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## G = C3×D4.F5order 480 = 25·3·5

### Direct product of C3 and D4.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×D4.F5
 Chief series C1 — C5 — C10 — Dic5 — C3×Dic5 — C3×C5⋊C8 — C6×C5⋊C8 — C3×D4.F5
 Lower central C5 — C10 — C3×D4.F5
 Upper central C1 — C6 — C3×D4

Generators and relations for C3×D4.F5
G = < a,b,c,d,e | a3=b4=c2=d5=1, e4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 328 in 124 conjugacy classes, 68 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, D4, D4, Q8, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, M4(2), C4○D4, Dic5, Dic5, C20, D10, C2×C10, C24, C2×C12, C3×D4, C3×D4, C3×Q8, C3×D5, C30, C30, C8○D4, C5⋊C8, C5⋊C8, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C2×C24, C3×M4(2), C3×C4○D4, C3×Dic5, C3×Dic5, C60, C6×D5, C2×C30, D5⋊C8, C4.F5, C2×C5⋊C8, C22.F5, D42D5, C3×C8○D4, C3×C5⋊C8, C3×C5⋊C8, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, D4×C15, D4.F5, C3×D5⋊C8, C3×C4.F5, C6×C5⋊C8, C3×C22.F5, C3×D42D5, C3×D4.F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, F5, C2×C12, C22×C6, C8○D4, C2×F5, C22×C12, C3×F5, C22×F5, C3×C8○D4, C6×F5, D4.F5, C2×C6×F5, C3×D4.F5

Smallest permutation representation of C3×D4.F5
On 240 points
Generators in S240
(1 101 169)(2 102 170)(3 103 171)(4 104 172)(5 97 173)(6 98 174)(7 99 175)(8 100 176)(9 57 111)(10 58 112)(11 59 105)(12 60 106)(13 61 107)(14 62 108)(15 63 109)(16 64 110)(17 147 87)(18 148 88)(19 149 81)(20 150 82)(21 151 83)(22 152 84)(23 145 85)(24 146 86)(25 79 215)(26 80 216)(27 73 209)(28 74 210)(29 75 211)(30 76 212)(31 77 213)(32 78 214)(33 222 126)(34 223 127)(35 224 128)(36 217 121)(37 218 122)(38 219 123)(39 220 124)(40 221 125)(41 95 137)(42 96 138)(43 89 139)(44 90 140)(45 91 141)(46 92 142)(47 93 143)(48 94 144)(49 157 201)(50 158 202)(51 159 203)(52 160 204)(53 153 205)(54 154 206)(55 155 207)(56 156 208)(65 119 161)(66 120 162)(67 113 163)(68 114 164)(69 115 165)(70 116 166)(71 117 167)(72 118 168)(129 179 231)(130 180 232)(131 181 225)(132 182 226)(133 183 227)(134 184 228)(135 177 229)(136 178 230)(185 239 193)(186 240 194)(187 233 195)(188 234 196)(189 235 197)(190 236 198)(191 237 199)(192 238 200)
(1 193 5 197)(2 194 6 198)(3 195 7 199)(4 196 8 200)(9 159 13 155)(10 160 14 156)(11 153 15 157)(12 154 16 158)(17 223 21 219)(18 224 22 220)(19 217 23 221)(20 218 24 222)(25 119 29 115)(26 120 30 116)(27 113 31 117)(28 114 32 118)(33 82 37 86)(34 83 38 87)(35 84 39 88)(36 85 40 81)(41 135 45 131)(42 136 46 132)(43 129 47 133)(44 130 48 134)(49 105 53 109)(50 106 54 110)(51 107 55 111)(52 108 56 112)(57 203 61 207)(58 204 62 208)(59 205 63 201)(60 206 64 202)(65 211 69 215)(66 212 70 216)(67 213 71 209)(68 214 72 210)(73 163 77 167)(74 164 78 168)(75 165 79 161)(76 166 80 162)(89 179 93 183)(90 180 94 184)(91 181 95 177)(92 182 96 178)(97 189 101 185)(98 190 102 186)(99 191 103 187)(100 192 104 188)(121 145 125 149)(122 146 126 150)(123 147 127 151)(124 148 128 152)(137 229 141 225)(138 230 142 226)(139 231 143 227)(140 232 144 228)(169 239 173 235)(170 240 174 236)(171 233 175 237)(172 234 176 238)
(9 13)(10 14)(11 15)(12 16)(33 37)(34 38)(35 39)(36 40)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(161 165)(162 166)(163 167)(164 168)(177 181)(178 182)(179 183)(180 184)(185 189)(186 190)(187 191)(188 192)(193 197)(194 198)(195 199)(196 200)(217 221)(218 222)(219 223)(220 224)(225 229)(226 230)(227 231)(228 232)(233 237)(234 238)(235 239)(236 240)
(1 137 17 77 49)(2 78 138 50 18)(3 51 79 19 139)(4 20 52 140 80)(5 141 21 73 53)(6 74 142 54 22)(7 55 75 23 143)(8 24 56 144 76)(9 69 125 133 191)(10 134 70 192 126)(11 185 135 127 71)(12 128 186 72 136)(13 65 121 129 187)(14 130 66 188 122)(15 189 131 123 67)(16 124 190 68 132)(25 81 89 171 203)(26 172 82 204 90)(27 205 173 91 83)(28 92 206 84 174)(29 85 93 175 207)(30 176 86 208 94)(31 201 169 95 87)(32 96 202 88 170)(33 58 184 116 238)(34 117 59 239 177)(35 240 118 178 60)(36 179 233 61 119)(37 62 180 120 234)(38 113 63 235 181)(39 236 114 182 64)(40 183 237 57 115)(41 147 213 157 101)(42 158 148 102 214)(43 103 159 215 149)(44 216 104 150 160)(45 151 209 153 97)(46 154 152 98 210)(47 99 155 211 145)(48 212 100 146 156)(105 193 229 223 167)(106 224 194 168 230)(107 161 217 231 195)(108 232 162 196 218)(109 197 225 219 163)(110 220 198 164 226)(111 165 221 227 199)(112 228 166 200 222)
(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)

