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## G = C3×SD16⋊3D5order 480 = 25·3·5

### Direct product of C3 and SD16⋊3D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×SD16⋊3D5
 Chief series C1 — C5 — C10 — C20 — C60 — D5×C12 — C3×D4⋊2D5 — C3×SD16⋊3D5
 Lower central C5 — C10 — C20 — C3×SD16⋊3D5
 Upper central C1 — C6 — C12 — C3×SD16

Generators and relations for C3×SD163D5
G = < a,b,c,d,e | a3=b8=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b3, bd=db, be=eb, cd=dc, ece=b4c, ede=d-1 >

Subgroups: 432 in 124 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C12, C12, C2×C6, C15, C2×C8, D8, SD16, SD16, Q16, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C24, C24, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×D5, C30, C30, C4○D8, C52C8, C40, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C2×C24, C3×D8, C3×SD16, C3×SD16, C3×Q16, C3×C4○D4, C3×Dic5, C3×Dic5, C60, C60, C6×D5, C6×D5, C2×C30, C8×D5, C40⋊C2, D4⋊D5, C5⋊Q16, C5×SD16, D42D5, Q82D5, C3×C4○D8, C3×C52C8, C120, C3×Dic10, D5×C12, D5×C12, C3×D20, C3×D20, C6×Dic5, C3×C5⋊D4, D4×C15, Q8×C15, SD163D5, D5×C24, C3×C40⋊C2, C3×D4⋊D5, C3×C5⋊Q16, C15×SD16, C3×D42D5, C3×Q82D5, C3×SD163D5
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, SD163D5, C3×D4×D5, C3×SD163D5

Smallest permutation representation of C3×SD163D5
On 240 points
Generators in S240
(1 87 193)(2 88 194)(3 81 195)(4 82 196)(5 83 197)(6 84 198)(7 85 199)(8 86 200)(9 69 231)(10 70 232)(11 71 225)(12 72 226)(13 65 227)(14 66 228)(15 67 229)(16 68 230)(17 64 98)(18 57 99)(19 58 100)(20 59 101)(21 60 102)(22 61 103)(23 62 104)(24 63 97)(25 90 106)(26 91 107)(27 92 108)(28 93 109)(29 94 110)(30 95 111)(31 96 112)(32 89 105)(33 219 114)(34 220 115)(35 221 116)(36 222 117)(37 223 118)(38 224 119)(39 217 120)(40 218 113)(41 177 127)(42 178 128)(43 179 121)(44 180 122)(45 181 123)(46 182 124)(47 183 125)(48 184 126)(49 185 132)(50 186 133)(51 187 134)(52 188 135)(53 189 136)(54 190 129)(55 191 130)(56 192 131)(73 142 235)(74 143 236)(75 144 237)(76 137 238)(77 138 239)(78 139 240)(79 140 233)(80 141 234)(145 161 203)(146 162 204)(147 163 205)(148 164 206)(149 165 207)(150 166 208)(151 167 201)(152 168 202)(153 169 216)(154 170 209)(155 171 210)(156 172 211)(157 173 212)(158 174 213)(159 175 214)(160 176 215)
(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 19)(2 22)(3 17)(4 20)(5 23)(6 18)(7 21)(8 24)(9 108)(10 111)(11 106)(12 109)(13 112)(14 107)(15 110)(16 105)(25 71)(26 66)(27 69)(28 72)(29 67)(30 70)(31 65)(32 68)(33 80)(34 75)(35 78)(36 73)(37 76)(38 79)(39 74)(40 77)(41 146)(42 149)(43 152)(44 147)(45 150)(46 145)(47 148)(48 151)(49 157)(50 160)(51 155)(52 158)(53 153)(54 156)(55 159)(56 154)(57 84)(58 87)(59 82)(60 85)(61 88)(62 83)(63 86)(64 81)(89 230)(90 225)(91 228)(92 231)(93 226)(94 229)(95 232)(96 227)(97 200)(98 195)(99 198)(100 193)(101 196)(102 199)(103 194)(104 197)(113 239)(114 234)(115 237)(116 240)(117 235)(118 238)(119 233)(120 236)(121 202)(122 205)(123 208)(124 203)(125 206)(126 201)(127 204)(128 207)(129 211)(130 214)(131 209)(132 212)(133 215)(134 210)(135 213)(136 216)(137 223)(138 218)(139 221)(140 224)(141 219)(142 222)(143 217)(144 220)(161 182)(162 177)(163 180)(164 183)(165 178)(166 181)(167 184)(168 179)(169 189)(170 192)(171 187)(172 190)(173 185)(174 188)(175 191)(176 186)
(1 226 77 48 55)(2 227 78 41 56)(3 228 79 42 49)(4 229 80 43 50)(5 230 73 44 51)(6 231 74 45 52)(7 232 75 46 53)(8 225 76 47 54)(9 143 181 188 84)(10 144 182 189 85)(11 137 183 190 86)(12 138 184 191 87)(13 139 177 192 88)(14 140 178 185 81)(15 141 179 186 82)(16 142 180 187 83)(17 91 38 149 157)(18 92 39 150 158)(19 93 40 151 159)(20 94 33 152 160)(21 95 34 145 153)(22 96 35 146 154)(23 89 36 147 155)(24 90 37 148 156)(25 118 206 211 97)(26 119 207 212 98)(27 120 208 213 99)(28 113 201 214 100)(29 114 202 215 101)(30 115 203 216 102)(31 116 204 209 103)(32 117 205 210 104)(57 108 217 166 174)(58 109 218 167 175)(59 110 219 168 176)(60 111 220 161 169)(61 112 221 162 170)(62 105 222 163 171)(63 106 223 164 172)(64 107 224 165 173)(65 240 127 131 194)(66 233 128 132 195)(67 234 121 133 196)(68 235 122 134 197)(69 236 123 135 198)(70 237 124 136 199)(71 238 125 129 200)(72 239 126 130 193)
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 181)(10 182)(11 183)(12 184)(13 177)(14 178)(15 179)(16 180)(17 153)(18 154)(19 155)(20 156)(21 157)(22 158)(23 159)(24 160)(25 202)(26 203)(27 204)(28 205)(29 206)(30 207)(31 208)(32 201)(33 37)(34 38)(35 39)(36 40)(41 227)(42 228)(43 229)(44 230)(45 231)(46 232)(47 225)(48 226)(57 170)(58 171)(59 172)(60 173)(61 174)(62 175)(63 176)(64 169)(65 127)(66 128)(67 121)(68 122)(69 123)(70 124)(71 125)(72 126)(81 185)(82 186)(83 187)(84 188)(85 189)(86 190)(87 191)(88 192)(89 151)(90 152)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(97 215)(98 216)(99 209)(100 210)(101 211)(102 212)(103 213)(104 214)(105 167)(106 168)(107 161)(108 162)(109 163)(110 164)(111 165)(112 166)(113 117)(114 118)(115 119)(116 120)(129 200)(130 193)(131 194)(132 195)(133 196)(134 197)(135 198)(136 199)(217 221)(218 222)(219 223)(220 224)

