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## G = C2×Dic3.F5order 480 = 25·3·5

### Direct product of C2 and Dic3.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×Dic3.F5
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C3×C5⋊C8 — Dic3.F5 — C2×Dic3.F5
 Lower central C15 — C30 — C2×Dic3.F5
 Upper central C1 — C22

Generators and relations for C2×Dic3.F5
G = < a,b,c,d,e | a2=b6=d5=1, c2=e4=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d3 >

Subgroups: 692 in 136 conjugacy classes, 54 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, D15, C30, C30, C2×M4(2), C5⋊C8, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3, C3×Dic5, D30, D30, C2×C30, C4.F5, C2×C5⋊C8, C2×C5⋊C8, C2×C4×D5, C2×C8⋊S3, C3×C5⋊C8, C15⋊C8, D30.C2, C6×Dic5, C10×Dic3, C22×D15, C2×C4.F5, Dic3.F5, C6×C5⋊C8, C2×C15⋊C8, C2×D30.C2, C2×Dic3.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, C4.F5, C22×F5, C2×C8⋊S3, S3×F5, C2×C4.F5, Dic3.F5, C2×S3×F5, C2×Dic3.F5

Smallest permutation representation of C2×Dic3.F5
On 240 points
Generators in S240
(1 186)(2 187)(3 188)(4 189)(5 190)(6 191)(7 192)(8 185)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 227)(18 228)(19 229)(20 230)(21 231)(22 232)(23 225)(24 226)(25 155)(26 156)(27 157)(28 158)(29 159)(30 160)(31 153)(32 154)(33 127)(34 128)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 49)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 96)(66 89)(67 90)(68 91)(69 92)(70 93)(71 94)(72 95)(73 184)(74 177)(75 178)(76 179)(77 180)(78 181)(79 182)(80 183)(97 234)(98 235)(99 236)(100 237)(101 238)(102 239)(103 240)(104 233)(105 214)(106 215)(107 216)(108 209)(109 210)(110 211)(111 212)(112 213)(113 220)(114 221)(115 222)(116 223)(117 224)(118 217)(119 218)(120 219)(137 166)(138 167)(139 168)(140 161)(141 162)(142 163)(143 164)(144 165)(145 170)(146 171)(147 172)(148 173)(149 174)(150 175)(151 176)(152 169)(193 202)(194 203)(195 204)(196 205)(197 206)(198 207)(199 208)(200 201)
(1 160 119 5 156 115)(2 153 120 6 157 116)(3 154 113 7 158 117)(4 155 114 8 159 118)(9 203 128 13 207 124)(10 204 121 14 208 125)(11 205 122 15 201 126)(12 206 123 16 202 127)(17 130 178 21 134 182)(18 131 179 22 135 183)(19 132 180 23 136 184)(20 133 181 24 129 177)(25 221 185 29 217 189)(26 222 186 30 218 190)(27 223 187 31 219 191)(28 224 188 32 220 192)(33 84 197 37 88 193)(34 85 198 38 81 194)(35 86 199 39 82 195)(36 87 200 40 83 196)(41 237 72 45 233 68)(42 238 65 46 234 69)(43 239 66 47 235 70)(44 240 67 48 236 71)(49 99 94 53 103 90)(50 100 95 54 104 91)(51 101 96 55 97 92)(52 102 89 56 98 93)(57 74 230 61 78 226)(58 75 231 62 79 227)(59 76 232 63 80 228)(60 77 225 64 73 229)(105 162 149 109 166 145)(106 163 150 110 167 146)(107 164 151 111 168 147)(108 165 152 112 161 148)(137 170 214 141 174 210)(138 171 215 142 175 211)(139 172 216 143 176 212)(140 173 209 144 169 213)
