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## G = C2×D6.F5order 480 = 25·3·5

### Direct product of C2 and D6.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D6.F5
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C3×C5⋊C8 — D6.F5 — C2×D6.F5
 Lower central C15 — C30 — C2×D6.F5
 Upper central C1 — C22

Generators and relations for C2×D6.F5
G = < a,b,c,d,e | a2=b6=c2=d5=1, e4=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d3 >

Subgroups: 564 in 136 conjugacy classes, 54 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C23, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, Dic5, C2×C10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C30, C2×M4(2), C5⋊C8, C5⋊C8, C2×Dic5, C2×Dic5, C22×C10, C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, C3×Dic5, Dic15, S3×C10, S3×C10, C2×C30, C2×C5⋊C8, C2×C5⋊C8, C22.F5, C22×Dic5, C2×C8⋊S3, C3×C5⋊C8, C15⋊C8, S3×Dic5, C6×Dic5, C2×Dic15, S3×C2×C10, C2×C22.F5, D6.F5, C6×C5⋊C8, C2×C15⋊C8, C2×S3×Dic5, C2×D6.F5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, F5, C4×S3, C22×S3, C2×M4(2), C2×F5, C8⋊S3, S3×C2×C4, C22.F5, C22×F5, C2×C8⋊S3, S3×F5, C2×C22.F5, D6.F5, C2×S3×F5, C2×D6.F5