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

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

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 D4 D4 D5 D10 D10 D10 C3×D4 C3×D4 C3×D5 C4○D8 C6×D5 C6×D5 C6×D5 C3×C4○D8 D4×D5 SD16⋊3D5 C3×D4×D5 C3×SD16⋊3D5 kernel C3×SD16⋊3D5 D5×C24 C3×C40⋊C2 C3×D4⋊D5 C3×C5⋊Q16 C15×SD16 C3×D4⋊2D5 C3×Q8⋊2D5 SD16⋊3D5 C8×D5 C40⋊C2 D4⋊D5 C5⋊Q16 C5×SD16 D4⋊2D5 Q8⋊2D5 C3×Dic5 C6×D5 C3×SD16 C24 C3×D4 C3×Q8 Dic5 D10 SD16 C15 C8 D4 Q8 C5 C6 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 2 2 2 2 2 2 4 4 4 4 4 8 2 4 4 8

Matrix representation of C3×SD163D5 in GL4(𝔽241) generated by

 225 0 0 0 0 225 0 0 0 0 15 0 0 0 0 15
,
 1 0 0 0 0 1 0 0 0 0 211 0 0 0 208 233
,
 1 0 0 0 0 1 0 0 0 0 90 181 0 0 139 151
,
 0 240 0 0 1 51 0 0 0 0 1 0 0 0 0 1
,
 190 240 0 0 190 51 0 0 0 0 1 0 0 0 3 240
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,15,0,0,0,0,15],[1,0,0,0,0,1,0,0,0,0,211,208,0,0,0,233],[1,0,0,0,0,1,0,0,0,0,90,139,0,0,181,151],[0,1,0,0,240,51,0,0,0,0,1,0,0,0,0,1],[190,190,0,0,240,51,0,0,0,0,1,3,0,0,0,240] >;

C3×SD163D5 in GAP, Magma, Sage, TeX

C_3\times {\rm SD}_{16}\rtimes_3D_5
% in TeX

G:=Group("C3xSD16:3D5");
// GroupNames label

G:=SmallGroup(480,709);
// by ID

G=gap.SmallGroup(480,709);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,1094,303,268,1271,648,102,18822]);
// 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^3,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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