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G = D6⋊Dic5.C2order 480 = 25·3·5

6th non-split extension by D6⋊Dic5 of C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6⋊C4.5D5, (C4×Dic15)⋊9C2, (C2×C20).183D6, D6⋊Dic5.6C2, C6.Dic107C2, C30.Q88C2, C159(C422C2), C30.33(C4○D4), C6.25(C4○D20), C10.D410S3, C6.7(D42D5), (C2×C12).182D10, (C2×C30).57C23, (C2×Dic5).15D6, (C22×S3).6D10, C10.29(C4○D12), (C2×C60).160C22, (C2×Dic3).15D10, C33(C23.D10), C2.11(D12⋊D5), C10.67(D42S3), C10.26(Q83S3), (C6×Dic5).34C22, C2.18(D6.D10), C2.13(C30.C23), (C10×Dic3).34C22, (C2×Dic15).187C22, C56(C4⋊C4⋊S3), (C5×D6⋊C4).5C2, (C2×C4).173(S3×D5), (S3×C2×C10).6C22, C22.144(C2×S3×D5), (C2×C6).69(C22×D5), (C3×C10.D4)⋊10C2, (C2×C10).69(C22×S3), SmallGroup(480,443)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D6⋊Dic5.C2
C1C5C15C30C2×C30C6×Dic5D6⋊Dic5 — D6⋊Dic5.C2
C15C2×C30 — D6⋊Dic5.C2
C1C22C2×C4

Generators and relations for D6⋊Dic5.C2
 G = < a,b,c,d,e | a6=b2=c10=1, d2=c5, e2=a3c5, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=a3b, ebe-1=bc5, dcd-1=ece-1=c-1, ede-1=a3c5d >

Subgroups: 524 in 120 conjugacy classes, 44 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×6], C22, C22 [×3], C5, S3, C6 [×3], C2×C4, C2×C4 [×5], C23, C10 [×3], C10, Dic3 [×3], C12 [×3], D6 [×3], C2×C6, C15, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×3], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C5×S3, C30 [×3], C422C2, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20, C22×C10, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4 [×2], C3×C4⋊C4, C5×Dic3, C3×Dic5 [×2], Dic15 [×2], C60, S3×C10 [×3], C2×C30, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C23.D5 [×2], C5×C22⋊C4, C4⋊C4⋊S3, C6×Dic5 [×2], C10×Dic3, C2×Dic15 [×2], C2×C60, S3×C2×C10, C23.D10, D6⋊Dic5 [×2], C30.Q8, C6.Dic10, C3×C10.D4, C5×D6⋊C4, C4×Dic15, D6⋊Dic5.C2
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4 [×3], D10 [×3], C22×S3, C422C2, C22×D5, C4○D12, D42S3, Q83S3, S3×D5, C4○D20, D42D5 [×2], C4⋊C4⋊S3, C2×S3×D5, C23.D10, D12⋊D5, D6.D10, C30.C23, D6⋊Dic5.C2