Smallest permutation representation of C2×D6.F5
On 240 points
Generators in S240
(1 56)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 38)(10 39)(11 40)(12 33)(13 34)(14 35)(15 36)(16 37)(17 238)(18 239)(19 240)(20 233)(21 234)(22 235)(23 236)(24 237)(25 184)(26 177)(27 178)(28 179)(29 180)(30 181)(31 182)(32 183)(41 62)(42 63)(43 64)(44 57)(45 58)(46 59)(47 60)(48 61)(65 116)(66 117)(67 118)(68 119)(69 120)(70 113)(71 114)(72 115)(73 154)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 153)(81 194)(82 195)(83 196)(84 197)(85 198)(86 199)(87 200)(88 193)(89 223)(90 224)(91 217)(92 218)(93 219)(94 220)(95 221)(96 222)(97 136)(98 129)(99 130)(100 131)(101 132)(102 133)(103 134)(104 135)(105 229)(106 230)(107 231)(108 232)(109 225)(110 226)(111 227)(112 228)(121 143)(122 144)(123 137)(124 138)(125 139)(126 140)(127 141)(128 142)(145 208)(146 201)(147 202)(148 203)(149 204)(150 205)(151 206)(152 207)(161 190)(162 191)(163 192)(164 185)(165 186)(166 187)(167 188)(168 189)(169 212)(170 213)(171 214)(172 215)(173 216)(174 209)(175 210)(176 211)
(1 115 40 5 119 36)(2 116 33 6 120 37)(3 117 34 7 113 38)(4 118 35 8 114 39)(9 50 66 13 54 70)(10 51 67 14 55 71)(11 52 68 15 56 72)(12 53 69 16 49 65)(17 63 197 21 59 193)(18 64 198 22 60 194)(19 57 199 23 61 195)(20 58 200 24 62 196)(25 158 220 29 154 224)(26 159 221 30 155 217)(27 160 222 31 156 218)(28 153 223 32 157 219)(41 83 233 45 87 237)(42 84 234 46 88 238)(43 85 235 47 81 239)(44 86 236 48 82 240)(73 90 184 77 94 180)(74 91 177 78 95 181)(75 92 178 79 96 182)(76 93 179 80 89 183)(97 225 137 101 229 141)(98 226 138 102 230 142)(99 227 139 103 231 143)(100 228 140 104 232 144)(105 127 136 109 123 132)(106 128 129 110 124 133)(107 121 130 111 125 134)(108 122 131 112 126 135)(145 166 175 149 162 171)(146 167 176 150 163 172)(147 168 169 151 164 173)(148 161 170 152 165 174)(185 216 202 189 212 206)(186 209 203 190 213 207)(187 210 204 191 214 208)(188 211 205 192 215 201)
(1 36)(2 33)(3 38)(4 35)(5 40)(6 37)(7 34)(8 39)(9 50)(10 55)(11 52)(12 49)(13 54)(14 51)(15 56)(16 53)(17 197)(18 194)(19 199)(20 196)(21 193)(22 198)(23 195)(24 200)(26 30)(28 32)(41 45)(43 47)(58 62)(60 64)(66 70)(68 72)(73 94)(74 91)(75 96)(76 93)(77 90)(78 95)(79 92)(80 89)(81 239)(82 236)(83 233)(84 238)(85 235)(86 240)(87 237)(88 234)(97 229)(98 226)(99 231)(100 228)(101 225)(102 230)(103 227)(104 232)(105 136)(106 133)(107 130)(108 135)(109 132)(110 129)(111 134)(112 131)(113 117)(115 119)(122 126)(124 128)(138 142)(140 144)(145 175)(146 172)(147 169)(148 174)(149 171)(150 176)(151 173)(152 170)(153 223)(154 220)(155 217)(156 222)(157 219)(158 224)(159 221)(160 218)(161 165)(163 167)(177 181)(179 183)(186 190)(188 192)(201 215)(202 212)(203 209)(204 214)(205 211)(206 216)(207 213)(208 210)
(1 203 20 89 228)(2 90 204 229 21)(3 230 91 22 205)(4 23 231 206 92)(5 207 24 93 232)(6 94 208 225 17)(7 226 95 18 201)(8 19 227 202 96)(9 133 155 85 176)(10 86 134 169 156)(11 170 87 157 135)(12 158 171 136 88)(13 129 159 81 172)(14 82 130 173 160)(15 174 83 153 131)(16 154 175 132 84)(25 162 127 46 65)(26 47 163 66 128)(27 67 48 121 164)(28 122 68 165 41)(29 166 123 42 69)(30 43 167 70 124)(31 71 44 125 168)(32 126 72 161 45)(33 77 214 97 193)(34 98 78 194 215)(35 195 99 216 79)(36 209 196 80 100)(37 73 210 101 197)(38 102 74 198 211)(39 199 103 212 75)(40 213 200 76 104)(49 224 149 105 234)(50 106 217 235 150)(51 236 107 151 218)(52 152 237 219 108)(53 220 145 109 238)(54 110 221 239 146)(55 240 111 147 222)(56 148 233 223 112)(57 139 189 182 114)(58 183 140 115 190)(59 116 184 191 141)(60 192 117 142 177)(61 143 185 178 118)(62 179 144 119 186)(63 120 180 187 137)(64 188 113 138 181)
