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## G = C3×Q16⋊D5order 480 = 25·3·5

### Direct product of C3 and Q16⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×Q16⋊D5
 Chief series C1 — C5 — C10 — C20 — C60 — D5×C12 — C3×Q8×D5 — C3×Q16⋊D5
 Lower central C5 — C10 — C20 — C3×Q16⋊D5
 Upper central C1 — C6 — C12 — C3×Q16

Generators and relations for C3×Q16⋊D5
G = < a,b,c,d,e | a3=b8=d5=e2=1, c2=b4, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=b5, cd=dc, ece=b4c, ede=d-1 >

Subgroups: 416 in 120 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D4, Q8, Q8, D5, C10, C12, C12, C2×C6, C15, M4(2), SD16, Q16, Q16, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C24, C24, C2×C12, C3×D4, C3×Q8, C3×Q8, C3×D5, C30, C8.C22, C52C8, C40, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C5×Q8, C3×M4(2), C3×SD16, C3×Q16, C3×Q16, C6×Q8, C3×C4○D4, C3×Dic5, C3×Dic5, C60, C60, C6×D5, C6×D5, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q82D5, C3×C8.C22, C3×C52C8, C120, C3×Dic10, C3×Dic10, D5×C12, D5×C12, C3×D20, C3×D20, Q8×C15, Q16⋊D5, C3×C8⋊D5, C3×C40⋊C2, C3×Q8⋊D5, C3×C5⋊Q16, C15×Q16, C3×Q8×D5, C3×Q82D5, C3×Q16⋊D5
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C8.C22, C22×D5, C6×D4, C6×D5, D4×D5, C3×C8.C22, D5×C2×C6, Q16⋊D5, C3×D4×D5, C3×Q16⋊D5

Smallest permutation representation of C3×Q16⋊D5
On 240 points
Generators in S240
(1 87 184)(2 88 177)(3 81 178)(4 82 179)(5 83 180)(6 84 181)(7 85 182)(8 86 183)(9 73 231)(10 74 232)(11 75 225)(12 76 226)(13 77 227)(14 78 228)(15 79 229)(16 80 230)(17 218 100)(18 219 101)(19 220 102)(20 221 103)(21 222 104)(22 223 97)(23 224 98)(24 217 99)(25 147 113)(26 148 114)(27 149 115)(28 150 116)(29 151 117)(30 152 118)(31 145 119)(32 146 120)(33 95 111)(34 96 112)(35 89 105)(36 90 106)(37 91 107)(38 92 108)(39 93 109)(40 94 110)(41 165 205)(42 166 206)(43 167 207)(44 168 208)(45 161 201)(46 162 202)(47 163 203)(48 164 204)(49 198 72)(50 199 65)(51 200 66)(52 193 67)(53 194 68)(54 195 69)(55 196 70)(56 197 71)(57 188 132)(58 189 133)(59 190 134)(60 191 135)(61 192 136)(62 185 129)(63 186 130)(64 187 131)(121 139 238)(122 140 239)(123 141 240)(124 142 233)(125 143 234)(126 144 235)(127 137 236)(128 138 237)(153 169 214)(154 170 215)(155 171 216)(156 172 209)(157 173 210)(158 174 211)(159 175 212)(160 176 213)
(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)
(1 208 5 204)(2 207 6 203)(3 206 7 202)(4 205 8 201)(9 115 13 119)(10 114 14 118)(11 113 15 117)(12 120 16 116)(17 125 21 121)(18 124 22 128)(19 123 23 127)(20 122 24 126)(25 79 29 75)(26 78 30 74)(27 77 31 73)(28 76 32 80)(33 71 37 67)(34 70 38 66)(35 69 39 65)(36 68 40 72)(41 86 45 82)(42 85 46 81)(43 84 47 88)(44 83 48 87)(49 90 53 94)(50 89 54 93)(51 96 55 92)(52 95 56 91)(57 160 61 156)(58 159 62 155)(59 158 63 154)(60 157 64 153)(97 237 101 233)(98 236 102 240)(99 235 103 239)(100 234 104 238)(105 195 109 199)(106 194 110 198)(107 193 111 197)(108 200 112 196)(129 216 133 212)(130 215 134 211)(131 214 135 210)(132 213 136 209)(137 220 141 224)(138 219 142 223)(139 218 143 222)(140 217 144 221)(145 231 149 227)(146 230 150 226)(147 229 151 225)(148 228 152 232)(161 179 165 183)(162 178 166 182)(163 177 167 181)(164 184 168 180)(169 191 173 187)(170 190 174 186)(171 189 175 185)(172 188 176 192)
(1 54 128 226 63)(2 55 121 227 64)(3 56 122 228 57)(4 49 123 229 58)(5 50 124 230 59)(6 51 125 231 60)(7 52 126 232 61)(8 53 127 225 62)(9 191 84 200 143)(10 192 85 193 144)(11 185 86 194 137)(12 186 87 195 138)(13 187 88 196 139)(14 188 81 197 140)(15 189 82 198 141)(16 190 83 199 142)(17 145 153 207 92)(18 146 154 208 93)(19 147 155 201 94)(20 148 156 202 95)(21 149 157 203 96)(22 150 158 204 89)(23 151 159 205 90)(24 152 160 206 91)(25 216 161 40 102)(26 209 162 33 103)(27 210 163 34 104)(28 211 164 35 97)(29 212 165 36 98)(30 213 166 37 99)(31 214 167 38 100)(32 215 168 39 101)(41 106 224 117 175)(42 107 217 118 176)(43 108 218 119 169)(44 109 219 120 170)(45 110 220 113 171)(46 111 221 114 172)(47 112 222 115 173)(48 105 223 116 174)(65 233 80 134 180)(66 234 73 135 181)(67 235 74 136 182)(68 236 75 129 183)(69 237 76 130 184)(70 238 77 131 177)(71 239 78 132 178)(72 240 79 133 179)
(1 63)(2 60)(3 57)(4 62)(5 59)(6 64)(7 61)(8 58)(9 196)(10 193)(11 198)(12 195)(13 200)(14 197)(15 194)(16 199)(18 22)(20 24)(25 40)(26 37)(27 34)(28 39)(29 36)(30 33)(31 38)(32 35)(41 175)(42 172)(43 169)(44 174)(45 171)(46 176)(47 173)(48 170)(49 225)(50 230)(51 227)(52 232)(53 229)(54 226)(55 231)(56 228)(65 80)(66 77)(67 74)(68 79)(69 76)(70 73)(71 78)(72 75)(81 188)(82 185)(83 190)(84 187)(85 192)(86 189)(87 186)(88 191)(89 146)(90 151)(91 148)(92 145)(93 150)(94 147)(95 152)(96 149)(97 101)(99 103)(105 120)(106 117)(107 114)(108 119)(109 116)(110 113)(111 118)(112 115)(121 125)(123 127)(129 179)(130 184)(131 181)(132 178)(133 183)(134 180)(135 177)(136 182)(137 141)(139 143)(153 207)(154 204)(155 201)(156 206)(157 203)(158 208)(159 205)(160 202)(161 216)(162 213)(163 210)(164 215)(165 212)(166 209)(167 214)(168 211)(217 221)(219 223)(234 238)(236 240)

