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## G = C2×D15⋊Q8order 480 = 25·3·5

### Direct product of C2 and D15⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D15⋊Q8
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — D30.C2 — C2×D30.C2 — C2×D15⋊Q8
 Lower central C15 — C30 — C2×D15⋊Q8
 Upper central C1 — C22 — C2×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, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, C12, D6, C2×C6, C15, C22×C4, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, Dic6, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, D15, C30, C30, C22×Q8, Dic10, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C2×Dic6, C2×Dic6, S3×C2×C4, S3×Q8, C6×Q8, C5×Dic3, C3×Dic5, Dic15, C60, D30, C2×C30, C2×Dic10, C2×Dic10, C2×C4×D5, Q8×D5, Q8×C10, C2×S3×Q8, D30.C2, C15⋊Q8, C3×Dic10, C6×Dic5, C5×Dic6, C10×Dic3, C4×D15, C2×Dic15, C2×C60, C22×D15, C2×Q8×D5, D15⋊Q8, C2×D30.C2, C2×C15⋊Q8, C6×Dic10, C10×Dic6, C2×C4×D15, C2×D15⋊Q8
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C24, D10, C22×S3, C22×Q8, C22×D5, S3×Q8, S3×C23, S3×D5, Q8×D5, C23×D5, C2×S3×Q8, C2×S3×D5, C2×Q8×D5, D15⋊Q8, C22×S3×D5, C2×D15⋊Q8

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

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 6A 6B 6C 10A ··· 10F 12A 12B 12C 12D 12E 12F 15A 15B 20A 20B 20C 20D 20E ··· 20L 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 12 12 12 12 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 15 15 15 15 2 2 2 6 6 6 6 10 10 10 10 30 30 2 2 2 2 2 2 ··· 2 4 4 20 20 20 20 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + - + + + + + + + - + - + + image C1 C2 C2 C2 C2 C2 C2 S3 Q8 D5 D6 D6 D6 D10 D10 D10 S3×Q8 S3×D5 Q8×D5 C2×S3×D5 C2×S3×D5 D15⋊Q8 kernel C2×D15⋊Q8 D15⋊Q8 C2×D30.C2 C2×C15⋊Q8 C6×Dic10 C10×Dic6 C2×C4×D15 C2×Dic10 D30 C2×Dic6 Dic10 C2×Dic5 C2×C20 Dic6 C2×Dic3 C2×C12 C10 C2×C4 C6 C4 C22 C2 # reps 1 8 2 2 1 1 1 1 4 2 4 2 1 8 4 2 2 2 4 4 2 8

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

 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 60 0 0 0 0 1 60 0 0 0 0 0 0 17 18 0 0 0 0 44 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 1 0 0 0 0 0 1 0 0 0 0 0 0 17 60 0 0 0 0 44 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 59 0 0 0 0 1 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 33 0 0 0 0 48 60

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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