×S3 | C22×C6 | C2×C10 | C2×C10 | C2×C10 | C2×C6 | C23 | C22 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 4 | 2 | 2 | 1 | 8 | 4 | 2 | 4 | 4 | 4 | 16 | 2 | 4 | 4 | 4 | 2 |
Matrix representation of C2×D6⋊Dic5 ►in GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 60 | 1 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 37 | 1 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 41 | 0 |
| 0 | 0 | 0 | 29 | 3 |
| 50 | 0 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 38 | 7 |
| 0 | 0 | 0 | 55 | 23 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,60,60,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,60,0,0,0,0,0,0,60,37,0,0,0,0,1],[60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,41,29,0,0,0,0,3],[50,0,0,0,0,0,50,0,0,0,0,0,50,0,0,0,0,0,38,55,0,0,0,7,23] >;
C2×D6⋊Dic5 in GAP, Magma, Sage, TeX
C_2\times D_6\rtimes {\rm Dic}_5 % in TeX
G:=Group("C2xD6:Dic5"); // GroupNames label
G:=SmallGroup(480,614);
// by ID
G=gap.SmallGroup(480,614);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^2=d^10=1,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^-1>;
// generators/relations