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## G = C2×Dic6⋊D5order 480 = 25·3·5

### Direct product of C2 and Dic6⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C2×Dic6⋊D5
 Chief series C1 — C5 — C15 — C30 — C60 — C3×C5⋊2C8 — Dic6⋊D5 — C2×Dic6⋊D5
 Lower central C15 — C30 — C60 — C2×Dic6⋊D5
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×Dic6⋊D5
G = < a,b,c,d,e | a2=b12=d5=e2=1, c2=b6, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b3c, ede=d-1 >

Subgroups: 924 in 136 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, D5 [×2], C10, C10 [×2], Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, C20 [×2], C20 [×2], D10 [×4], C2×C10, C24 [×2], Dic6 [×2], Dic6, D12 [×3], C2×Dic3, C2×C12, C22×S3, D15 [×2], C30, C30 [×2], C2×SD16, C52C8 [×2], D20 [×3], C2×C20, C2×C20, C5×Q8 [×3], C22×D5, C24⋊C2 [×4], C2×C24, C2×Dic6, C2×D12, C5×Dic3 [×2], C60 [×2], D30 [×4], C2×C30, C2×C52C8, Q8⋊D5 [×4], C2×D20, Q8×C10, C2×C24⋊C2, C3×C52C8 [×2], C5×Dic6 [×2], C5×Dic6, C10×Dic3, D60 [×2], D60, C2×C60, C22×D15, C2×Q8⋊D5, Dic6⋊D5 [×4], C6×C52C8, C10×Dic6, C2×D60, C2×Dic6⋊D5
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], SD16 [×2], C2×D4, D10 [×3], D12 [×2], C22×S3, C2×SD16, C5⋊D4 [×2], C22×D5, C24⋊C2 [×2], C2×D12, S3×D5, Q8⋊D5 [×2], C2×C5⋊D4, C2×C24⋊C2, C5⋊D12 [×2], C2×S3×D5, C2×Q8⋊D5, Dic6⋊D5 [×2], C2×C5⋊D12, C2×Dic6⋊D5