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

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

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

75 conjugacy classes

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

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 8 8 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C6 C6 C12 C12 C12 C8○D4 C3×C8○D4 F5 C2×F5 C2×F5 C3×F5 C6×F5 C6×F5 D4.F5 C3×D4.F5 kernel C3×D4.F5 C3×D5⋊C8 C3×C4.F5 C6×C5⋊C8 C3×C22.F5 C3×D4⋊2D5 D4.F5 C3×Dic10 C3×C5⋊D4 D4×C15 D5⋊C8 C4.F5 C2×C5⋊C8 C22.F5 D4⋊2D5 Dic10 C5⋊D4 C5×D4 C15 C5 C3×D4 C12 C2×C6 D4 C4 C22 C3 C1 # reps 1 1 1 2 2 1 2 2 4 2 2 2 4 4 2 4 8 4 4 8 1 1 2 2 2 4 1 2

Matrix representation of C3×D4.F5 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 15 0 0 0 0 0 0 15 0 0 0 0 0 0 15 0 0 0 0 0 0 15
,
 64 2 0 0 0 0 0 177 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 0 177 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 1 0 0 0 0 240 0 1 0 0 0 240 0 0 1 0 0 240 0 0 0
,
 233 0 0 0 0 0 0 233 0 0 0 0 0 0 148 121 171 36 0 0 78 157 177 184 0 0 84 64 57 114 0 0 205 235 93 120

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[64,0,0,0,0,0,2,177,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,177,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[233,0,0,0,0,0,0,233,0,0,0,0,0,0,148,78,84,205,0,0,121,157,64,235,0,0,171,177,57,93,0,0,36,184,114,120] >;

C3×D4.F5 in GAP, Magma, Sage, TeX

C_3\times D_4.F_5
% in TeX

G:=Group("C3xD4.F5");
// GroupNames label

G:=SmallGroup(480,1053);
// by ID

G=gap.SmallGroup(480,1053);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,555,102,9414,818]);
// Polycyclic

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

׿
×
𝔽