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G = (C2×Dic6)⋊D5order 480 = 25·3·5

6th semidirect product of C2×Dic6 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×Dic6)⋊6D5, C30.150(C2×D4), (C2×C20).231D6, (C2×C12).27D10, C10.137(S3×D4), D10⋊C418S3, D303C421C2, D304C419C2, (C10×Dic6)⋊16C2, C30.89(C4○D4), C32(C20.23D4), C1510(C4.4D4), (C5×Dic3).10D4, (C22×D5).16D6, (Dic3×Dic5)⋊23C2, C10.16(C4○D12), D10⋊Dic316C2, (C2×C60).324C22, (C2×C30).145C23, C6.18(Q82D5), (C2×Dic5).119D6, Dic3.2(C5⋊D4), C56(C23.11D6), C10.15(D42S3), C2.18(C12.28D10), C2.18(D20⋊S3), (C2×Dic3).113D10, (C6×Dic5).87C22, (C10×Dic3).88C22, (C22×D15).50C22, (C2×Dic15).112C22, (C2×C4).60(S3×D5), C2.18(S3×C5⋊D4), C6.39(C2×C5⋊D4), (C2×C3⋊D20).9C2, (D5×C2×C6).30C22, C22.197(C2×S3×D5), (C3×D10⋊C4)⋊23C2, (C2×C6).157(C22×D5), (C2×C10).157(C22×S3), SmallGroup(480,531)

Series: Derived Chief Lower central Upper central

C1C2×C30 — (C2×Dic6)⋊D5
C1C5C15C30C2×C30D5×C2×C6C2×C3⋊D20 — (C2×Dic6)⋊D5
C15C2×C30 — (C2×Dic6)⋊D5
C1C22C2×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=b-1, bd=db, ebe=ab7, cd=dc, ece=b6c, ede=d-1 >

Subgroups: 876 in 152 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×6], C22, C22 [×6], C5, S3, C6 [×3], C6, C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], D5 [×2], C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], D6 [×3], C2×C6, C2×C6 [×3], C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×4], D10 [×6], C2×C10, Dic6 [×2], C2×Dic3 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12, C2×C12, C22×S3, C22×C6, C3×D5, D15, C30 [×3], C4.4D4, D20 [×2], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C22×D5, C4×Dic3, D6⋊C4 [×2], C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, C5×Dic3 [×2], C5×Dic3, C3×Dic5, Dic15, C60, C6×D5 [×3], D30 [×3], C2×C30, C4×Dic5, D10⋊C4, D10⋊C4 [×3], C2×D20, Q8×C10, C23.11D6, C3⋊D20 [×2], C6×Dic5, C5×Dic6 [×2], C10×Dic3 [×2], C2×Dic15, C2×C60, D5×C2×C6, C22×D15, C20.23D4, Dic3×Dic5, D10⋊Dic3, D304C4, C3×D10⋊C4, D303C4, C2×C3⋊D20, C10×Dic6, (C2×Dic6)⋊D5
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C22×S3, C4.4D4, C5⋊D4 [×2], C22×D5, C4○D12, S3×D4, D42S3, S3×D5, Q82D5 [×2], C2×C5⋊D4, C23.11D6, C2×S3×D5, C20.23D4, D20⋊S3, C12.28D10, S3×C5⋊D4, (C2×Dic6)⋊D5

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

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E10A···10F12A12B12C12D15A15B20A20B20C20D20E···20L30A···30F60A···60H
order122222344444444556666610···101212121215152020202020···2030···3060···60
size11112060246610101230302222220202···244202044444412···124···44···4

60 irreducible representations

dim111111112222222222244444444
type+++++++++++++++++-++++
imageC1C2C2C2C2C2C2C2S3D4D5D6D6D6C4○D4D10D10C5⋊D4C4○D12S3×D4D42S3S3×D5Q82D5C2×S3×D5D20⋊S3C12.28D10S3×C5⋊D4
kernel(C2×Dic6)⋊D5Dic3×Dic5D10⋊Dic3D304C4C3×D10⋊C4D303C4C2×C3⋊D20C10×Dic6D10⋊C4C5×Dic3C2×Dic6C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12Dic3C10C10C10C2×C4C6C22C2C2C2
# reps111111111221114428411242444

Matrix representation of (C2×Dic6)⋊D5 in GL4(𝔽61) generated by

60000
06000
00600
00060
,
233800
234600
003117
004430
,
18900
524300
00600
00060
,
1000
0100
0001
006043
,
524300
18900
00060
00600
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[23,23,0,0,38,46,0,0,0,0,31,44,0,0,17,30],[18,52,0,0,9,43,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,43],[52,18,0,0,43,9,0,0,0,0,0,60,0,0,60,0] >;

(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(480,531);
// by ID

G=gap.SmallGroup(480,531);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,254,219,100,1356,18822]);
// 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=b^-1,b*d=d*b,e*b*e=a*b^7,c*d=d*c,e*c*e=b^6*c,e*d*e=d^-1>;
// generators/relations

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