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G = C3×D8⋊3D5order 480 = 25·3·5

Direct product of C3 and D8⋊3D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D8⋊3D5
 Chief series C1 — C5 — C10 — C20 — C60 — D5×C12 — C3×D4⋊2D5 — C3×D8⋊3D5
 Lower central C5 — C10 — C20 — C3×D8⋊3D5
 Upper central C1 — C6 — C12 — C3×D8

Generators and relations for C3×D83D5
G = < a,b,c,d,e | a3=b8=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b4c, ede=d-1 >

Subgroups: 400 in 124 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D4, D4, Q8, D5, C10, C10, C12, C12, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, Dic5, C20, D10, C2×C10, C24, C24, C2×C12, C3×D4, C3×D4, C3×Q8, C3×D5, C30, C30, C4○D8, C52C8, C40, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×C4○D4, C3×Dic5, C3×Dic5, C60, C6×D5, C2×C30, C8×D5, Dic20, D4.D5, C5×D8, D42D5, C3×C4○D8, C3×C52C8, C120, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, D4×C15, D83D5, D5×C24, C3×Dic20, C3×D4.D5, C15×D8, C3×D42D5, C3×D83D5
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C4○D8, C22×D5, C6×D4, C6×D5, D4×D5, C3×C4○D8, D5×C2×C6, D83D5, C3×D4×D5, C3×D83D5

