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G = C2×D15⋊Q8order 480 = 25·3·5

Direct product of C2 and D15⋊Q8

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

Aliases: C2×D15⋊Q8, D3011Q8, Dic621D10, Dic1021D6, C30.10C24, C60.162C23, D30.34C23, Dic15.36C23, C61(Q8×D5), C101(S3×Q8), C303(C2×Q8), D151(C2×Q8), C15⋊Q88C22, C153(C22×Q8), (C2×Dic6)⋊14D5, (C2×C20).167D6, C6.10(C23×D5), (C2×Dic10)⋊14S3, (C6×Dic10)⋊13C2, (C10×Dic6)⋊13C2, (C2×C12).167D10, C10.10(S3×C23), C20.127(C22×S3), (C2×C30).229C23, (C2×C60).209C22, (C2×Dic5).138D6, (C5×Dic6)⋊27C22, (C4×D15).59C22, C12.128(C22×D5), D30.C2.9C22, Dic3.6(C22×D5), Dic5.7(C22×S3), (C5×Dic3).6C23, (C3×Dic5).7C23, (C2×Dic3).130D10, (C3×Dic10)⋊27C22, (C6×Dic5).130C22, (C2×Dic15).233C22, (C10×Dic3).129C22, (C22×D15).118C22, C51(C2×S3×Q8), C31(C2×Q8×D5), (C2×C15⋊Q8)⋊21C2, C4.134(C2×S3×D5), (C2×C4×D15).17C2, C22.98(C2×S3×D5), C2.14(C22×S3×D5), (C2×C4).219(S3×D5), (C2×D30.C2).9C2, (C2×C6).239(C22×D5), (C2×C10).239(C22×S3), SmallGroup(480,1082)

Series: Derived Chief Lower central Upper central

C1C30 — C2×D15⋊Q8
C1C5C15C30C3×Dic5D30.C2C2×D30.C2 — C2×D15⋊Q8
C15C30 — C2×D15⋊Q8
C1C22C2×C4

Generators and relations for C2×D15⋊Q8
 G = < a,b,c,d,e | a2=b15=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b11, cd=dc, ece-1=b10c, ede-1=d-1 >

Subgroups: 1404 in 312 conjugacy classes, 124 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×10], C22, C22 [×6], C5, S3 [×4], C6, C6 [×2], C2×C4, C2×C4 [×17], Q8 [×16], C23, D5 [×4], C10, C10 [×2], Dic3 [×4], Dic3 [×2], C12 [×2], C12 [×4], D6 [×6], C2×C6, C15, C22×C4 [×3], C2×Q8 [×12], Dic5 [×4], Dic5 [×2], C20 [×2], C20 [×4], D10 [×6], C2×C10, Dic6 [×4], Dic6 [×8], C4×S3 [×12], C2×Dic3 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C3×Q8 [×4], C22×S3, D15 [×4], C30, C30 [×2], C22×Q8, Dic10 [×4], Dic10 [×8], C4×D5 [×12], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×4], C22×D5, C2×Dic6, C2×Dic6 [×2], S3×C2×C4 [×3], S3×Q8 [×8], C6×Q8, C5×Dic3 [×4], C3×Dic5 [×4], Dic15 [×2], C60 [×2], D30 [×6], C2×C30, C2×Dic10, C2×Dic10 [×2], C2×C4×D5 [×3], Q8×D5 [×8], Q8×C10, C2×S3×Q8, D30.C2 [×8], C15⋊Q8 [×8], C3×Dic10 [×4], C6×Dic5 [×2], C5×Dic6 [×4], C10×Dic3 [×2], C4×D15 [×4], C2×Dic15, C2×C60, C22×D15, C2×Q8×D5, D15⋊Q8 [×8], C2×D30.C2 [×2], C2×C15⋊Q8 [×2], C6×Dic10, C10×Dic6, C2×C4×D15, C2×D15⋊Q8
Quotients: C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D5, D6 [×7], C2×Q8 [×6], C24, D10 [×7], C22×S3 [×7], C22×Q8, C22×D5 [×7], S3×Q8 [×2], S3×C23, S3×D5, Q8×D5 [×2], C23×D5, C2×S3×Q8, C2×S3×D5 [×3], C2×Q8×D5, D15⋊Q8 [×2], C22×S3×D5, C2×D15⋊Q8