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

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

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

75 conjugacy classes

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

75 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 4 4 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 C3×D4 C3×D4 C3×D5 C6×D5 C6×D5 C8.C22 D4×D5 C3×C8.C22 Q16⋊D5 C3×D4×D5 C3×Q16⋊D5 kernel C3×Q16⋊D5 C3×C8⋊D5 C3×C40⋊C2 C3×Q8⋊D5 C3×C5⋊Q16 C15×Q16 C3×Q8×D5 C3×Q8⋊2D5 Q16⋊D5 C8⋊D5 C40⋊C2 Q8⋊D5 C5⋊Q16 C5×Q16 Q8×D5 Q8⋊2D5 C3×Dic5 C6×D5 C3×Q16 C24 C3×Q8 Dic5 D10 Q16 C8 Q8 C15 C6 C5 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 2 2 4 2 2 4 4 8 1 2 2 4 4 8

Matrix representation of C3×Q16⋊D5 in GL4(𝔽241) generated by

 15 0 0 0 0 15 0 0 0 0 15 0 0 0 0 15
,
 225 0 16 16 77 0 0 164 112 169 129 144 113 72 113 128
,
 227 122 172 172 240 83 0 76 96 173 202 119 188 190 53 211
,
 190 240 0 0 191 240 0 0 1 0 189 240 190 240 1 0
,
 0 189 0 0 190 0 0 0 1 0 189 240 189 189 52 52
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[225,77,112,113,0,0,169,72,16,0,129,113,16,164,144,128],[227,240,96,188,122,83,173,190,172,0,202,53,172,76,119,211],[190,191,1,190,240,240,0,240,0,0,189,1,0,0,240,0],[0,190,1,189,189,0,0,189,0,0,189,52,0,0,240,52] >;

C3×Q16⋊D5 in GAP, Magma, Sage, TeX

C_3\times Q_{16}\rtimes D_5
% in TeX

G:=Group("C3xQ16:D5");
// GroupNames label

G:=SmallGroup(480,711);
// by ID

G=gap.SmallGroup(480,711);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,1094,303,268,1271,648,102,18822]);
// Polycyclic

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

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