Smallest permutation representation of C2×Dic6⋊D5
On 240 points
Generators in S240
(1 148)(2 149)(3 150)(4 151)(5 152)(6 153)(7 154)(8 155)(9 156)(10 145)(11 146)(12 147)(13 224)(14 225)(15 226)(16 227)(17 228)(18 217)(19 218)(20 219)(21 220)(22 221)(23 222)(24 223)(25 214)(26 215)(27 216)(28 205)(29 206)(30 207)(31 208)(32 209)(33 210)(34 211)(35 212)(36 213)(37 196)(38 197)(39 198)(40 199)(41 200)(42 201)(43 202)(44 203)(45 204)(46 193)(47 194)(48 195)(49 188)(50 189)(51 190)(52 191)(53 192)(54 181)(55 182)(56 183)(57 184)(58 185)(59 186)(60 187)(61 143)(62 144)(63 133)(64 134)(65 135)(66 136)(67 137)(68 138)(69 139)(70 140)(71 141)(72 142)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 129)(80 130)(81 131)(82 132)(83 121)(84 122)(85 162)(86 163)(87 164)(88 165)(89 166)(90 167)(91 168)(92 157)(93 158)(94 159)(95 160)(96 161)(97 110)(98 111)(99 112)(100 113)(101 114)(102 115)(103 116)(104 117)(105 118)(106 119)(107 120)(108 109)(169 229)(170 230)(171 231)(172 232)(173 233)(174 234)(175 235)(176 236)(177 237)(178 238)(179 239)(180 240)
(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)
(1 216 7 210)(2 215 8 209)(3 214 9 208)(4 213 10 207)(5 212 11 206)(6 211 12 205)(13 168 19 162)(14 167 20 161)(15 166 21 160)(16 165 22 159)(17 164 23 158)(18 163 24 157)(25 156 31 150)(26 155 32 149)(27 154 33 148)(28 153 34 147)(29 152 35 146)(30 151 36 145)(37 144 43 138)(38 143 44 137)(39 142 45 136)(40 141 46 135)(41 140 47 134)(42 139 48 133)(49 132 55 126)(50 131 56 125)(51 130 57 124)(52 129 58 123)(53 128 59 122)(54 127 60 121)(61 203 67 197)(62 202 68 196)(63 201 69 195)(64 200 70 194)(65 199 71 193)(66 198 72 204)(73 191 79 185)(74 190 80 184)(75 189 81 183)(76 188 82 182)(77 187 83 181)(78 186 84 192)(85 224 91 218)(86 223 92 217)(87 222 93 228)(88 221 94 227)(89 220 95 226)(90 219 96 225)(97 172 103 178)(98 171 104 177)(99 170 105 176)(100 169 106 175)(101 180 107 174)(102 179 108 173)(109 233 115 239)(110 232 116 238)(111 231 117 237)(112 230 118 236)(113 229 119 235)(114 240 120 234)
(1 99 86 72 75)(2 100 87 61 76)(3 101 88 62 77)(4 102 89 63 78)(5 103 90 64 79)(6 104 91 65 80)(7 105 92 66 81)(8 106 93 67 82)(9 107 94 68 83)(10 108 95 69 84)(11 97 96 70 73)(12 98 85 71 74)(13 46 51 28 231)(14 47 52 29 232)(15 48 53 30 233)(16 37 54 31 234)(17 38 55 32 235)(18 39 56 33 236)(19 40 57 34 237)(20 41 58 35 238)(21 42 59 36 239)(22 43 60 25 240)(23 44 49 26 229)(24 45 50 27 230)(109 160 139 122 145)(110 161 140 123 146)(111 162 141 124 147)(112 163 142 125 148)(113 164 143 126 149)(114 165 144 127 150)(115 166 133 128 151)(116 167 134 129 152)(117 168 135 130 153)(118 157 136 131 154)(119 158 137 132 155)(120 159 138 121 156)(169 222 203 188 215)(170 223 204 189 216)(171 224 193 190 205)(172 225 194 191 206)(173 226 195 192 207)(174 227 196 181 208)(175 228 197 182 209)(176 217 198 183 210)(177 218 199 184 211)(178 219 200 185 212)(179 220 201 186 213)(180 221 202 187 214)
(1 75)(2 74)(3 73)(4 84)(5 83)(6 82)(7 81)(8 80)(9 79)(10 78)(11 77)(12 76)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)(25 49)(26 60)(27 59)(28 58)(29 57)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 235)(38 234)(39 233)(40 232)(41 231)(42 230)(43 229)(44 240)(45 239)(46 238)(47 237)(48 236)(61 98)(62 97)(63 108)(64 107)(65 106)(66 105)(67 104)(68 103)(69 102)(70 101)(71 100)(72 99)(85 87)(88 96)(89 95)(90 94)(91 93)(109 133)(110 144)(111 143)(112 142)(113 141)(114 140)(115 139)(116 138)(117 137)(118 136)(119 135)(120 134)(121 152)(122 151)(123 150)(124 149)(125 148)(126 147)(127 146)(128 145)(129 156)(130 155)(131 154)(132 153)(158 168)(159 167)(160 166)(161 165)(162 164)(169 202)(170 201)(171 200)(172 199)(173 198)(174 197)(175 196)(176 195)(177 194)(178 193)(179 204)(180 203)(181 209)(182 208)(183 207)(184 206)(185 205)(186 216)(187 215)(188 214)(189 213)(190 212)(191 211)(192 210)(217 226)(218 225)(219 224)(220 223)(221 222)(227 228)