Smallest permutation representation of C2×D15⋊Q8
On 240 points
Generators in S240
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 61)(15 62)(16 81)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 76)(27 77)(28 78)(29 79)(30 80)(31 100)(32 101)(33 102)(34 103)(35 104)(36 105)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(46 113)(47 114)(48 115)(49 116)(50 117)(51 118)(52 119)(53 120)(54 106)(55 107)(56 108)(57 109)(58 110)(59 111)(60 112)(121 190)(122 191)(123 192)(124 193)(125 194)(126 195)(127 181)(128 182)(129 183)(130 184)(131 185)(132 186)(133 187)(134 188)(135 189)(136 198)(137 199)(138 200)(139 201)(140 202)(141 203)(142 204)(143 205)(144 206)(145 207)(146 208)(147 209)(148 210)(149 196)(150 197)(151 213)(152 214)(153 215)(154 216)(155 217)(156 218)(157 219)(158 220)(159 221)(160 222)(161 223)(162 224)(163 225)(164 211)(165 212)(166 228)(167 229)(168 230)(169 231)(170 232)(171 233)(172 234)(173 235)(174 236)(175 237)(176 238)(177 239)(178 240)(179 226)(180 227)
(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 62)(2 61)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 89)(17 88)(18 87)(19 86)(20 85)(21 84)(22 83)(23 82)(24 81)(25 80)(26 79)(27 78)(28 77)(29 76)(30 90)(31 105)(32 104)(33 103)(34 102)(35 101)(36 100)(37 99)(38 98)(39 97)(40 96)(41 95)(42 94)(43 93)(44 92)(45 91)(46 118)(47 117)(48 116)(49 115)(50 114)(51 113)(52 112)(53 111)(54 110)(55 109)(56 108)(57 107)(58 106)(59 120)(60 119)(121 189)(122 188)(123 187)(124 186)(125 185)(126 184)(127 183)(128 182)(129 181)(130 195)(131 194)(132 193)(133 192)(134 191)(135 190)(136 200)(137 199)(138 198)(139 197)(140 196)(141 210)(142 209)(143 208)(144 207)(145 206)(146 205)(147 204)(148 203)(149 202)(150 201)(151 219)(152 218)(153 217)(154 216)(155 215)(156 214)(157 213)(158 212)(159 211)(160 225)(161 224)(162 223)(163 222)(164 221)(165 220)(166 238)(167 237)(168 236)(169 235)(170 234)(171 233)(172 232)(173 231)(174 230)(175 229)(176 228)(177 227)(178 226)(179 240)(180 239)
(1 49 28 34)(2 50 29 35)(3 51 30 36)(4 52 16 37)(5 53 17 38)(6 54 18 39)(7 55 19 40)(8 56 20 41)(9 57 21 42)(10 58 22 43)(11 59 23 44)(12 60 24 45)(13 46 25 31)(14 47 26 32)(15 48 27 33)(61 114 76 101)(62 115 77 102)(63 116 78 103)(64 117 79 104)(65 118 80 105)(66 119 81 91)(67 120 82 92)(68 106 83 93)(69 107 84 94)(70 108 85 95)(71 109 86 96)(72 110 87 97)(73 111 88 98)(74 112 89 99)(75 113 90 100)(121 162 145 179)(122 163 146 180)(123 164 147 166)(124 165 148 167)(125 151 149 168)(126 152 150 169)(127 153 136 170)(128 154 137 171)(129 155 138 172)(130 156 139 173)(131 157 140 174)(132 158 141 175)(133 159 142 176)(134 160 143 177)(135 161 144 178)(181 215 198 232)(182 216 199 233)(183 217 200 234)(184 218 201 235)(185 219 202 236)(186 220 203 237)(187 221 204 238)(188 222 205 239)(189 223 206 240)(190 224 207 226)(191 225 208 227)(192 211 209 228)(193 212 210 229)(194 213 196 230)(195 214 197 231)
(1 145 28 121)(2 141 29 132)(3 137 30 128)(4 148 16 124)(5 144 17 135)(6 140 18 131)(7 136 19 127)(8 147 20 123)(9 143 21 134)(10 139 22 130)(11 150 23 126)(12 146 24 122)(13 142 25 133)(14 138 26 129)(15 149 27 125)(31 176 46 159)(32 172 47 155)(33 168 48 151)(34 179 49 162)(35 175 50 158)(36 171 51 154)(37 167 52 165)(38 178 53 161)(39 174 54 157)(40 170 55 153)(41 166 56 164)(42 177 57 160)(43 173 58 156)(44 169 59 152)(45 180 60 163)(61 200 76 183)(62 196 77 194)(63 207 78 190)(64 203 79 186)(65 199 80 182)(66 210 81 193)(67 206 82 189)(68 202 83 185)(69 198 84 181)(70 209 85 192)(71 205 86 188)(72 201 87 184)(73 197 88 195)(74 208 89 191)(75 204 90 187)(91 229 119 212)(92 240 120 223)(93 236 106 219)(94 232 107 215)(95 228 108 211)(96 239 109 222)(97 235 110 218)(98 231 111 214)(99 227 112 225)(100 238 113 221)(101 234 114 217)(102 230 115 213)(103 226 116 224)(104 237 117 220)(105 233 118 216)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222222234444444444445566610···1012121212121215152020202020···2030···3060···60
size1111151515152226666101010103030222222···2442020202044444412···124···44···4

66 irreducible representations

dim1111111222222222444444
type++++++++-+++++++-+-++
imageC1C2C2C2C2C2C2S3Q8D5D6D6D6D10D10D10S3×Q8S3×D5Q8×D5C2×S3×D5C2×S3×D5D15⋊Q8
kernelC2×D15⋊Q8D15⋊Q8C2×D30.C2C2×C15⋊Q8C6×Dic10C10×Dic6C2×C4×D15C2×Dic10D30C2×Dic6Dic10C2×Dic5C2×C20Dic6C2×Dic3C2×C12C10C2×C4C6C4C22C2
# reps1822111142421842224428

Matrix representation of C2×D15⋊Q8 in GL6(𝔽61)

6000000
0600000
0060000
0006000
0000600
0000060
,
0600000
1600000
00171800
0044000
000010
000001
,
6010000
010000
00176000
00444400
0000600
0000060
,
6000000
0600000
0060000
0006000
0000159
0000160
,
0600000
6000000
0060000
0006000
0000133
00004860

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,1,0,0,0,0,60,60,0,0,0,0,0,0,17,44,0,0,0,0,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,1,1,0,0,0,0,0,0,17,44,0,0,0,0,60,44,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,1,0,0,0,0,59,60],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,48,0,0,0,0,33,60] >;

C2×D15⋊Q8 in GAP, Magma, Sage, TeX

C_2\times D_{15}\rtimes Q_8
% in TeX

G:=Group("C2xD15:Q8");
// GroupNames label

G:=SmallGroup(480,1082);
// by ID

G=gap.SmallGroup(480,1082);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^15=c^2=d^4=1,e^2=d^2,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,e*b*e^-1=b^11,c*d=d*c,e*c*e^-1=b^10*c,e*d*e^-1=d^-1>;
// generators/relations

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