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## G = C2×Q8⋊2D15order 480 = 25·3·5

### Direct product of C2 and Q8⋊2D15

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×Q82D15
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 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×2], Q8, C23, D5 [×2], C10, C10 [×2], C12 [×2], C12 [×2], D6 [×4], C2×C6, C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, C20 [×2], C20 [×2], D10 [×4], C2×C10, C3⋊C8 [×2], D12 [×3], C2×C12, C2×C12, C3×Q8 [×2], C3×Q8, C22×S3, D15 [×2], C30, C30 [×2], C2×SD16, C52C8 [×2], D20 [×3], C2×C20, C2×C20, C5×Q8 [×2], C5×Q8, C22×D5, C2×C3⋊C8, Q82S3 [×4], C2×D12, C6×Q8, C60 [×2], C60 [×2], D30 [×4], C2×C30, C2×C52C8, Q8⋊D5 [×4], C2×D20, Q8×C10, C2×Q82S3, C153C8 [×2], D60 [×2], D60, C2×C60, C2×C60, Q8×C15 [×2], Q8×C15, C22×D15, C2×Q8⋊D5, C2×C153C8, Q82D15 [×4], C2×D60, Q8×C30, C2×Q82D15
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], SD16 [×2], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, D15, C2×SD16, C5⋊D4 [×2], C22×D5, Q82S3 [×2], C2×C3⋊D4, D30 [×3], Q8⋊D5 [×2], C2×C5⋊D4, C2×Q82S3, C157D4 [×2], C22×D15, C2×Q8⋊D5, Q82D15 [×2], C2×C157D4, C2×Q82D15

Smallest permutation representation of C2×Q82D15
On 240 points
Generators in S240
(1 124)(2 125)(3 126)(4 127)(5 128)(6 129)(7 130)(8 131)(9 132)(10 133)(11 134)(12 135)(13 121)(14 122)(15 123)(16 145)(17 146)(18 147)(19 148)(20 149)(21 150)(22 136)(23 137)(24 138)(25 139)(26 140)(27 141)(28 142)(29 143)(30 144)(31 155)(32 156)(33 157)(34 158)(35 159)(36 160)(37 161)(38 162)(39 163)(40 164)(41 165)(42 151)(43 152)(44 153)(45 154)(46 174)(47 175)(48 176)(49 177)(50 178)(51 179)(52 180)(53 166)(54 167)(55 168)(56 169)(57 170)(58 171)(59 172)(60 173)(61 185)(62 186)(63 187)(64 188)(65 189)(66 190)(67 191)(68 192)(69 193)(70 194)(71 195)(72 181)(73 182)(74 183)(75 184)(76 197)(77 198)(78 199)(79 200)(80 201)(81 202)(82 203)(83 204)(84 205)(85 206)(86 207)(87 208)(88 209)(89 210)(90 196)(91 213)(92 214)(93 215)(94 216)(95 217)(96 218)(97 219)(98 220)(99 221)(100 222)(101 223)(102 224)(103 225)(104 211)(105 212)(106 229)(107 230)(108 231)(109 232)(110 233)(111 234)(112 235)(113 236)(114 237)(115 238)(116 239)(117 240)(118 226)(119 227)(120 228)
(1 60 20 37)(2 46 21 38)(3 47 22 39)(4 48 23 40)(5 49 24 41)(6 50 25 42)(7 51 26 43)(8 52 27 44)(9 53 28 45)(10 54 29 31)(11 55 30 32)(12 56 16 33)(13 57 17 34)(14 58 18 35)(15 59 19 36)(61 103 76 106)(62 104 77 107)(63 105 78 108)(64 91 79 109)(65 92 80 110)(66 93 81 111)(67 94 82 112)(68 95 83 113)(69 96 84 114)(70 97 85 115)(71 98 86 116)(72 99 87 117)(73 100 88 118)(74 101 89 119)(75 102 90 120)(121 170 146 158)(122 171 147 159)(123 172 148 160)(124 173 149 161)(125 174 150 162)(126 175 136 163)(127 176 137 164)(128 177 138 165)(129 178 139 151)(130 179 140 152)(131 180 141 153)(132 166 142 154)(133 167 143 155)(134 168 144 156)(135 169 145 157)(181 221 208 240)(182 222 209 226)(183 223 210 227)(184 224 196 228)(185 225 197 229)(186 211 198 230)(187 212 199 231)(188 213 200 232)(189 214 201 233)(190 215 202 234)(191 216 203 235)(192 217 204 236)(193 218 205 237)(194 219 206 238)(195 220 207 239)
(1 199 20 187)(2 200 21 188)(3 201 22 189)(4 202 23 190)(5 203 24 191)(6 204 25 192)(7 205 26 193)(8 206 27 194)(9 207 28 195)(10 208 29 181)(11 209 30 182)(12 210 16 183)(13 196 17 184)(14 197 18 185)(15 198 19 186)(31 240 54 221)(32 226 55 222)(33 227 56 223)(34 228 57 224)(35 229 58 225)(36 230 59 211)(37 231 60 212)(38 232 46 213)(39 233 47 214)(40 234 48 215)(41 235 49 216)(42 236 50 217)(43 237 51 218)(44 238 52 219)(45 239 53 220)(61 122 76 147)(62 123 77 148)(63 124 78 149)(64 125 79 150)(65 126 80 136)(66 127 81 137)(67 128 82 138)(68 129 83 139)(69 130 84 140)(70 131 85 141)(71 132 86 142)(72 133 87 143)(73 134 88 144)(74 135 89 145)(75 121 90 146)(91 162 109 174)(92 163 110 175)(93 164 111 176)(94 165 112 177)(95 151 113 178)(96 152 114 179)(97 153 115 180)(98 154 116 166)(99 155 117 167)(100 156 118 168)(101 157 119 169)(102 158 120 170)(103 159 106 171)(104 160 107 172)(105 161 108 173)
(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 23)(17 22)(18 21)(19 20)(24 30)(25 29)(26 28)(31 50)(32 49)(33 48)(34 47)(35 46)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(61 91)(62 105)(63 104)(64 103)(65 102)(66 101)(67 100)(68 99)(69 98)(70 97)(71 96)(72 95)(73 94)(74 93)(75 92)(76 109)(77 108)(78 107)(79 106)(80 120)(81 119)(82 118)(83 117)(84 116)(85 115)(86 114)(87 113)(88 112)(89 111)(90 110)(121 126)(122 125)(123 124)(127 135)(128 134)(129 133)(130 132)(136 146)(137 145)(138 144)(139 143)(140 142)(147 150)(148 149)(151 167)(152 166)(153 180)(154 179)(155 178)(156 177)(157 176)(158 175)(159 174)(160 173)(161 172)(162 171)(163 170)(164 169)(165 168)(181 217)(182 216)(183 215)(184 214)(185 213)(186 212)(187 211)(188 225)(189 224)(190 223)(191 222)(192 221)(193 220)(194 219)(195 218)(196 233)(197 232)(198 231)(199 230)(200 229)(201 228)(202 227)(203 226)(204 240)(205 239)(206 238)(207 237)(208 236)(209 235)(210 234)

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

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

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

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×Q82D15 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×Q82D15 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

׿
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𝔽