G:=sub<Sym(240)| (1,148)(2,149)(3,150)(4,151)(5,152)(6,153)(7,154)(8,155)(9,156)(10,145)(11,146)(12,147)(13,224)(14,225)(15,226)(16,227)(17,228)(18,217)(19,218)(20,219)(21,220)(22,221)(23,222)(24,223)(25,214)(26,215)(27,216)(28,205)(29,206)(30,207)(31,208)(32,209)(33,210)(34,211)(35,212)(36,213)(37,196)(38,197)(39,198)(40,199)(41,200)(42,201)(43,202)(44,203)(45,204)(46,193)(47,194)(48,195)(49,188)(50,189)(51,190)(52,191)(53,192)(54,181)(55,182)(56,183)(57,184)(58,185)(59,186)(60,187)(61,143)(62,144)(63,133)(64,134)(65,135)(66,136)(67,137)(68,138)(69,139)(70,140)(71,141)(72,142)(73,123)(74,124)(75,125)(76,126)(77,127)(78,128)(79,129)(80,130)(81,131)(82,132)(83,121)(84,122)(85,162)(86,163)(87,164)(88,165)(89,166)(90,167)(91,168)(92,157)(93,158)(94,159)(95,160)(96,161)(97,110)(98,111)(99,112)(100,113)(101,114)(102,115)(103,116)(104,117)(105,118)(106,119)(107,120)(108,109)(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)(177,237)(178,238)(179,239)(180,240), (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), (1,216,7,210)(2,215,8,209)(3,214,9,208)(4,213,10,207)(5,212,11,206)(6,211,12,205)(13,168,19,162)(14,167,20,161)(15,166,21,160)(16,165,22,159)(17,164,23,158)(18,163,24,157)(25,156,31,150)(26,155,32,149)(27,154,33,148)(28,153,34,147)(29,152,35,146)(30,151,36,145)(37,144,43,138)(38,143,44,137)(39,142,45,136)(40,141,46,135)(41,140,47,134)(42,139,48,133)(49,132,55,126)(50,131,56,125)(51,130,57,124)(52,129,58,123)(53,128,59,122)(54,127,60,121)(61,203,67,197)(62,202,68,196)(63,201,69,195)(64,200,70,194)(65,199,71,193)(66,198,72,204)(73,191,79,185)(74,190,80,184)(75,189,81,183)(76,188,82,182)(77,187,83,181)(78,186,84,192)(85,224,91,218)(86,223,92,217)(87,222,93,228)(88,221,94,227)(89,220,95,226)(90,219,96,225)(97,172,103,178)(98,171,104,177)(99,170,105,176)(100,169,106,175)(101,180,107,174)(102,179,108,173)(109,233,115,239)(110,232,116,238)(111,231,117,237)(112,230,118,236)(113,229,119,235)(114,240,120,234), (1,99,86,72,75)(2,100,87,61,76)(3,101,88,62,77)(4,102,89,63,78)(5,103,90,64,79)(6,104,91,65,80)(7,105,92,66,81)(8,106,93,67,82)(9,107,94,68,83)(10,108,95,69,84)(11,97,96,70,73)(12,98,85,71,74)(13,46,51,28,231)(14,47,52,29,232)(15,48,53,30,233)(16,37,54,31,234)(17,38,55,32,235)(18,39,56,33,236)(19,40,57,34,237)(20,41,58,35,238)(21,42,59,36,239)(22,43,60,25,240)(23,44,49,26,229)(24,45,50,27,230)(109,160,139,122,145)(110,161,140,123,146)(111,162,141,124,147)(112,163,142,125,148)(113,164,143,126,149)(114,165,144,127,150)(115,166,133,128,151)(116,167,134,129,152)(117,168,135,130,153)(118,157,136,131,154)(119,158,137,132,155)(120,159,138,121,156)(169,222,203,188,215)(170,223,204,189,216)(171,224,193,190,205)(172,225,194,191,206)(173,226,195,192,207)(174,227,196,181,208)(175,228,197,182,209)(176,217,198,183,210)(177,218,199,184,211)(178,219,200,185,212)(179,220,201,186,213)(180,221,202,187,214), (1,75)(2,74)(3,73)(4,84)(5,83)(6,82)(7,81)(8,80)(9,79)(10,78)(11,77)(12,76)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,49)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,235)(38,234)(39,233)(40,232)(41,231)(42,230)(43,229)(44,240)(45,239)(46,238)(47,237)(48,236)(61,98)(62,97)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(85,87)(88,96)(89,95)(90,94)(91,93)(109,133)(110,144)(111,143)(112,142)(113,141)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134)(121,152)(122,151)(123,150)(124,149)(125,148)(126,147)(127,146)(128,145)(129,156)(130,155)(131,154)(132,153)(158,168)(159,167)(160,166)(161,165)(162,164)(169,202)(170,201)(171,200)(172,199)(173,198)(174,197)(175,196)(176,195)(177,194)(178,193)(179,204)(180,203)(181,209)(182,208)(183,207)(184,206)(185,205)(186,216)(187,215)(188,214)(189,213)(190,212)(191,211)(192,210)(217,226)(218,225)(219,224)(220,223)(221,222)(227,228)>;