(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)| (1,56)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,38)(10,39)(11,40)(12,33)(13,34)(14,35)(15,36)(16,37)(17,238)(18,239)(19,240)(20,233)(21,234)(22,235)(23,236)(24,237)(25,184)(26,177)(27,178)(28,179)(29,180)(30,181)(31,182)(32,183)(41,62)(42,63)(43,64)(44,57)(45,58)(46,59)(47,60)(48,61)(65,116)(66,117)(67,118)(68,119)(69,120)(70,113)(71,114)(72,115)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,153)(81,194)(82,195)(83,196)(84,197)(85,198)(86,199)(87,200)(88,193)(89,223)(90,224)(91,217)(92,218)(93,219)(94,220)(95,221)(96,222)(97,136)(98,129)(99,130)(100,131)(101,132)(102,133)(103,134)(104,135)(105,229)(106,230)(107,231)(108,232)(109,225)(110,226)(111,227)(112,228)(121,143)(122,144)(123,137)(124,138)(125,139)(126,140)(127,141)(128,142)(145,208)(146,201)(147,202)(148,203)(149,204)(150,205)(151,206)(152,207)(161,190)(162,191)(163,192)(164,185)(165,186)(166,187)(167,188)(168,189)(169,212)(170,213)(171,214)(172,215)(173,216)(174,209)(175,210)(176,211), (1,115,40,5,119,36)(2,116,33,6,120,37)(3,117,34,7,113,38)(4,118,35,8,114,39)(9,50,66,13,54,70)(10,51,67,14,55,71)(11,52,68,15,56,72)(12,53,69,16,49,65)(17,63,197,21,59,193)(18,64,198,22,60,194)(19,57,199,23,61,195)(20,58,200,24,62,196)(25,158,220,29,154,224)(26,159,221,30,155,217)(27,160,222,31,156,218)(28,153,223,32,157,219)(41,83,233,45,87,237)(42,84,234,46,88,238)(43,85,235,47,81,239)(44,86,236,48,82,240)(73,90,184,77,94,180)(74,91,177,78,95,181)(75,92,178,79,96,182)(76,93,179,80,89,183)(97,225,137,101,229,141)(98,226,138,102,230,142)(99,227,139,103,231,143)(100,228,140,104,232,144)(105,127,136,109,123,132)(106,128,129,110,124,133)(107,121,130,111,125,134)(108,122,131,112,126,135)(145,166,175,149,162,171)(146,167,176,150,163,172)(147,168,169,151,164,173)(148,161,170,152,165,174)(185,216,202,189,212,206)(186,209,203,190,213,207)(187,210,204,191,214,208)(188,211,205,192,215,201), (1,36)(2,33)(3,38)(4,35)(5,40)(6,37)(7,34)(8,39)(9,50)(10,55)(11,52)(12,49)(13,54)(14,51)(15,56)(16,53)(17,197)(18,194)(19,199)(20,196)(21,193)(22,198)(23,195)(24,200)(26,30)(28,32)(41,45)(43,47)(58,62)(60,64)(66,70)(68,72)(73,94)(74,91)(75,96)(76,93)(77,90)(78,95)(79,92)(80,89)(81,239)(82,236)(83,233)(84,238)(85,235)(86,240)(87,237)(88,234)(97,229)(98,226)(99,231)(100,228)(101,225)(102,230)(103,227)(104,232)(105,136)(106,133)(107,130)(108,135)(109,132)(110,129)(111,134)(112,131)(113,117)(115,119)(122,126)(124,128)(138,142)(140,144)(145,175)(146,172)(147,169)(148,174)(149,171)(150,176)(151,173)(152,170)(153,223)(154,220)(155,217)(156,222)(157,219)(158,224)(159,221)(160,218)(161,165)(163,167)(177,181)(179,183)(186,190)(188,192)(201,215)(202,212)(203,209)(204,214)(205,211)(206,216)(207,213)(208,210), (1,203,20,89,228)(2,90,204,229,21)(3,230,91,22,205)(4,23,231,206,92)(5,207,24,93,232)(6,94,208,225,17)(7,226,95,18,201)(8,19,227,202,96)(9,133,155,85,176)(10,86,134,169,156)(11,170,87,157,135)(12,158,171,136,88)(13,129,159,81,172)(14,82,130,173,160)(15,174,83,153,131)(16,154,175,132,84)(25,162,127,46,65)(26,47,163,66,128)(27,67,48,121,164)(28,122,68,165,41)(29,166,123,42,69)(30,43,167,70,124)(31,71,44,125,168)(32,126,72,161,45)(33,77,214,97,193)(34,98,78,194,215)(35,195,99,216,79)(36,209,196,80,100)(37,73,210,101,197)(38,102,74,198,211)(39,199,103,212,75)(40,213,200,76,104)(49,224,149,105,234)(50,106,217,235,150)(51,236,107,151,218)(52,152,237,219,108)(53,220,145,109,238)(54,110,221,239,146)(55,240,111,147,222)(56,148,233,223,112)(57,139,189,182,114)(58,183,140,115,190)(59,116,184,191,141)(60,192,117,142,177)(61,143,185,178,118)(62,179,144,119,186)(63,120,180,187,137)(64,188,113,138,181), (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( (1,56)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,38)(10,39)(11,40)(12,33)(13,34)(14,35)(15,36)(16,37)(17