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G = C2×D6⋊Dic5order 480 = 25·3·5

Direct product of C2 and D6⋊Dic5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D6⋊Dic5, C104(D6⋊C4), D65(C2×Dic5), (C2×C30).69D4, C305(C22⋊C4), (C2×Dic5)⋊19D6, C30.218(C2×D4), (C2×C10).44D12, C10.64(C2×D12), C61(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, C55(C2×D6⋊C4), (S3×C2×C10)⋊7C4, C159(C2×C22⋊C4), (C2×C6×Dic5)⋊1C2, C32(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

C1C30 — C2×D6⋊Dic5
C1C5C15C30C2×C30C6×Dic5D6⋊Dic5 — C2×D6⋊Dic5
C15C30 — C2×D6⋊Dic5
C1C23

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 [×3], C2 [×4], C2 [×4], C3, C4 [×4], C22, C22 [×6], C22 [×16], C5, S3 [×4], C6 [×3], C6 [×4], C2×C4 [×8], C23, C23 [×10], C10 [×3], C10 [×4], C10 [×4], Dic3 [×2], C12 [×2], D6 [×4], D6 [×12], C2×C6, C2×C6 [×6], C15, C22⋊C4 [×4], C22×C4 [×2], C24, Dic5 [×4], C2×C10, C2×C10 [×6], C2×C10 [×16], C2×Dic3 [×4], C2×C12 [×4], C22×S3 [×6], C22×S3 [×4], C22×C6, C5×S3 [×4], C30 [×3], C30 [×4], C2×C22⋊C4, C2×Dic5 [×2], C2×Dic5 [×6], C22×C10, C22×C10 [×10], D6⋊C4 [×4], C22×Dic3, C22×C12, S3×C23, C3×Dic5 [×2], Dic15 [×2], S3×C10 [×4], S3×C10 [×12], C2×C30, C2×C30 [×6], C23.D5 [×4], C22×Dic5, C22×Dic5, C23×C10, C2×D6⋊C4, C6×Dic5 [×2], C6×Dic5 [×2], C2×Dic15 [×2], C2×Dic15 [×2], S3×C2×C10 [×6], S3×C2×C10 [×4], C22×C30, C2×C23.D5, D6⋊Dic5 [×4], C2×C6×Dic5, C22×Dic15, S3×C22×C10, C2×D6⋊Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D5, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, S3×D5, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C2×D6⋊C4, S3×Dic5 [×2], C15⋊D4 [×2], C5⋊D12 [×2], C2×S3×D5, C2×C23.D5, D6⋊Dic5 [×4], C2×S3×Dic5, C2×C15⋊D4, C2×C5⋊D12, C2×D6⋊Dic5