G:=Group( (1,148)(2,149)(3,150)(4,151)(5,152)(6,153)(7,154)(8,155)(9,156)(10,145)(11,146)(12,147)(13,224)(14,225)(15,226)(16,227)(17,228)(18,217)(19,218)(20,219)(21,220)(22,221)(23,222)(24,223)(25,214)(26,215)(27,216)(28,205)(29,206)(30,207)(31,208)(32,209)(33,210)(34,211)(35,212)(36,213)(37,196)(38,197)(39,198)(40,199)(41,200)(42,201)(43,202)(44,203)(45,204)(46,193)(47,194)(48,195)(49,188)(50,189)(51,190)(52,191)(53,192)(54,181)(55,182)(56,183)(57,184)(58,185)(59,186)(60,187)(61,143)(62,144)(63,133)(64,134)(65,135)(66,136)(67,137)(68,138)(69,139)(70,140)(71,141)(72,142)(73,123)(74,124)(75,125)(76,126)(77,127)(78,128)(79,129)(80,130)(81,131)(82,132)(83,121)(84,122)(85,162)(86,163)(87,164)(88,165)(89,166)(90,167)(91,168)(92,157)(93,158)(94,159)(95,160)(96,161)(97,110)(98,111)(99,112)(100,113)(101,114)(102,115)(103,116)(104,117)(105,118)(106,119)(107,120)(108,109)(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)(177,237)(178,238)(179,239)(180,240), (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), (1,216,7,210)(2,215,8,209)(3,214,9,208)(4,213,10,207)(5,212,11,206)(6,211,12,205)(13,168,19,162)(14,167,20,161)(15,166,21,160)(16,165,22,159)(17,164,23,158)(18,163,24,157)(25,156,31,150)(26,155,32,149)(27,154,33,148)(28,153,34,147)(29,152,35,146)(30,151,36,145)(37,144,43,138)(38,143,44,137)(39,142,45,136)(40,141,46,135)(41,140,47,134)(42,139,48,133)(49,132,55,126)(50,131,56,125)(51,130,57,124)(52,129,58,123)(53,128,59,122)(54,127,60,121)(61,203,67,197)(62,202,68,196)(63,201,69,195)(64,200,70,194)(65,199,71,193)(66,198,72,204)(73,191,79,185)(74,190,80,184)(75,189,81,183)(76,188,82,182)(77,187,83,181)(78,186,84,192)(85,224,91,218)(86,223,92,217)(87,222,93,228)(88,221,94,227)(89,220,95,226)(90,219,96,225)(97,172,103,178)(98,171,104,177)(99,170,105,176)(100,169,106,175)(101,180,107,174)(102,179,108,173)(109,233,115,239)(110,232,116,238)(111,231,117,237)(112,230,118,236)(113,229,119,235)(114,240,120,234), (1,99,86,72,75)(2,100,87,61,76)(3,101,88,62,77)(4,102,89,63,78)(5,103,90,64,79)(6,104,91,65,80)(7,105,92,66,81)(8,106,93,67,82)(9,107,94,68,83)(10,108,95,69,84)(11,97,96,70,73)(12,98,85,71,74)(13,46,51,28,231)(14,47,52,29,232)(15,48,53,30,233)(16,37,54,31,234)(17,38,55,32,235)(18,39,56,33,236)(19,40,57,34,237)(20,41,58,35,238)(21,42,59,36,239)(22,43,60,25,240)(23,44,49,26,229)(24,45,50,27,230)(109,160,139,122,145)(110,161,140,123,146)(111,162,141,124,147)(112,163,142,125,148)(113,164,143,126,149)(114,165,144,127,150)(115,166,133,128,151)(116,167,134,129,152)(117,168,135,130,153)(118,157,136,131,154)(119,158,137,132,155)(120,159,138,121,156)(169,222,203,188,215)(170,223,204,189,216)(171,224,193,190,205)(172,225,194,191,206)(173,226,195,192,207)(174,227,196,181,208)(175,228,197,182,209)(176,217,198,183,210)(177,218,199,184,211)(178,219,200,185,212)(179,220,201,186,213)(180,221,202,187,214), (1,75)(2,74)(3,73)(4,84)(5,83)(6,82)(7,81)(8,80)(9,79)(10,78)(11,77)(12,76)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,49)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,235)(38,234)(39,233)(40,232)(41,231)(42,230)(43,229)(44,240)(45,239)(46,238)(47,237)(48,236)(61,98)(62,97)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(85,87)(88,96)(89,95)(90,94)(91,93)(109,133)(110,144)(111,143)(112,142)(113,141)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134)(121,152)(122,151)(123,150)(124,149)(125,148)(126,147)(127,146)(128,145)(129,156)(130,155)(131,154)(132,153)(158,168)(159,167)(160,166)(161,165)(162,164)(169,202)(170,201)(171,200)(172,199)(173,198)(174,197)(175,196)(176,195)(177,194)(178,193)(179,204)(180,203)(181,209)(182,208)(183,207)(184,206)(185,205)(186,216)(187,215)(188,214)(189,213)(190,212)(191,211)(192,210)(217,226)(218,225)(219,224)(220,223)(221,222)(227,228) );

