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G = C6×C22.F5order 480 = 25·3·5

Direct product of C6 and C22.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C6×C22.F5
 Chief series C1 — C5 — C10 — Dic5 — C3×Dic5 — C3×C5⋊C8 — C6×C5⋊C8 — C6×C22.F5
 Lower central C5 — C10 — C6×C22.F5
 Upper central C1 — C2×C6 — C22×C6

Generators and relations for C6×C22.F5
G = < a,b,c,d,e | a6=b2=c2=d5=1, e4=c, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 328 in 136 conjugacy classes, 76 normal (28 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C8, C2×C4, C23, C10, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, Dic5, C2×C10, C2×C10, C2×C10, C24, C2×C12, C22×C6, C30, C30, C30, C2×M4(2), C5⋊C8, C2×Dic5, C2×Dic5, C22×C10, C2×C24, C3×M4(2), C22×C12, C3×Dic5, C3×Dic5, C2×C30, C2×C30, C2×C30, C2×C5⋊C8, C22.F5, C22×Dic5, C6×M4(2), C3×C5⋊C8, C6×Dic5, C6×Dic5, C22×C30, C2×C22.F5, C6×C5⋊C8, C3×C22.F5, C2×C6×Dic5, C6×C22.F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, M4(2), C22×C4, F5, C2×C12, C22×C6, C2×M4(2), C2×F5, C3×M4(2), C22×C12, C3×F5, C22.F5, C22×F5, C6×M4(2), C6×F5, C2×C22.F5, C3×C22.F5, C2×C6×F5, C6×C22.F5

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 type + + + + + + - image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 M4(2) C3×M4(2) F5 C2×F5 C3×F5 C22.F5 C6×F5 C3×C22.F5 kernel C6×C22.F5 C6×C5⋊C8 C3×C22.F5 C2×C6×Dic5 C2×C22.F5 C6×Dic5 C22×C30 C2×C5⋊C8 C22.F5 C22×Dic5 C2×Dic5 C22×C10 C30 C10 C22×C6 C2×C6 C23 C6 C22 C2 # reps 1 2 4 1 2 6 2 4 8 2 12 4 4 8 1 3 2 4 6 8

Matrix representation of C6×C22.F5 in GL6(𝔽241)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 225 0 0 0 0 0 0 225 0 0 0 0 0 0 225 0 0 0 0 0 0 225
,
 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 0 239 102 240 0 0 0 39 204 0 240
,
 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 0 0 0 0 0 0 240
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 189 1 0 0 0 0 240 0 0 0 0 0 103 0 52 189 0 0 40 0 52 240
,
 0 1 0 0 0 0 64 0 0 0 0 0 0 0 239 102 239 0 0 0 39 204 0 239 0 0 104 137 2 139 0 0 3 102 202 37

G:=sub<GL(6,GF(241))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,225,0,0,0,0,0,0,225,0,0,0,0,0,0,225,0,0,0,0,0,0,225],[240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,239,39,0,0,0,1,102,204,0,0,0,0,240,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,189,240,103,40,0,0,1,0,0,0,0,0,0,0,52,52,0,0,0,0,189,240],[0,64,0,0,0,0,1,0,0,0,0,0,0,0,239,39,104,3,0,0,102,204,137,102,0,0,239,0,2,202,0,0,0,239,139,37] >;

C6×C22.F5 in GAP, Magma, Sage, TeX

C_6\times C_2^2.F_5
% in TeX

G:=Group("C6xC2^2.F5");
// GroupNames label

G:=SmallGroup(480,1058);
// by ID

G=gap.SmallGroup(480,1058);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,1094,102,9414,818]);
// Polycyclic

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

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