(1 7 5 3)(2 4 6 8)(9 110 13 106)(10 107 14 111)(11 112 15 108)(12 109 16 105)(17 44 21 48)(18 41 22 45)(19 46 23 42)(20 43 24 47)(25 219 29 223)(26 224 30 220)(27 221 31 217)(28 218 32 222)(33 137 37 141)(34 142 38 138)(35 139 39 143)(36 144 40 140)(49 227 53 231)(50 232 54 228)(51 229 55 225)(52 226 56 230)(57 89 61 93)(58 94 62 90)(59 91 63 95)(60 96 64 92)(65 136 69 132)(66 133 70 129)(67 130 71 134)(68 135 72 131)(73 97 77 101)(74 102 78 98)(75 99 79 103)(76 104 80 100)(81 211 85 215)(82 216 86 212)(83 213 87 209)(84 210 88 214)(113 156 117 160)(114 153 118 157)(115 158 119 154)(116 155 120 159)(121 168 125 164)(122 165 126 161)(123 162 127 166)(124 167 128 163)(145 206 149 202)(146 203 150 207)(147 208 151 204)(148 205 152 201)(169 200 173 196)(170 197 174 193)(171 194 175 198)(172 199 176 195)(177 239 181 235)(178 236 182 240)(179 233 183 237)(180 238 184 234)(185 187 189 191)(186 192 190 188)
(1 173 104 74 198)(2 75 174 199 97)(3 200 76 98 175)(4 99 193 176 77)(5 169 100 78 194)(6 79 170 195 101)(7 196 80 102 171)(8 103 197 172 73)(9 218 165 41 133)(10 42 219 134 166)(11 135 43 167 220)(12 168 136 221 44)(13 222 161 45 129)(14 46 223 130 162)(15 131 47 163 224)(16 164 132 217 48)(17 105 121 65 27)(18 66 106 28 122)(19 29 67 123 107)(20 124 30 108 68)(21 109 125 69 31)(22 70 110 32 126)(23 25 71 127 111)(24 128 26 112 72)(33 212 225 155 94)(34 156 213 95 226)(35 96 157 227 214)(36 228 89 215 158)(37 216 229 159 90)(38 160 209 91 230)(39 92 153 231 210)(40 232 93 211 154)(49 88 143 60 118)(50 61 81 119 144)(51 120 62 137 82)(52 138 113 83 63)(53 84 139 64 114)(54 57 85 115 140)(55 116 58 141 86)(56 142 117 87 59)(145 204 238 191 182)(146 192 205 183 239)(147 184 185 240 206)(148 233 177 207 186)(149 208 234 187 178)(150 188 201 179 235)(151 180 189 236 202)(152 237 181 203 190)
(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)| (1,186)(2,187)(3,188)(4,189)(5,190)(6,191)(7,192)(8,185)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,227)(18,228)(19,229)(20,230)(21,231)(22,232)(23,225)(24,226)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,153)(32,154)(33,127)(34,128)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,49)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,96)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95)(73,184)(74,177)(75,178)(76,179)(77,180)(78,181)(79,182)(80,183)(97,234)(98,235)(99,236)(100,237)(101,238)(102,239)(103,240)(104,233)(105,214)(106,215)(107,216)(108,209)(109,210)(110,211)(111,212)(112,213)(113,220)(114,221)(115,222)(116,223)(117,224)(118,217)(119,218)(120,219)(137,166)(138,167)(139,168)(140,161)(141,162)(142,163)(143,164)(144,165)(145,170)(146,171)(147,172)(148,173)(149,174)(150,175)(151,176)(152,169)(193,202)(194,203)(195,204)(196,205)(197,206)(198,207)(199,208)(200,201), (1,160,119,5,156,115)(2,153,120,6,157,116)(3,154,113,7,158,117)(4,155,114,8,159,118)(9,203,128,13,207,124)(10,204,121,14,208,125)(11,205,122,15,201,126)(12,206,123,16,202,127)(17,130,178,21,134,182)(18,131,179,22,135,183)(19,132,180,23,136,184)(20,133,181,24,129,177)(25,221,185,29,217,189)(26,222,186,30,218,190)(