,238)(18,239)(19,240)(20,233)(21,234)(22,235)(23,236)(24,237)(25,184)(26,177)(27,178)(28,179)(29,180)(30,181)(31,182)(32,183)(41,62)(42,63)(43,64)(44,57)(45,58)(46,59)(47,60)(48,61)(65,116)(66,117)(67,118)(68,119)(69,120)(70,113)(71,114)(72,115)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,153)(81,194)(82,195)(83,196)(84,197)(85,198)(86,199)(87,200)(88,193)(89,223)(90,224)(91,217)(92,218)(93,219)(94,220)(95,221)(96,222)(97,136)(98,129)(99,130)(100,131)(101,132)(102,133)(103,134)(104,135)(105,229)(106,230)(107,231)(108,232)(109,225)(110,226)(111,227)(112,228)(121,143)(122,144)(123,137)(124,138)(125,139)(126,140)(127,141)(128,142)(145,208)(146,201)(147,202)(148,203)(149,204)(150,205)(151,206)(152,207)(161,190)(162,191)(163,192)(164,185)(165,186)(166,187)(167,188)(168,189)(169,212)(170,213)(171,214)(172,215)(173,216)(174,209)(175,210)(176,211), (1,115,40,5,119,36)(2,116,33,6,120,37)(3,117,34,7,113,38)(4,118,35,8,114,39)(9,50,66,13,54,70)(10,51,67,14,55,71)(11,52,68,15,56,72)(12,53,69,16,49,65)(17,63,197,21,59,193)(18,64,198,22,60,194)(19,57,199,23,61,195)(20,58,200,24,62,196)(25,158,220,29,154,224)(26,159,221,30,155,217)(27,160,222,31,156,218)(28,153,223,32,157,219)(41,83,233,45,87,237)(42,84,234,46,88,238)(43,85,235,47,81,239)(44,86,236,48,82,240)(73,90,184,77,94,180)(74,91,177,78,95,181)(75,92,178,79,96,182)(76,93,179,80,89,183)(97,225,137,101,229,141)(98,226,138,102,230,142)(99,227,139,103,231,143)(100,228,140,104,232,144)(105,127,136,109,123,132)(106,128,129,110,124,133)(107,121,130,111,125,134)(108,122,131,112,126,135)(145,166,175,149,162,171)(146,167,176,150,163,172)(147,168,169,151,164,173)(148,161,170,152,165,174)(185,216,202,189,212,206)(186,209,203,190,213,207)(187,210,204,191,214,208)(188,211,205,192,215,201), (1,36)(2,33)(3,38)(4,35)(5,40)(6,37)(7,34)(8,39)(9,50)(10,55)(11,52)(12,49)(13,54)(14,51)(15,56)(16,53)(17,197)(18,194)(19,199)(20,196)(21,193)(22,198)(23,195)(24,200)(26,30)(28,32)(41,45)(43,47)(58,62)(60,64)(66,70)(68,72)(73,94)(74,91)(75,96)(76,93)(77,90)(78,95)(79,92)(80,89)(81,239)(82,236)(83,233)(84,238)(85,235)(86,240)(87,237)(88,234)(97,229)(98,226)(99,231)(100,228)(101,225)(102,230)(103,227)(104,232)(105,136)(106,133)(107,130)(108,135)(109,132)(110,129)(111,134)(112,131)(113,117)(115,119)(122,126)(124,128)(138,142)(140,144)(145,175)(146,172)(147,169)(148,174)(149,171)(150,176)(151,173)(152,170)(153,223)(154,220)(155,217)(156,222)(157,219)(158,224)(159,221)(160,218)(161,165)(163,167)(177,181)(179,183)(186,190)(188,192)(201,215)(202,212)(203,209)(204,214)(205,211)(206,216)(207,213)(208,210), (1,203,20,89,228)(2,90,204,229,21)(3,230,91,22,205)(4,23,231,206,92)(5,207,24,93,232)(6,94,208,225,17)(7,226,95,18,201)(8,19,227,202,96)(9,133,155,85,176)(10,86,134,169,156)(11,170,87,157,135)(12,158,171,136,88)(13,129,159,81,172)(14,82,130,173,160)(15,174,83,153,131)(16,154,175,132,84)(25,162,127,46,65)(26,47,163,66,128)(27,67,48,121,164)(28,122,68,165,41)(29,166,123,42,69)(30,43,167,70,124)(31,71,44,125,168)(32,126,72,161,45)(33,77,214,97,193)(34,98,78,194,215)(35,195,99,216,79)(36,209,196,80,100)(37,73,210,101,197)(38,102,74,198,211)(39,199,103,212,75)(40,213,200,76,104)(49,224,149,105,234)(50,106,217,235,150)(51,236,107,151,218)(52,152,237,219,108)(53,220,145,109,238)(54,110,221,239,146)(55,240,111,147,222)(56,148,233,223,112)(57,139,189,182,114)(58,183,140,115,190)(59,116,184,191,141)(60,192,117,142,177)(61,143,185,178,118)(62,179,144,119,186)(63,120,180,187,137)(64,188,113,138,181), (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([[(1,56),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,38),(10,39),(11,40),(12,33),(13,34),(14,35),(15,36),(16,37),(17,238),(18,239),(19,240),(20,233),(21,234),(22,235),(23,236),(24,237),(25,184),(26,177),(27