Smallest permutation representation of 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 90)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 99)(30 100)(31 168)(32 169)(33 170)(34 161)(35 162)(36 163)(37 164)(38 165)(39 166)(40 167)(41 129)(42 130)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 127)(50 128)(51 78)(52 79)(53 80)(54 71)(55 72)(56 73)(57 74)(58 75)(59 76)(60 77)(61 106)(62 107)(63 108)(64 109)(65 110)(66 101)(67 102)(68 103)(69 104)(70 105)(131 233)(132 234)(133 235)(134 236)(135 237)(136 238)(137 239)(138 240)(139 231)(140 232)(141 216)(142 217)(143 218)(144 219)(145 220)(146 211)(147 212)(148 213)(149 214)(150 215)(151 206)(152 207)(153 208)(154 209)(155 210)(156 201)(157 202)(158 203)(159 204)(160 205)(171 199)(172 200)(173 191)(174 192)(175 193)(176 194)(177 195)(178 196)(179 197)(180 198)(181 221)(182 222)(183 223)(184 224)(185 225)(186 226)(187 227)(188 228)(189 229)(190 230)
(1 67 27 51 47 13)(2 68 28 52 48 14)(3 69 29 53 49 15)(4 70 30 54 50 16)(5 61 21 55 41 17)(6 62 22 56 42 18)(7 63 23 57 43 19)(8 64 24 58 44 20)(9 65 25 59 45 11)(10 66 26 60 46 12)(31 208 239 223 218 192)(32 209 240 224 219 193)(33 210 231 225 220 194)(34 201 232 226 211 195)(35 202 233 227 212 196)(36 203 234 228 213 197)(37 204 235 229 214 198)(38 205 236 230 215 199)(39 206 237 221 216 200)(40 207 238 222 217 191)(71 128 85 119 105 100)(72 129 86 120 106 91)(73 130 87 111 107 92)(74 121 88 112 108 93)(75 122 89 113 109 94)(76 123 90 114 110 95)(77 124 81 115 101 96)(78 125 82 116 102 97)(79 126 83 117 103 98)(80 127 84 118 104 99)(131 187 147 178 162 157)(132 188 148 179 163 158)(133 189 149 180 164 159)(134 190 150 171 165 160)(135 181 141 172 166 151)(136 182 142 173 167 152)(137 183 143 174 168 153)(138 184 144 175 169 154)(139 185 145 176 170 155)(140 186 146 177 161 156)
(1 97)(2 98)(3 99)(4 100)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(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)(41 129)(42 130)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 127)(50 128)(51 82)(52 83)(53 84)(54 85)(55 86)(56 87)(57 88)(58 89)(59 90)(60 81)(61 106)(62 107)(63 108)(64 109)(65 110)(66 101)(67 102)(68 103)(69 104)(70 105)(131 202)(132 203)(133 204)(134 205)(135 206)(136 207)(137 208)(138 209)(139 210)(140 201)(141 200)(142 191)(143 192)(144 193)(145 194)(146 195)(147 196)(148 197)(149 198)(150 199)(151 237)(152 238)(153 239)(154 240)(155 231)(156 232)(157 233)(158 234)(159 235)(160 236)(161 226)(162 227)(163 228)(164 229)(165 230)(166 221)(167 222)(168 223)(169 224)(170 225)(171 215)(172 216)(173 217)(174 218)(175 219)(176 220)(177 211)(178 212)(179 213)(180 214)
(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)
(1 197 6 192)(2 196 7 191)(3 195 8 200)(4 194 9 199)(5 193 10 198)(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)(11,90)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,98)(29,99)(30,100)(31,168)(32,169)(33,170)(34,161)(35,162)(36,163)(37,164)(38,165)(39,166)(40,167)(41,129)(42,130)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,127)(50,128)(51,78)(52,79)(53,80)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77)(61,106)(62,107)(63,108)(64,109)(65,110)(66,101)(67,102)(68,103)(69,104)(70,105)(131,233)(132,234)(133,235)(134,236)(135,237)(136,238)(137,239)(138,240)(139,231)(140,232)(141,216)(142,217)(143,218)(144,219)(145,220)(146,211)(147,212)(148,213)(149,214)(150,215)(151,206)(152,207)(153,208)(154,209)(155,210)(156,201)(157,202)(158,203)(159,204)(160,205)(171,199)(172,200)(173,191)(174,192)(175,193)(176,194)(177,195)(178,196)(179,197)(180,198)(181,221)(182,222)(183,223)(184,224)(185,225)(186,226)(187,227)(188,228)(189,229)(190,230), (1,67,27,51,47,13)(2,68,28,52,48,14)(3,69,29,53,49,15)(4,70,30,54,50,16)(5,61,21,55,41,17)(6,62,22,56,42,18)(7,63,23,57,43,19)(8,64,24,58,44,20)(9,65,25,59,45,11)(10,66,26,60,46,12)(31,208,239,223,218,192)(32,209,240,224,219,193)(33,210,231,225,220,194)(34,201,232,226,211,195)(35,202,233,227,212,196)(36,203,234,228,213,197)(37,204,235,229,214,198)(38,205,236,230,215,199)(39,206,237,221,216,200)(40,207,238,