27,223,187,31,219,191)(28,224,188,32,220,192)(33,84,197,37,88,193)(34,85,198,38,81,194)(35,86,199,39,82,195)(36,87,200,40,83,196)(41,237,72,45,233,68)(42,238,65,46,234,69)(43,239,66,47,235,70)(44,240,67,48,236,71)(49,99,94,53,103,90)(50,100,95,54,104,91)(51,101,96,55,97,92)(52,102,89,56,98,93)(57,74,230,61,78,226)(58,75,231,62,79,227)(59,76,232,63,80,228)(60,77,225,64,73,229)(105,162,149,109,166,145)(106,163,150,110,167,146)(107,164,151,111,168,147)(108,165,152,112,161,148)(137,170,214,141,174,210)(138,171,215,142,175,211)(139,172,216,143,176,212)(140,173,209,144,169,213), (1,7,5,3)(2,4,6,8)(9,110,13,106)(10,107,14,111)(11,112,15,108)(12,109,16,105)(17,44,21,48)(18,41,22,45)(19,46,23,42)(20,43,24,47)(25,219,29,223)(26,224,30,220)(27,221,31,217)(28,218,32,222)(33,137,37,141)(34,142,38,138)(35,139,39,143)(36,144,40,140)(49,227,53,231)(50,232,54,228)(51,229,55,225)(52,226,56,230)(57,89,61,93)(58,94,62,90)(59,91,63,95)(60,96,64,92)(65,136,69,132)(66,133,70,129)(67,130,71,134)(68,135,72,131)(73,97,77,101)(74,102,78,98)(75,99,79,103)(76,104,80,100)(81,211,85,215)(82,216,86,212)(83,213,87,209)(84,210,88,214)(113,156,117,160)(114,153,118,157)(115,158,119,154)(116,155,120,159)(121,168,125,164)(122,165,126,161)(123,162,127,166)(124,167,128,163)(145,206,149,202)(146,203,150,207)(147,208,151,204)(148,205,152,201)(169,200,173,196)(170,197,174,193)(171,194,175,198)(172,199,176,195)(177,239,181,235)(178,236,182,240)(179,233,183,237)(180,238,184,234)(185,187,189,191)(186,192,190,188), (1,173,104,74,198)(2,75,174,199,97)(3,200,76,98,175)(4,99,193,176,77)(5,169,100,78,194)(6,79,170,195,101)(7,196,80,102,171)(8,103,197,172,73)(9,218,165,41,133)(10,42,219,134,166)(11,135,43,167,220)(12,168,136,221,44)(13,222,161,45,129)(14,46,223,130,162)(15,131,47,163,224)(16,164,132,217,48)(17,105,121,65,27)(18,66,106,28,122)(19,29,67,123,107)(20,124,30,108,68)(21,109,125,69,31)(22,70,110,32,126)(23,25,71,127,111)(24,128,26,112,72)(33,212,225,155,94)(34,156,213,95,226)(35,96,157,227,214)(36,228,89,215,158)(37,216,229,159,90)(38,160,209,91,230)(39,92,153,231,210)(40,232,93,211,154)(49,88,143,60,118)(50,61,81,119,144)(51,120,62,137,82)(52,138,113,83,63)(53,84,139,64,114)(54,57,85,115,140)(55,116,58,141,86)(56,142,117,87,59)(145,204,238,191,182)(146,192,205,183,239)(147,184,185,240,206)(148,233,177,207,186)(149,208,234,187,178)(150,188,201,179,235)(151,180,189,236,202)(152,237,181,203,190), (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( (1,186)(2,187)(3,188)(4,189)(5,190)(6,191)(7,192)(8,185)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,227)(18,228)(19,229)(20,230)(21,231)(22,232)(23,225)(24,226)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,153)(32,154)(33,127)(34,128)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,49)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,96)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95)(73,184)(74,177)(75,178)(76,179)(77,180)(78,181)(79,182)(80,183)(97,234)(98,235)(99,236)(100,237)(101,238