C2xQ8:2D15 - GroupNames

G = C₂×Q₈⋊₂D₁₅ order 480 = 2⁵·3·5

Direct product of C₂ and Q₈⋊₂D₁₅

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

Aliases: C₂×Q₈⋊₂D₁₅, Q₈⋊₄D₃₀, C₆₀.₁₈D₄, C₃₀⋊₁₃SD₁₆, C₆₀.₇₇C₂³, D₆₀.₃₈C₂², (C₆×Q₈)⋊₁D₅, C₆⋊₃(Q₈⋊D₅), (Q₈×C₁₀)⋊₅S₃, (C₅×Q₈)⋊₁₉D₆, (C₂×Q₈)⋊₃D₁₅, (Q₈×C₃₀)⋊₁C₂, (C₃×Q₈)⋊₁₆D₁₀, (C₂×D₆₀).₉C₂, (C₂×C₄).₅₃D₃₀, C₁₅⋊₂₅(C₂×SD₁₆), (C₂×C₃₀).₁₄₈D₄, C₃₀.₃₈₅(C₂×D₄), (C₂×C₂₀).₁₅₁D₆, C₄.₈(C₁₅⋊₇D₄), C₁₅⋊₃C₈⋊₃₁C₂², C₁₀⋊₃(Q₈⋊₂S₃), (C₂×C₁₂).₁₅₀D₁₀, C₂₀.₄₃(C₃⋊D₄), C₁₂.₄₅(C₅⋊D₄), (C₂×C₆₀).₇₇C₂², (Q₈×C₁₅)⋊₁₈C₂², C₄.₁₄(C₂²×D₁₅), C₂₀.₁₁₅(C₂²×S₃), C₁₂.₁₁₅(C₂²×D₅), C₂².₂₃(C₁₅⋊₇D₄), C₃⋊₄(C₂×Q₈⋊D₅), C₅⋊₄(C₂×Q₈⋊₂S₃), (C₂×C₁₅⋊₃C₈)⋊₆C₂, C₂.₁₇(C₂×C₁₅⋊₇D₄), C₆.₁₁₂(C₂×C₅⋊D₄), C₁₀.₁₁₂(C₂×C₃⋊D₄), (C₂×C₆).₈₀(C₅⋊D₄), (C₂×C₁₀).₈₀(C₃⋊D₄), SmallGroup(480,906)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆₀ — C₂×Q₈⋊₂D₁₅

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₆₀ — D₆₀ — C₂×D₆₀ — C₂×Q₈⋊₂D₁₅

Lower central

C₁₅ — C₃₀ — C₆₀ — C₂×Q₈⋊₂D₁₅

Upper central

C₁ — C₂² — C₂×C₄ — C₂×Q₈

Generators and relations for C₂×Q₈⋊₂D₁₅
G = < a,b,c,d,e | a²=b⁴=d¹⁵=e²=1, c²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=ebe=b^-1, bd=db, cd=dc, ece=b^-1c, ede=d^-1 >

Subgroups: 916 in 136 conjugacy classes, 55 normal (31 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₅, S₃, C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, D₅, C₁₀, C₁₀, C₁₂, C₁₂, D₆, C₂×C₆, C₁₅, C₂×C₈, SD₁₆, C₂×D₄, C₂×Q₈, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₃⋊C₈, D₁₂, C₂×C₁₂, C₂×C₁₂, C₃×Q₈, C₃×Q₈, C₂²×S₃, D₁₅, C₃₀, C₃₀, C₂×SD₁₆, C₅⋊₂C₈, D₂₀, C₂×C₂₀, C₂×C₂₀, C₅×Q₈, C₅×Q₈, C₂²×D₅, C₂×C₃⋊C₈, Q₈⋊₂S₃, C₂×D₁₂, C₆×Q₈, C₆₀, C₆₀, D₃₀, C₂×C₃₀, C₂×C₅⋊₂C₈, Q₈⋊D₅, C₂×D₂₀, Q₈×C₁₀, C₂×Q₈⋊₂S₃, C₁₅⋊₃C₈, D₆₀, D₆₀, C₂×C₆₀, C₂×C₆₀, Q₈×C₁₅, Q₈×C₁₅, C₂²×D₁₅, C₂×Q₈⋊D₅, C₂×C₁₅⋊₃C₈, Q₈⋊₂D₁₅, C₂×D₆₀, Q₈×C₃₀, C₂×Q₈⋊₂D₁₅
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₅, D₆, SD₁₆, C₂×D₄, D₁₀, C₃⋊D₄, C₂²×S₃, D₁₅, C₂×SD₁₆, C₅⋊D₄, C₂²×D₅, Q₈⋊₂S₃, C₂×C₃⋊D₄, D₃₀, Q₈⋊D₅, C₂×C₅⋊D₄, C₂×Q₈⋊₂S₃, C₁₅⋊₇D₄, C₂²×D₁₅, C₂×Q₈⋊D₅, Q₈⋊₂D₁₅, C₂×C₁₅⋊₇D₄, C₂×Q₈⋊₂D₁₅