,178),(28,179),(29,180),(30,181),(31,182),(32,183),(41,62),(42,63),(43,64),(44,57),(45,58),(46,59),(47,60),(48,61),(65,116),(66,117),(67,118),(68,119),(69,120),(70,113),(71,114),(72,115),(73,154),(74,155),(75,156),(76,157),(77,158),(78,159),(79,160),(80,153),(81,194),(82,195),(83,196),(84,197),(85,198),(86,199),(87,200),(88,193),(89,223),(90,224),(91,217),(92,218),(93,219),(94,220),(95,221),(96,222),(97,136),(98,129),(99,130),(100,131),(101,132),(102,133),(103,134),(104,135),(105,229),(106,230),(107,231),(108,232),(109,225),(110,226),(111,227),(112,228),(121,143),(122,144),(123,137),(124,138),(125,139),(126,140),(127,141),(128,142),(145,208),(146,201),(147,202),(148,203),(149,204),(150,205),(151,206),(152,207),(161,190),(162,191),(163,192),(164,185),(165,186),(166,187),(167,188),(168,189),(169,212),(170,213),(171,214),(172,215),(173,216),(174,209),(175,210),(176,211)], [(1,115,40,5,119,36),(2,116,33,6,120,37),(3,117,34,7,113,38),(4,118,35,8,114,39),(9,50,66,13,54,70),(10,51,67,14,55,71),(11,52,68,15,56,72),(12,53,69,16,49,65),(17,63,197,21,59,193),(18,64,198,22,60,194),(19,57,199,23,61,195),(20,58,200,24,62,196),(25,158,220,29,154,224),(26,159,221,30,155,217),(27,160,222,31,156,218),(28,153,223,32,157,219),(41,83,233,45,87,237),(42,84,234,46,88,238),(43,85,235,47,81,239),(44,86,236,48,82,240),(73,90,184,77,94,180),(74,91,177,78,95,181),(75,92,178,79,96,182),(76,93,179,80,89,183),(97,225,137,101,229,141),(98,226,138,102,230,142),(99,227,139,103,231,143),(100,228,140,104,232,144),(105,127,136,109,123,132),(106,128,129,110,124,133),(107,121,130,111,125,134),(108,122,131,112,126,135),(145,166,175,149,162,171),(146,167,176,150,163,172),(147,168,169,151,164,173),(148,161,170,152,165,174),(185,216,202,189,212,206),(186,209,203,190,213,207),(187,210,204,191,214,208),(188,211,205,192,215,201)], [(1,36),(2,33),(3,38),(4,35),(5,40),(6,37),(7,34),(8,39),(9,50),(10,55),(11,52),(12,49),(13,54),(14,51),(15,56),(16,53),(17,197),(18,194),(19,199),(20,196),(21,193),(22,198),(23,195),(24,200),(26,30),(28,32),(41,45),(43,47),(58,62),(60,64),(66,70),(68,72),(73,94),(74,91),(75,96),(76,93),(77,90),(78,95),(79,92),(80,89),(81,239),(82,236),(83,233),(84,238),(85,235),(86,240),(87,237),(88,234),(97,229),(98,226),(99,231),(100,228),(101,225),(102,230),(103,227),(104,232),(105,136),(106,133),(107,130),(108,135),(109,132),(110,129),(111,134),(112,131),(113,117),(115,119),(122,126),(124,128),(138,142),(140,144),(145,175),(146,172),(147,169),(148,174),(149,171),(150,176),(151,173),(152,170),(153,223),(154,220),(155,217),(156,222),(157,219),(158,224),(159,221),(160,218),(161,165),(163,167),(177,181),(179,183),(186,190),(188,192),(201,215),(202,212),(203,209),(204,214),(205,211),(206,216),(207,213),(208,210)], [(1,203,20,89,228),(2,90,204,229,21),(3,230,91,22,205),(4,23,231,206,92),(5,207,24,93,232),(6,94,208,225,17),(7,226,95,18,201),(8,19,227,202,96),(9,133,155,85,176),(10,86,134,169,156),(11,170,87,157,135),(12,158,171,136,88),(13,129,159,81,172),(14,82,130,173,160),(15,174,83,153,131),(16,154,175,132,84),(25,162,127,46,65),(26,47,163,66,128),(27,67,48,121,164),(28,122,68,165,41),(29,166,123,42,69),(30,43,167,70,124),(31,71,44,125,168),(32,126,72,161,45),(33,77,214,97,193),(34,98,78,194,215),(35,195,99,216,79),(36,209,196,80,100),(37,73,210,101,197),(38,102,74,198,211),(39,199,103,212,75),(40,213,200,76,104),(49,224,149,105,234),(50,106,217,235,150),(51,236,107,151,218),(52,152,237,219,108),(53,220,145,109,238),(54,110,221,239,146),(55,240,111,147,222),(56,148,233,223,112),(57,139,189,182,114),(58,183,140,115,190),(59,116,184,191,141),(60,192,117,142,177),(61,143,185,178,118),(62,179,144,119,186),(63,120,180,187,137),(64,188,113,138,181)], [(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)]])