)(102,239)(103,240)(104,233)(105,214)(106,215)(107,216)(108,209)(109,210)(110,211)(111,212)(112,213)(113,220)(114,221)(115,222)(116,223)(117,224)(118,217)(119,218)(120,219)(137,166)(138,167)(139,168)(140,161)(141,162)(142,163)(143,164)(144,165)(145,170)(146,171)(147,172)(148,173)(149,174)(150,175)(151,176)(152,169)(193,202)(194,203)(195,204)(196,205)(197,206)(198,207)(199,208)(200,201), (1,160,119,5,156,115)(2,153,120,6,157,116)(3,154,113,7,158,117)(4,155,114,8,159,118)(9,203,128,13,207,124)(10,204,121,14,208,125)(11,205,122,15,201,126)(12,206,123,16,202,127)(17,130,178,21,134,182)(18,131,179,22,135,183)(19,132,180,23,136,184)(20,133,181,24,129,177)(25,221,185,29,217,189)(26,222,186,30,218,190)(27,223,187,31,219,191)(28,224,188,32,220,192)(33,84,197,37,88,193)(34,85,198,38,81,194)(35,86,199,39,82,195)(36,87,200,40,83,196)(41,237,72,45,233,68)(42,238,65,46,234,69)(43,239,66,47,235,70)(44,240,67,48,236,71)(49,99,94,53,103,90)(50,100,95,54,104,91)(51,101,96,55,97,92)(52,102,89,56,98,93)(57,74,230,61,78,226)(58,75,231,62,79,227)(59,76,232,63,80,228)(60,77,225,64,73,229)(105,162,149,109,166,145)(106,163,150,110,167,146)(107,164,151,111,168,147)(108,165,152,112,161,148)(137,170,214,141,174,210)(138,171,215,142,175,211)(139,172,216,143,176,212)(140,173,209,144,169,213), (1,7,5,3)(2,4,6,8)(9,110,13,106)(10,107,14,111)(11,112,15,108)(12,109,16,105)(17,44,21,48)(18,41,22,45)(19,46,23,42)(20,43,24,47)(25,219,29,223)(26,224,30,220)(27,221,31,217)(28,218,32,222)(33,137,37,141)(34,142,38,138)(35,139,39,143)(36,144,40,140)(49,227,53,231)(50,232,54,228)(51,229,55,225)(52,226,56,230)(57,89,61,93)(58,94,62,90)(59,91,63,95)(60,96,64,92)(65,136,69,132)(66,133,70,129)(67,130,71,134)(68,135,72,131)(73,97,77,101)(74,102,78,98)(75,99,79,103)(76,104,80,100)(81,211,85,215)(82,216,86,212)(83,213,87,209)(84,210,88,214)(113,156,117,160)(114,153,118,157)(115,158,119,154)(116,155,120,159)(121,168,125,164)(122,165,126,161)(123,162,127,166)(124,167,128,163)(145,206,149,202)(146,203,150,207)(147,208,151,204)(148,205,152,201)(169,200,173,196)(170,197,174,193)(171,194,175,198)(172,199,176,195)(177,239,181,235)(178,236,182,240)(179,233,183,237)(180,238,184,234)(185,187,189,191)(186,192,190,188), (1,173,104,74,198)(2,75,174,199,97)(3,200,76,98,175)(4,99,193,176,77)(5,169,100,78,194)(6,79,170,195,101)(7,196,80,102,171)(8,103,197,172,73)(9,218,165,41,133)(10,42,219,134,166)(11,135,43,167,220)(12,168,136,221,44)(13,222,161,45,129)(14,46,223,130,162)(15,131,47,163,224)(16,164,132,217,48)(17,105,121,65,27)(18,66,106,28,122)(19,29,67,123,107)(20,124,30,108,68)(21,109,125,69,31)(22,70,110,32,126)(23,25,71,127,111)(24,128,26,112,72)(33,212,225,155,94)(34,156,213,95,226)(35,96,157,227,214)(36,228,89,215,158)(37,216,229,159,90)(38,160,209,91,230)(39,92,153,231,210)(40,232,93,211,154)(49,88,143,60,118)(50,61,81,119,144)(51,120,62,137,82)(52,138,113,83,63)(53,84,139,64,114)(54,57,85,115,140)(55,116,58,141,86)(56,142,117,87,59)(145,204,238,191,182)(146,192,205,183,239)(147