Smallest permutation representation of C₂×Q₈⋊₂D₁₅
►On 240 points

Generators in S₂₄₀

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

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

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

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

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

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

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

84 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	···	12F	15A	15B	15C	15D	20A	···	20L	30A	···	30L	60A	···	60X
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	15	15	15	15	20	···	20	30	···	30	60	···	60
size	1	1	1	1	60	60	2	2	2	4	4	2	2	2	2	2	30	30	30	30	2	···	2	4	···	4	2	2	2	2	4	···	4	2	···	2	4	···	4

84 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+		+	+			+			+	+			+	+	+
image	C₁	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₅	D₆	D₆	SD₁₆	D₁₀	D₁₀	C₃⋊D₄	C₃⋊D₄	D₁₅	C₅⋊D₄	C₅⋊D₄	D₃₀	D₃₀	C₁₅⋊₇D₄	C₁₅⋊₇D₄	Q₈⋊₂S₃	Q₈⋊D₅	Q₈⋊₂D₁₅
kernel	C₂×Q₈⋊₂D₁₅	C₂×C₁₅⋊₃C₈	Q₈⋊₂D₁₅	C₂×D₆₀	Q₈×C₃₀	Q₈×C₁₀	C₆₀	C₂×C₃₀	C₆×Q₈	C₂×C₂₀	C₅×Q₈	C₃₀	C₂×C₁₂	C₃×Q₈	C₂₀	C₂×C₁₀	C₂×Q₈	C₁₂	C₂×C₆	C₂×C₄	Q₈	C₄	C₂²	C₁₀	C₆	C₂
# reps	1	1	4	1	1	1	1	1	2	1	2	4	2	4	2	2	4	4	4	4	8	8	8	2	4	8

Matrix representation of C₂×Q₈⋊₂D₁₅ ►in GL₆(𝔽₂₄₁)

240	0	0	0	0	0
0	240	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

1	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	49
0	0	0	0	118	240

1	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	0	33
0	0	0	0	73	0

207	187	0	0	0	0
177	33	0	0	0	0
0	0	173	16	0	0
0	0	225	64	0	0
0	0	0	0	1	0
0	0	0	0	0	1

207	187	0	0	0	0
8	34	0	0	0	0
0	0	173	16	0	0
0	0	178	68	0	0
0	0	0	0	240	192
0	0	0	0	0	1

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,118,0,0,0,0,49,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,73,0,0,0,0,33,0],[207,177,0,0,0,0,187,33,0,0,0,0,0,0,173,225,0,0,0,0,16,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[207,8,0,0,0,0,187,34,0,0,0,0,0,0,173,178,0,0,0,0,16,68,0,0,0,0,0,0,240,0,0,0,0,0,192,1] >;

C₂×Q₈⋊₂D₁₅ in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_2D_{15}

% in TeX

G:=Group("C2xQ8:2D15");

// GroupNames label

G:=SmallGroup(480,906);

// by ID

G=gap.SmallGroup(480,906);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,100,675,185,80,2693,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^15=e^2=1,c^2=b^2,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^-1*c,e*d*e=d^-1>;

// generators/relations

G = C2×Q8⋊2D15 order 480 = 25·3·5

Direct product of C2 and Q8⋊2D15

G = C₂×Q₈⋊₂D₁₅ order 480 = 2⁵·3·5

Direct product of C₂ and Q₈⋊₂D₁₅