222,217,191)(71,128,85,119,105,100)(72,129,86,120,106,91)(73,130,87,111,107,92)(74,121,88,112,108,93)(75,122,89,113,109,94)(76,123,90,114,110,95)(77,124,81,115,101,96)(78,125,82,116,102,97)(79,126,83,117,103,98)(80,127,84,118,104,99)(131,187,147,178,162,157)(132,188,148,179,163,158)(133,189,149,180,164,159)(134,190,150,171,165,160)(135,181,141,172,166,151)(136,182,142,173,167,152)(137,183,143,174,168,153)(138,184,144,175,169,154)(139,185,145,176,170,155)(140,186,146,177,161,156), (1,97)(2,98)(3,99)(4,100)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(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)(41,129)(42,130)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,127)(50,128)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,81)(61,106)(62,107)(63,108)(64,109)(65,110)(66,101)(67,102)(68,103)(69,104)(70,105)(131,202)(132,203)(133,204)(134,205)(135,206)(136,207)(137,208)(138,209)(139,210)(140,201)(141,200)(142,191)(143,192)(144,193)(145,194)(146,195)(147,196)(148,197)(149,198)(150,199)(151,237)(152,238)(153,239)(154,240)(155,231)(156,232)(157,233)(158,234)(159,235)(160,236)(161,226)(162,227)(163,228)(164,229)(165,230)(166,221)(167,222)(168,223)(169,224)(170,225)(171,215)(172,216)(173,217)(174,218)(175,219)(176,220)(177,211)(178,212)(179,213)(180,214), (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), (1,197,6,192)(2,196,7,191)(3,195,8,200)(4,194,9,199)(5,193,10,198)(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)(11,90)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,98)(29,99)(30,100)(31,168)(32,169)(33,170)(34,161)(35,162)(36,163)(37,164)(38,165)(39,166)(40,167)(41,129)(42,130)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,127)(50,128)(51,78)(52,79)(53,80)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77)(61,106)(62,107)(63,108)(64,109)(65,110)(66,101)(67,102)(68,103)(69,104)(70,105)(131,233)(132,234)(133,235)(134,236)(135,237)(136,238)(137,239)(138,240)(139,231)(140,232)(141,216)(142,217)(143,218)(144,219)(145,220)(146,211)(147,212)(148,213)(149,214)(150,215)(151,206)(152,207)(153,208)(154,209)(155,210)(156,201)(157,202)(158,203)(159,204)(160,205)(171,199)(172,200)(173,191)(174,192)(175,193)(176,194)(177,195)(178,196)(179,197)(180,198)(181,221)(182,222)(183,223)(184,224)(185,225)(186,226)(187,227)(188,228)(189,229)(190,230), (1,67,27,51,47,13)(2,68,28,52,48,14)(3,69,29,53,49,15)(4,70,30,54,50,16)(5,61,21,55,41,17)(6,62,22,56,42,18)(7,63,23,57,43,19)(8,64,24,58,44,20)(9,65,25,59,45,11)(10,66,26,60,46,12)(31,208,239,223,218,192)(32,209,240,224,219,193)(33,210,231,225,220,194)(34,201,232,226,211,195)(35,202,233,227,212,196)(36,203,234,228,213,197)(37,204,235,229,214,198)(38,205,236,230,215,199)(39,206,237,221,216,200)(40,207,238,222,217,191)(71,128,85,119,105,100)(72,129,86,120,106,91)(73,130,87,111,107,92)(74,121,88,112,108,93)(75,122,89,113,109,94)(76,123,90,114,110,95)(77,124,81,115,101,96)(78,125,82,116,102,97)(79,126,83,117,103,98)(80,127,84,118,104,99)(131,187,147,178,162,157)(132,188,148,179,163,158)(133,189,149,180,164,159)(134,190,150,171,165,160)(135,181,141,172,166,151)(136,182,142,173,167,152)(137,183,143,174,168,153)(138,184,144,175,169,154)(139,185,145,176,170,155)(140,186,146,177,161,156), (1,97)(2,98)(3,99)(4,100)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(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)(41,129)(42,130)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,127)(50,128)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,81)(61,106)(62,107)(63,108)(64,109)(65,110)(66,101)(67,102)(68,103)(69,104)(70,105)(131,202)(132,203)(133,204)(134,205)(135,206)(136,207)(137,208)(138,209)(139,210)(140,201)(141,200)(142,191)(143,192)(144,193)(145,194)(146,195)(147,196)(148,197)(149,198)(150,199)(151,237)(152,238)(153,239)(154,240)(155,231)(156,232)(157,233)(158,234)(159,235)(160,236)(161,226)(162,227)(163,228)(164,229)(165,230)(166,221)(167,222)(168,223)(169,224)(170,225)(171,215)(172,216)(173,217)(174,218)(175,219)(176,220)(177,211)(178,212)(179,213)(180,214), (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