,184,185,240,206)(148,233,177,207,186)(149,208,234,187,178)(150,188,201,179,235)(151,180,189,236,202)(152,237,181,203,190), (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([[(1,186),(2,187),(3,188),(4,189),(5,190),(6,191),(7,192),(8,185),(9,81),(10,82),(11,83),(12,84),(13,85),(14,86),(15,87),(16,88),(17,227),(18,228),(19,229),(20,230),(21,231),(22,232),(23,225),(24,226),(25,155),(26,156),(27,157),(28,158),(29,159),(30,160),(31,153),(32,154),(33,127),(34,128),(35,121),(36,122),(37,123),(38,124),(39,125),(40,126),(41,50),(42,51),(43,52),(44,53),(45,54),(46,55),(47,56),(48,49),(57,129),(58,130),(59,131),(60,132),(61,133),(62,134),(63,135),(64,136),(65,96),(66,89),(67,90),(68,91),(69,92),(70,93),(71,94),(72,95),(73,184),(74,177),(75,178),(76,179),(77,180),(78,181),(79,182),(80,183),(97,234),(98,235),(99,236),(100,237),(101,238),(102,239),(103,240),(104,233),(105,214),(106,215),(107,216),(108,209),(109,210),(110,211),(111,212),(112,213),(113,220),(114,221),(115,222),(116,223),(117,224),(118,217),(119,218),(120,219),(137,166),(138,167),(139,168),(140,161),(141,162),(142,163),(143,164),(144,165),(145,170),(146,171),(147,172),(148,173),(149,174),(150,175),(151,176),(152,169),(193,202),(194,203),(195,204),(196,205),(197,206),(198,207),(199,208),(200,201)], [(1,160,119,5,156,115),(2,153,120,6,157,116),(3,154,113,7,158,117),(4,155,114,8,159,118),(9,203,128,13,207,124),(10,204,121,14,208,125),(11,205,122,15,201,126),(12,206,123,16,202,127),(17,130,178,21,134,182),(18,131,179,22,135,183),(19,132,180,23,136,184),(20,133,181,24,129,177),(25,221,185,29,217,189),(26,222,186,30,218,190),(27,223,187,31,219,191),(28,224,188,32,220,192),(33,84,197,37,88,193),(34,85,198,38,81,194),(35,86,199,39,82,195),(36,87,200,40,83,196),(41,237,72,45,233,68),(42,238,65,46,234,69),(43,239,66,47,235,70),(44,240,67,48,236,71),(49,99,94,53,103,90),(50,100,95,54,104,91),(51,101,96,55,97,92),(52,102,89,56,98,93),(57,74,230,61,78,226),(58,75,231,62,79,227),(59,76,232,63,80,228),(60,77,225,64,73,229),(105,162,149,109,166,145),(106,163,150,110,167,146),(107,164,151,111,168,147),(108,165,152,112,161,148),(137,170,214,141,174,210),(138,171,215,142,175,211),(139,172,216,143,176,212),(140,173,209,144,169,213)], [(1,7,5,3),(2,4,6,8),(9,110,13,106),(10,107,14,111),(11,112,15,108),(12,109,16,105),(17,44,21,48),(18,41,22,45),(19,46,23,42),(20,43,24,47),(25,219,29,223),(26,224,30,220),(27,221,31,217),(28,218,32,222),(33,137,37,141),(34,142,38,138),(35,139,39,143),(36,144,40,140),(49,227,53,231),(50,232,54,228),(51,229,55,225),(52,226,56,230),(57,89,61,93),(58,94,62,90),(59,91,63,95),(60,96,64,92),(65,136,69,132),(66,133,70,129),(67,130,71,134),(68,135,72,131),(73,97,77,101),(74,102,78,98),(75,99,79,103),(76,104,80,100),(81,211,85,215),(82,216,86,212),(83,213,87,209),(84,210,88,214),(113,156,117,160),(114,153,118,157),(115,158,119,154),(116,155,120,159),(121,168,125,164),(122,165,126,161),(123,162,127,166),(124,167,128,163),(145,206,149,202),(146,203,150,207),(147,208,151