Smallest permutation representation of C3×D83D5
On 240 points
Generators in S240
(1 92 199)(2 93 200)(3 94 193)(4 95 194)(5 96 195)(6 89 196)(7 90 197)(8 91 198)(9 76 231)(10 77 232)(11 78 225)(12 79 226)(13 80 227)(14 73 228)(15 74 229)(16 75 230)(17 218 109)(18 219 110)(19 220 111)(20 221 112)(21 222 105)(22 223 106)(23 224 107)(24 217 108)(25 44 99)(26 45 100)(27 46 101)(28 47 102)(29 48 103)(30 41 104)(31 42 97)(32 43 98)(33 149 120)(34 150 113)(35 151 114)(36 152 115)(37 145 116)(38 146 117)(39 147 118)(40 148 119)(49 180 127)(50 181 128)(51 182 121)(52 183 122)(53 184 123)(54 177 124)(55 178 125)(56 179 126)(57 188 132)(58 189 133)(59 190 134)(60 191 135)(61 192 136)(62 185 129)(63 186 130)(64 187 131)(65 139 238)(66 140 239)(67 141 240)(68 142 233)(69 143 234)(70 144 235)(71 137 236)(72 138 237)(81 165 203)(82 166 204)(83 167 205)(84 168 206)(85 161 207)(86 162 208)(87 163 201)(88 164 202)(153 169 210)(154 170 211)(155 171 212)(156 172 213)(157 173 214)(158 174 215)(159 175 216)(160 176 209)
(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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 115)(10 114)(11 113)(12 120)(13 119)(14 118)(15 117)(16 116)(17 72)(18 71)(19 70)(20 69)(21 68)(22 67)(23 66)(24 65)(33 79)(34 78)(35 77)(36 76)(37 75)(38 74)(39 73)(40 80)(41 94)(42 93)(43 92)(44 91)(45 90)(46 89)(47 96)(48 95)(49 86)(50 85)(51 84)(52 83)(53 82)(54 81)(55 88)(56 87)(57 155)(58 154)(59 153)(60 160)(61 159)(62 158)(63 157)(64 156)(97 200)(98 199)(99 198)(100 197)(101 196)(102 195)(103 194)(104 193)(105 233)(106 240)(107 239)(108 238)(109 237)(110 236)(111 235)(112 234)(121 206)(122 205)(123 204)(124 203)(125 202)(126 201)(127 208)(128 207)(129 215)(130 214)(131 213)(132 212)(133 211)(134 210)(135 209)(136 216)(137 219)(138 218)(139 217)(140 224)(141 223)(142 222)(143 221)(144 220)(145 230)(146 229)(147 228)(148 227)(149 226)(150 225)(151 232)(152 231)(161 181)(162 180)(163 179)(164 178)(165 177)(166 184)(167 183)(168 182)(169 190)(170 189)(171 188)(172 187)(173 186)(174 185)(175 192)(176 191)
(1 229 72 51 63)(2 230 65 52 64)(3 231 66 53 57)(4 232 67 54 58)(5 225 68 55 59)(6 226 69 56 60)(7 227 70 49 61)(8 228 71 50 62)(9 140 184 188 94)(10 141 177 189 95)(11 142 178 190 96)(12 143 179 191 89)(13 144 180 192 90)(14 137 181 185 91)(15 138 182 186 92)(16 139 183 187 93)(17 84 157 32 146)(18 85 158 25 147)(19 86 159 26 148)(20 87 160 27 149)(21 88 153 28 150)(22 81 154 29 151)(23 82 155 30 152)(24 83 156 31 145)(33 112 201 209 101)(34 105 202 210 102)(35 106 203 211 103)(36 107 204 212 104)(37 108 205 213 97)(38 109 206 214 98)(39 110 207 215 99)(40 111 208 216 100)(41 115 224 166 171)(42 116 217 167 172)(43 117 218 168 173)(44 118 219 161 174)(45 119 220 162 175)(46 120 221 163 176)(47 113 222 164 169)(48 114 223 165 170)(73 236 128 129 198)(74 237 121 130 199)(75 238 122 131 200)(76 239 123 132 193)(77 240 124 133 194)(78 233 125 134 195)(79 234 126 135 196)(80 235 127 136 197)
(1 63)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 184)(10 177)(11 178)(12 179)(13 180)(14 181)(15 182)(16 183)(17 21)(18 22)(19 23)(20 24)(25 154)(26 155)(27 156)(28 157)(29 158)(30 159)(31 160)(32 153)(33 205)(34 206)(35 207)(36 208)(37 201)(38 202)(39 203)(40 204)(41 175)(42 176)(43 169)(44 170)(45 171)(46 172)(47 173)(48 174)(49 227)(50 228)(51 229)(52 230)(53 231)(54 232)(55 225)(56 226)(73 128)(74 121)(75 122)(76 123)(77 124)(78 125)(79 126)(80 127)(81 147)(82 148)(83 149)(84 150)(85 151)(86 152)(87 145)(88 146)(89 191)(90 192)(91 185)(92 186)(93 187)(94 188)(95 189)(96 190)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 109)(106 110)(107 111)(108 112)(113 168)(114 161)(115 162)(116 163)(117 164)(118 165)(119 166)(120 167)(129 198)(130 199)(131 200)(132 193)(133 194)(134 195)(135 196)(136 197)(217 221)(218 222)(219 223)(220 224)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 15A 15B 15C 15D 20A 20B 24A 24B 24C 24D 24E 24F 24G 24H 30A 30B 30C 30D 30E ··· 30L 40A 40B 40C 40D 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 2 3 3 4 4 4 4 4 5 5 6 6 6 6 6 6 6 6 8 8 8 8 10 10 10 10 10 10 12 12 12 12 12 12 12 12 12 12 15 15 15 15 20 20 24 24 24 24 24 24 24 24 30 30 30 30 30 ··· 30 40 40 40 40 60 60 60 60 120 ··· 120 size 1 1 4 4 10 1 1 2 5 5 20 20 2 2 1 1 4 4 4 4 10 10 2 2 10 10 2 2 8 8 8 8 2 2 5 5 5 5 20 20 20 20 2 2 2 2 4 4 2 2 2 2 10 10 10 10 2 2 2 2 8 ··· 8 4 4 4 4 4 4 4 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D5 D10 D10 C3×D4 C3×D4 C3×D5 C4○D8 C6×D5 C6×D5 C3×C4○D8 D4×D5 D8⋊3D5 C3×D4×D5 C3×D8⋊3D5 kernel C3×D8⋊3D5 D5×C24 C3×Dic20 C3×D4.D5 C15×D8 C3×D4⋊2D5 D8⋊3D5 C8×D5 Dic20 D4.D5 C5×D8 D4⋊2D5 C3×Dic5 C6×D5 C3×D8 C24 C3×D4 Dic5 D10 D8 C15 C8 D4 C5 C6 C3 C2 C1 # reps 1 1 1 2 1 2 2 2 2 4 2 4 1 1 2 2 4 2 2 4 4 4 8 8 2 4 4 8

Matrix representation of C3×D83D5 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 15 0 0 0 0 15
,
 8 0 0 0 19 211 0 0 0 0 240 0 0 0 0 240
,
 211 60 0 0 222 30 0 0 0 0 240 0 0 0 0 240
,
 1 0 0 0 0 1 0 0 0 0 0 240 0 0 1 189
,
 1 0 0 0 1 240 0 0 0 0 52 240 0 0 52 189
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,15,0,0,0,0,15],[8,19,0,0,0,211,0,0,0,0,240,0,0,0,0,240],[211,222,0,0,60,30,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,240,189],[1,1,0,0,0,240,0,0,0,0,52,52,0,0,240,189] >;

C3×D83D5 in GAP, Magma, Sage, TeX

C_3\times D_8\rtimes_3D_5
% in TeX

G:=Group("C3xD8:3D5");
// GroupNames label

G:=SmallGroup(480,705);
// by ID

G=gap.SmallGroup(480,705);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,1094,303,1271,648,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^8=c^2=d^5=e^2=1,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,b*e=e*b,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations

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