G=PermutationGroup([(1,148),(2,149),(3,150),(4,151),(5,152),(6,153),(7,154),(8,155),(9,156),(10,145),(11,146),(12,147),(13,224),(14,225),(15,226),(16,227),(17,228),(18,217),(19,218),(20,219),(21,220),(22,221),(23,222),(24,223),(25,214),(26,215),(27,216),(28,205),(29,206),(30,207),(31,208),(32,209),(33,210),(34,211),(35,212),(36,213),(37,196),(38,197),(39,198),(40,199),(41,200),(42,201),(43,202),(44,203),(45,204),(46,193),(47,194),(48,195),(49,188),(50,189),(51,190),(52,191),(53,192),(54,181),(55,182),(56,183),(57,184),(58,185),(59,186),(60,187),(61,143),(62,144),(63,133),(64,134),(65,135),(66,136),(67,137),(68,138),(69,139),(70,140),(71,141),(72,142),(73,123),(74,124),(75,125),(76,126),(77,127),(78,128),(79,129),(80,130),(81,131),(82,132),(83,121),(84,122),(85,162),(86,163),(87,164),(88,165),(89,166),(90,167),(91,168),(92,157),(93,158),(94,159),(95,160),(96,161),(97,110),(98,111),(99,112),(100,113),(101,114),(102,115),(103,116),(104,117),(105,118),(106,119),(107,120),(108,109),(169,229),(170,230),(171,231),(172,232),(173,233),(174,234),(175,235),(176,236),(177,237),(178,238),(179,239),(180,240)], [(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)], [(1,216,7,210),(2,215,8,209),(3,214,9,208),(4,213,10,207),(5,212,11,206),(6,211,12,205),(13,168,19,162),(14,167,20,161),(15,166,21,160),(16,165,22,159),(17,164,23,158),(18,163,24,157),(25,156,31,150),(26,155,32,149),(27,154,33,148),(28,153,34,147),(29,152,35,146),(30,151,36,145),(37,144,43,138),(38,143,44,137),(39,142,45,136),(40,141,46,135),(41,140,47,134),(42,139,48,133),(49,132,55,126),(50,131,56,125),(51,130,57,124),(52,129,58,123),(53,128,59,122),(54,127,60,121),(61,203,67,197),(62,202,68,196),(63,201,69,195),(64,200,70,194),(65,199,71,193),(66,198,72,204),(73,191,79,185),(74,190,80,184),(75,189,81,183),(76,188,82,182),(77,187,83,181),(78,186,84,192),(85,224,91,218),(86,223,92,217),(87,222,93,228),(88,221,94,227),(89,220,95,226),(90,219,96,225),(97,172,103,178),(98,171,104,177),(99,170,105,176),(100,169,106,175),(101,180,107,174),(102,179,108,173),(109,233,115,239),(110,232,116,238),(111,231,117,237),(112,230,118,236),(113,229,119,235),(114,240,120,234)], [(1,99,86,72,75),(2,100,87,61,76),(3,101,88,62,77),(4,102,89,63,78),(5,103,90,64,79),(6,104,91,65,80),(7,105,92,66,81),(8,106,93,67,82),(9,107,94,68,83),(10,108,95,69,84),(11,97,96,70,73),(12,98,85,71,74),(13,46,51,28,231),(14,47,52,29,232),(15,48,53,30,233),(16,37,54,31,234),(17,38,55,32,235),(18,39,56,33,236),(19,40,57,34,237),(20,41,58,35,238),(21,42,59,36,239),(22,43,60,25,240),(23,44,49,26,229),(24,45,50,27,230),(109,160,139,122,145),(110,161,140,123,146),(111,162,141,124,147),(112,163,142,125,148),(113,164,143,126,149),(114,165,144,127,150),(115,166,133,128,151),(116,167,134,129,152),(117,168,135,130,153),(118,157,136,131,154),(119,158,137,132,155),(120,159,138,121,156),(169,222,203,188,215),(170,223,204,189,216),(171,224,193,190,205),(172,225,194,191,206),(173,226,195,192,207),(174,227,196,181,208),(175,228,197,182,209),(176,217,198,183,210),(177,218,199,184,211),(178,219,200,185,212),(179,220,201,186,213),(180,221,202,187,214)], [(1,75),(2,74),(3,73),(4,84),(5,83),(6,82),(7,81),(8,80),(9,79),(10,78),(11,77),(12,76),(13,20),(14,19),(15,18),(16,17),(21,24),(22,23),(25,49),(26,60),(27,59),(28,58),(29,57),(30,56),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,235),(38,234),(39,233),(40,232),(41,231),(42,230),(43,229),(44,240),(45,239),(46,238),(47,237),(48,236),(61,98),(62,97),(63,108),(64,107),(65,106),(66,105),(67,104),(68,103),(69,102),(70,101),(71,100),(72,99),(85,87),(88,96),(89,95),(90,94),(91,93),(109,133),(110,144),(111,143),(112,142),(113,141),(114,140),(115,139),(116,138),(117,137),(118,136),(119,135),(120,134),(121,152),(122,151),(123,150),(124,149),(125,148),(126,147),(127,146),(128,145),(129,156),(130,155),(131,154),(132,153),(158,168),(159,167),(160,166),(161,165),(162,164),(169,202),(170,201),(171,200),(172,199),(173,198),(174,197),(175,196),(176,195),(177,194),(178,193),(179,204),(180,203),(181,209),(182,208),(183,207),(184,206),(185,205),(186,216),(187,215),(188,214),(189,213),(190,212),(191,211),(192,210),(217,226),(218,225),(219,224),(220,223),(221,222),(227,228)])