Smallest permutation representation of D6⋊Dic5.C2
On 240 points
Generators in S240
(1 62 27 56 16 48)(2 63 28 57 17 49)(3 64 29 58 18 50)(4 65 30 59 19 41)(5 66 21 60 20 42)(6 67 22 51 11 43)(7 68 23 52 12 44)(8 69 24 53 13 45)(9 70 25 54 14 46)(10 61 26 55 15 47)(31 203 239 193 226 220)(32 204 240 194 227 211)(33 205 231 195 228 212)(34 206 232 196 229 213)(35 207 233 197 230 214)(36 208 234 198 221 215)(37 209 235 199 222 216)(38 210 236 200 223 217)(39 201 237 191 224 218)(40 202 238 192 225 219)(71 105 121 83 112 97)(72 106 122 84 113 98)(73 107 123 85 114 99)(74 108 124 86 115 100)(75 109 125 87 116 91)(76 110 126 88 117 92)(77 101 127 89 118 93)(78 102 128 90 119 94)(79 103 129 81 120 95)(80 104 130 82 111 96)(131 164 158 186 141 177)(132 165 159 187 142 178)(133 166 160 188 143 179)(134 167 151 189 144 180)(135 168 152 190 145 171)(136 169 153 181 146 172)(137 170 154 182 147 173)(138 161 155 183 148 174)(139 162 156 184 149 175)(140 163 157 185 150 176)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 41)(10 42)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 61)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 51)(28 52)(29 53)(30 54)(31 36)(32 37)(33 38)(34 39)(35 40)(71 112)(72 113)(73 114)(74 115)(75 116)(76 117)(77 118)(78 119)(79 120)(80 111)(81 103)(82 104)(83 105)(84 106)(85 107)(86 108)(87 109)(88 110)(89 101)(90 102)(131 186)(132 187)(133 188)(134 189)(135 190)(136 181)(137 182)(138 183)(139 184)(140 185)(141 177)(142 178)(143 179)(144 180)(145 171)(146 172)(147 173)(148 174)(149 175)(150 176)(151 167)(152 168)(153 169)(154 170)(155 161)(156 162)(157 163)(158 164)(159 165)(160 166)(191 196)(192 197)(193 198)(194 199)(195 200)(201 213)(202 214)(203 215)(204 216)(205 217)(206 218)(207 219)(208 220)(209 211)(210 212)(221 239)(222 240)(223 231)(224 232)(225 233)(226 234)(227 235)(228 236)(229 237)(230 238)
(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)
(1 220 6 215)(2 219 7 214)(3 218 8 213)(4 217 9 212)(5 216 10 211)(11 198 16 193)(12 197 17 192)(13 196 18 191)(14 195 19 200)(15 194 20 199)(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)(41 223 46 228)(42 222 47 227)(43 221 48 226)(44 230 49 225)(45 229 50 224)(51 234 56 239)(52 233 57 238)(53 232 58 237)(54 231 59 236)(55 240 60 235)(71 165 76 170)(72 164 77 169)(73 163 78 168)(74 162 79 167)(75 161 80 166)(81 144 86 149)(82 143 87 148)(83 142 88 147)(84 141 89 146)(85 150 90 145)(91 138 96 133)(92 137 97 132)(93 136 98 131)(94 135 99 140)(95 134 100 139)(101 153 106 158)(102 152 107 157)(103 151 108 156)(104 160 109 155)(105 159 110 154)(111 179 116 174)(112 178 117 173)(113 177 118 172)(114 176 119 171)(115 175 120 180)(121 187 126 182)(122 186 127 181)(123 185 128 190)(124 184 129 189)(125 183 130 188)
(1 179 51 155)(2 178 52 154)(3 177 53 153)(4 176 54 152)(5 175 55 151)(6 174 56 160)(7 173 57 159)(8 172 58 158)(9 171 59 157)(10 180 60 156)(11 183 62 133)(12 182 63 132)(13 181 64 131)(14 190 65 140)(15 189 66 139)(16 188 67 138)(17 187 68 137)(18 186 69 136)(19 185 70 135)(20 184 61 134)(21 162 47 144)(22 161 48 143)(23 170 49 142)(24 169 50 141)(25 168 41 150)(26 167 42 149)(27 166 43 148)(28 165 44 147)(29 164 45 146)(30 163 46 145)(31 130 198 91)(32 129 199 100)(33 128 200 99)(34 127 191 98)(35 126 192 97)(36 125 193 96)(37 124 194 95)(38 123 195 94)(39 122 196 93)(40 121 197 92)(71 207 88 225)(72 206 89 224)(73 205 90 223)(74 204 81 222)(75 203 82 221)(76 202 83 230)(77 201 84 229)(78 210 85 228)(79 209 86 227)(80 208 87 226)(101 237 113 213)(102 236 114 212)(103 235 115 211)(104 234 116 220)(105 233 117 219)(106 232 118 218)(107 231 119 217)(108 240 120 216)(109 239 111 215)(110 238 112 214)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G4H4I5A5B6A6B6C10A···10F10G10H10I10J12A12B12C12D12E12F15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order1222234444444445566610···10101010101212121212121515202020202020202030···3060···60
size11111222212202030303030222222···2121212124420202020444444121212124···44···4

60 irreducible representations

dim1111111222222222244444444
type++++++++++++++-++-+-
imageC1C2C2C2C2C2C2S3D5D6D6C4○D4D10D10D10C4○D12C4○D20D42S3Q83S3S3×D5D42D5C2×S3×D5D12⋊D5D6.D10C30.C23
kernelD6⋊Dic5.C2D6⋊Dic5C30.Q8C6.Dic10C3×C10.D4C5×D6⋊C4C4×Dic15C10.D4D6⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3C10C6C10C10C2×C4C6C22C2C2C2
# reps1211111122162224811242444

Matrix representation of D6⋊Dic5.C2 in GL6(𝔽61)

6000000
0600000
0060000
0006000
0000601
0000600
,
100000
7600000
0060000
001100
0000600
0000601
,
300000
50410000
001000
000100
000010
000001
,
1350000
27480000
00112200
00505000
0000600
0000060
,
21550000
53400000
0011000
00505000
000010
000001

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[1,7,0,0,0,0,0,60,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,0,1],[3,50,0,0,0,0,0,41,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,27,0,0,0,0,5,48,0,0,0,0,0,0,11,50,0,0,0,0,22,50,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[21,53,0,0,0,0,55,40,0,0,0,0,0,0,11,50,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D6⋊Dic5.C2 in GAP, Magma, Sage, TeX

D_6\rtimes {\rm Dic}_5.C_2
% in TeX

G:=Group("D6:Dic5.C2");
// GroupNames label

G:=SmallGroup(480,443);
// by ID

G=gap.SmallGroup(480,443);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,422,219,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^2=c^10=1,d^2=c^5,e^2=a^3*c^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=a^3*b,e*b*e^-1=b*c^5,d*c*d^-1=e*c*e^-1=c^-1,e*d*e^-1=a^3*c^5*d>;
// generators/relations

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