), (1,197,6,192)(2,196,7,191)(3,195,8,200)(4,194,9,199)(5,193,10,198)(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),(11,90),(12,81),(13,82),(14,83),(15,84),(16,85),(17,86),(18,87),(19,88),(20,89),(21,91),(22,92),(23,93),(24,94),(25,95),(26,96),(27,97),(28,98),(29,99),(30,100),(31,168),(32,169),(33,170),(34,161),(35,162),(36,163),(37,164),(38,165),(39,166),(40,167),(41,129),(42,130),(43,121),(44,122),(45,123),(46,124),(47,125),(48,126),(49,127),(50,128),(51,78),(52,79),(53,80),(54,71),(55,72),(56,73),(57,74),(58,75),(59,76),(60,77),(61,106),(62,107),(63,108),(64,109),(65,110),(66,101),(67,102),(68,103),(69,104),(70,105),(131,233),(132,234),(133,235),(134,236),(135,237),(136,238),(137,239),(138,240),(139,231),(140,232),(141,216),(142,217),(143,218),(144,219),(145,220),(146,211),(147,212),(148,213),(149,214),(150,215),(151,206),(152,207),(153,208),(154,209),(155,210),(156,201),(157,202),(158,203),(159,204),(160,205),(171,199),(172,200),(173,191),(174,192),(175,193),(176,194),(177,195),(178,196),(179,197),(180,198),(181,221),(182,222),(183,223),(184,224),(185,225),(186,226),(187,227),(188,228),(189,229),(190,230)], [(1,67,27,51,47,13),(2,68,28,52,48,14),(3,69,29,53,49,15),(4,70,30,54,50,16),(5,61,21,55,41,17),(6,62,22,56,42,18),(7,63,23,57,43,19),(8,64,24,58,44,20),(9,65,25,59,45,11),(10,66,26,60,46,12),(31,208,239,223,218,192),(32,209,240,224,219,193),(33,210,231,225,220,194),(34,201,232,226,211,195),(35,202,233,227,212,196),(36,203,234,228,213,197),(37,204,235,229,214,198),(38,205,236,230,215,199),(39,206,237,221,216,200),(40,207,238,222,217,191),(71,128,85,119,105,100),(72,129,86,120,106,91),(73,130,87,111,107,92),(74,121,88,112,108,93),(75,122,89,113,109,94),(76,123,90,114,110,95),(77,124,81,115,101,96),(78,125,82,116,102,97),(79,126,83,117,103,98),(80,127,84,118,104,99),(131,187,147,178,162,157),(132,188,148,179,163,158),(133,189,149,180,164,159),(134,190,150,171,165,160),(135,181,141,172,166,151),(136,182,142,173,167,152),(137,183,143,174,168,153),(138,184,144,175,169,154),(139,185,145,176,170,155),(140,186,146,177,161,156)], [(1,97),(2,98),(3,99),(4,100),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(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),(41,129),(42,130),(43,121),(44,122),(45,123),(46,124),(47,125),(48,126),(49,127),(50,128),(51,82),(52,83),(53,84),(54,85),(55,86),(56,87),(57,88),(58,89),(59,90),(60,81),(61,106),(62,107),(63,108),(64,109),(65,110),(66,101),(67,102),(68,103),(69,104),(70,105),(131,202),(132,203),(133,204),(134,205),(135,206),(136,207),(137,208),(138,209),(139,210),(140,201),(141,200),(142,191),(143,192),(144,193),(145,194),(146,195),(147,196),(148,197),(149,198),(150,199),(151,237),(152,238),(153,239),(154,240),(155,231),(156,232),(157,233),(158,234),(159,235),(160,236),(161,226),(162,227),(163,228),(164,229),(165,230),(166,221),(167,222),(168,223),(169,224),(170,225),(171,215),(172,216),(173,217),(174,218),(175,219),(176,220),(177,211),(178,212),(179,213),(180,214)], [(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)], [(1,197,6,192),(2,196,7,191),(3,195,8,200),(4,194,9,199),(5,193,10,198),(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···2G2H2I2J2K 3 4A4B4C4D4E4F4G4H5A5B6A···6G10A···10N10O···10AD12A···12H15A15B30A···30N
order12···22222344444444556···610···1010···1012···12151530···30
size11···1666621010101030303030222···22···26···610···10444···4

84 irreducible representations

dim11111122222222222244444
type++++++++++-++++--++
imageC1C2C2C2C2C4S3D4D5D6D6Dic5D10D10C4×S3D12C3⋊D4C5⋊D4S3×D5S3×Dic5C15⋊D4C5⋊D12C2×S3×D5
kernelC2×D6⋊Dic5D6⋊Dic5C2×C6×Dic5C22×Dic15S3×C22×C10S3×C2×C10C22×Dic5C2×C30S3×C23C2×Dic5C22×C10C22×S3C22×S3C22×C6C2×C10C2×C10C2×C10C2×C6C23C22C22C22C22
# reps141118142218424441624442

Matrix representation of C2×D6⋊Dic5 in GL5(𝔽61)

10000
060000
006000
000600
000060
,
10000
060100
060000
000600
000060
,
600000
006000
060000
000600
000371
,
600000
060000
006000
000410
000293
,
500000
050000
005000
000387
0005523

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

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