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 8 8 8 type + + + + + + + + + + + - + - + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D6 D6 M4(2) C4×S3 C4×S3 C8⋊S3 F5 C2×F5 C2×F5 C22.F5 S3×F5 D6.F5 C2×S3×F5 kernel C2×D6.F5 D6.F5 C6×C5⋊C8 C2×C15⋊C8 C2×S3×Dic5 S3×Dic5 C2×Dic15 S3×C2×C10 C2×C5⋊C8 C5⋊C8 C2×Dic5 C30 Dic5 C2×C10 C10 C22×S3 D6 C2×C6 C6 C22 C2 C2 # reps 1 4 1 1 1 4 2 2 1 2 1 4 2 2 8 1 2 1 4 1 2 1

Matrix representation of C2×D6.F5 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
,
 240 1 0 0 0 0 240 0 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
,
 240 0 0 0 0 0 240 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 152 133 1 0 0 0 180 195 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 240 0 0 0 0 1 51 0 0 0 0 188 63 189 190 0 0 119 142 52 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 21 11 139 239 0 0 2 220 4 2 0 0 77 161 15 19 0 0 124 133 114 226

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],[240,240,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,240,0,0,0,0,0,1,0,0,0,0,0,0,240,0,152,180,0,0,0,240,133,195,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,188,119,0,0,240,51,63,142,0,0,0,0,189,52,0,0,0,0,190,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,2,77,124,0,0,11,220,161,133,0,0,139,4,15,114,0,0,239,2,19,226] >;

C2×D6.F5 in GAP, Magma, Sage, TeX

C_2\times D_6.F_5
% in TeX

G:=Group("C2xD6.F5");
// GroupNames label

G:=SmallGroup(480,1008);
// by ID

G=gap.SmallGroup(480,1008);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,80,1356,9414,2379]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=c^2=d^5=1,e^4=b^3,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^-1=b^3*c,e*d*e^-1=d^3>;
// generators/relations

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