,204),(148,205,152,201),(169,200,173,196),(170,197,174,193),(171,194,175,198),(172,199,176,195),(177,239,181,235),(178,236,182,240),(179,233,183,237),(180,238,184,234),(185,187,189,191),(186,192,190,188)], [(1,173,104,74,198),(2,75,174,199,97),(3,200,76,98,175),(4,99,193,176,77),(5,169,100,78,194),(6,79,170,195,101),(7,196,80,102,171),(8,103,197,172,73),(9,218,165,41,133),(10,42,219,134,166),(11,135,43,167,220),(12,168,136,221,44),(13,222,161,45,129),(14,46,223,130,162),(15,131,47,163,224),(16,164,132,217,48),(17,105,121,65,27),(18,66,106,28,122),(19,29,67,123,107),(20,124,30,108,68),(21,109,125,69,31),(22,70,110,32,126),(23,25,71,127,111),(24,128,26,112,72),(33,212,225,155,94),(34,156,213,95,226),(35,96,157,227,214),(36,228,89,215,158),(37,216,229,159,90),(38,160,209,91,230),(39,92,153,231,210),(40,232,93,211,154),(49,88,143,60,118),(50,61,81,119,144),(51,120,62,137,82),(52,138,113,83,63),(53,84,139,64,114),(54,57,85,115,140),(55,116,58,141,86),(56,142,117,87,59),(145,204,238,191,182),(146,192,205,183,239),(147,184,185,240,206),(148,233,177,207,186),(149,208,234,187,178),(150,188,201,179,235),(151,180,189,236,202),(152,237,181,203,190)], [(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 12A 12B 12C 12D 15 20A 20B 20C 20D 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 12 12 12 12 15 20 20 20 20 24 ··· 24 30 30 30 size 1 1 1 1 30 30 2 5 5 5 5 6 6 4 2 2 2 10 10 10 10 30 30 30 30 4 4 4 10 10 10 10 8 12 12 12 12 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 C4.F5 S3×F5 Dic3.F5 C2×S3×F5 kernel C2×Dic3.F5 Dic3.F5 C6×C5⋊C8 C2×C15⋊C8 C2×D30.C2 D30.C2 C10×Dic3 C22×D15 C2×C5⋊C8 C5⋊C8 C2×Dic5 C30 Dic5 C2×C10 C10 C2×Dic3 Dic3 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×Dic3.F5 in GL8(𝔽241)

 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 240 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 64 0 0 0 0 0 0 0 0 177 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 240 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 240 0 0 0 0 1 0 0 240 0 0 0 0 0 1 0 240 0 0 0 0 0 0 1 240
,
 0 1 0 0 0 0 0 0 177 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 157 84 17 0 0 0 0 0 174 84 0 157 0 0 0 0 157 0 84 174 0 0 0 0 0 17 84 157

G:=sub<GL(8,GF(241))| [240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[64,0,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240],[0,177,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,157,174,157,0,0,0,0,0,84,84,0,17,0,0,0,0,17,0,84,84,0,0,0,0,0,157,174,157] >;

C2×Dic3.F5 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3.F_5
% in TeX

G:=Group("C2xDic3.F5");
// GroupNames label

G:=SmallGroup(480,1009);
// by ID

G=gap.SmallGroup(480,1009);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,120,80,1356,9414,2379]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=d^5=1,c^2=e^4=b^3,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,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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