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 15A 15B 20A 20B 20C 20D 20E ··· 20L 24A ··· 24H 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 15 15 20 20 20 20 20 ··· 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 1 1 60 60 2 2 2 12 12 2 2 2 2 2 10 10 10 10 2 ··· 2 2 2 2 2 4 4 4 4 4 4 12 ··· 12 10 ··· 10 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 SD16 D10 D10 D12 D12 C5⋊D4 C5⋊D4 C24⋊C2 S3×D5 Q8⋊D5 C5⋊D12 C2×S3×D5 C5⋊D12 Dic6⋊D5 kernel C2×Dic6⋊D5 Dic6⋊D5 C6×C5⋊2C8 C10×Dic6 C2×D60 C2×C5⋊2C8 C60 C2×C30 C2×Dic6 C5⋊2C8 C2×C20 C30 Dic6 C2×C12 C20 C2×C10 C12 C2×C6 C10 C2×C4 C6 C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 1 2 2 1 4 4 2 2 2 4 4 8 2 4 2 2 2 8

Matrix representation of C2×Dic6⋊D5 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 239 49 0 0 0 0 177 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 103 5 0 0 0 0 47 138
,
 240 0 0 0 0 0 177 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 193 146 0 0 0 0 75 48
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 51 240 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 64 240 0 0 0 0 0 0 51 240 0 0 0 0 190 190 0 0 0 0 0 0 1 0 0 0 0 0 7 240

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[239,177,0,0,0,0,49,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,103,47,0,0,0,0,5,138],[240,177,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,193,75,0,0,0,0,146,48],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,51,1,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,64,0,0,0,0,0,240,0,0,0,0,0,0,51,190,0,0,0,0,240,190,0,0,0,0,0,0,1,7,0,0,0,0,0,240] >;

C2×Dic6⋊D5 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_6\rtimes D_5
% in TeX

G:=Group("C2xDic6:D5");
// GroupNames label

G:=SmallGroup(480,393);
// by ID

G=gap.SmallGroup(480,393);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,365,120,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^12=d^5=e^2=1,c^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b^3*c,e*d*e=d^-1>;
// generators/relations

׿
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𝔽