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## G = C3×Q8.F5order 480 = 25·3·5

### Direct product of C3 and Q8.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×Q8.F5
 Chief series C1 — C5 — C10 — Dic5 — C3×Dic5 — C3×C5⋊C8 — C3×D5⋊C8 — C3×Q8.F5
 Lower central C5 — C10 — C3×Q8.F5
 Upper central C1 — C6 — C3×Q8

Generators and relations for C3×Q8.F5
G = < a,b,c,d,e | a3=b4=d5=1, c2=e4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 360 in 124 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, Q8, D5, C10, C12, C12, C2×C6, C15, C2×C8, M4(2), C4○D4, Dic5, C20, D10, C24, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C8○D4, C5⋊C8, C5⋊C8, C4×D5, D20, C5×Q8, C2×C24, C3×M4(2), C3×C4○D4, C3×Dic5, C60, C6×D5, D5⋊C8, C4.F5, Q82D5, C3×C8○D4, C3×C5⋊C8, C3×C5⋊C8, D5×C12, C3×D20, Q8×C15, Q8.F5, C3×D5⋊C8, C3×C4.F5, C3×Q82D5, C3×Q8.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, Q8.F5, C2×C6×F5, C3×Q8.F5

Smallest permutation representation of C3×Q8.F5
On 240 points
Generators in S240
(1 101 145)(2 102 146)(3 103 147)(4 104 148)(5 97 149)(6 98 150)(7 99 151)(8 100 152)(9 57 111)(10 58 112)(11 59 105)(12 60 106)(13 61 107)(14 62 108)(15 63 109)(16 64 110)(17 155 207)(18 156 208)(19 157 201)(20 158 202)(21 159 203)(22 160 204)(23 153 205)(24 154 206)(25 79 215)(26 80 216)(27 73 209)(28 74 210)(29 75 211)(30 76 212)(31 77 213)(32 78 214)(33 169 85)(34 170 86)(35 171 87)(36 172 88)(37 173 81)(38 174 82)(39 175 83)(40 176 84)(41 95 137)(42 96 138)(43 89 139)(44 90 140)(45 91 141)(46 92 142)(47 93 143)(48 94 144)(49 191 237)(50 192 238)(51 185 239)(52 186 240)(53 187 233)(54 188 234)(55 189 235)(56 190 236)(65 119 161)(66 120 162)(67 113 163)(68 114 164)(69 115 165)(70 116 166)(71 117 167)(72 118 168)(121 197 217)(122 198 218)(123 199 219)(124 200 220)(125 193 221)(126 194 222)(127 195 223)(128 196 224)(129 179 231)(130 180 232)(131 181 225)(132 182 226)(133 183 227)(134 184 228)(135 177 229)(136 178 230)
(1 55 5 51)(2 56 6 52)(3 49 7 53)(4 50 8 54)(9 155 13 159)(10 156 14 160)(11 157 15 153)(12 158 16 154)(17 107 21 111)(18 108 22 112)(19 109 23 105)(20 110 24 106)(25 115 29 119)(26 116 30 120)(27 117 31 113)(28 118 32 114)(33 217 37 221)(34 218 38 222)(35 219 39 223)(36 220 40 224)(41 131 45 135)(42 132 46 136)(43 133 47 129)(44 134 48 130)(57 207 61 203)(58 208 62 204)(59 201 63 205)(60 202 64 206)(65 215 69 211)(66 216 70 212)(67 209 71 213)(68 210 72 214)(73 167 77 163)(74 168 78 164)(75 161 79 165)(76 162 80 166)(81 193 85 197)(82 194 86 198)(83 195 87 199)(84 196 88 200)(89 183 93 179)(90 184 94 180)(91 177 95 181)(92 178 96 182)(97 185 101 189)(98 186 102 190)(99 187 103 191)(100 188 104 192)(121 173 125 169)(122 174 126 170)(123 175 127 171)(124 176 128 172)(137 225 141 229)(138 226 142 230)(139 227 143 231)(140 228 144 232)(145 235 149 239)(146 236 150 240)(147 237 151 233)(148 238 152 234)
(1 7 5 3)(2 8 6 4)(9 135 13 131)(10 136 14 132)(11 129 15 133)(12 130 16 134)(17 141 21 137)(18 142 22 138)(19 143 23 139)(20 144 24 140)(25 87 29 83)(26 88 30 84)(27 81 31 85)(28 82 32 86)(33 73 37 77)(34 74 38 78)(35 75 39 79)(36 76 40 80)(41 155 45 159)(42 156 46 160)(43 157 47 153)(44 158 48 154)(49 51 53 55)(50 52 54 56)(57 177 61 181)(58 178 62 182)(59 179 63 183)(60 180 64 184)(65 123 69 127)(66 124 70 128)(67 125 71 121)(68 126 72 122)(89 201 93 205)(90 202 94 206)(91 203 95 207)(92 204 96 208)(97 103 101 99)(98 104 102 100)(105 231 109 227)(106 232 110 228)(107 225 111 229)(108 226 112 230)(113 193 117 197)(114 194 118 198)(115 195 119 199)(116 196 120 200)(145 151 149 147)(146 152 150 148)(161 219 165 223)(162 220 166 224)(163 221 167 217)(164 222 168 218)(169 209 173 213)(170 210 174 214)(171 211 175 215)(172 212 176 216)(185 187 189 191)(186 188 190 192)(233 235 237 239)(234 236 238 240)
(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)| (1,101,145)(2,102,146)(3,103,147)(4,104,148)(5,97,149)(6,98,150)(7,99,151)(8,100,152)(9,57,111)(10,58,112)(11,59,105)(12,60,106)(13,61,107)(14,62,108)(15,63,109)(16,64,110)(17,155,207)(18,156,208)(19,157,201)(20,158,202)(21,159,203)(22,160,204)(23,153,205)(24,154,206)(25,79,215)(26,80,216)(27,73,209)(28,74,210)(29,75,211)(30,76,212)(31,77,213)(32,78,214)(33,169,85)(34,170,86)(35,171,87)(36,172,88)(37,173,81)(38,174,82)(39,175,83)(40,176,84)(41,95,137)(42,96,138)(43,89,139)(44,90,140)(45,91,141)(46,92,142)(47,93,143)(48,94,144)(49,191,237)(50,192,238)(51,185,239)(52,186,240)(53,187,233)(54,188,234)(55,189,235)(56,190,236)(65,119,161)(66,120,162)(67,113,163)(68,114,164)(69,115,165)(70,116,166)(71,117,167)(72,118,168)(121,197,217)(122,198,218)(123,199,219)(124,200,220)(125,193,221)(126,194,222)(127,195,223)(128,196,224)(129,179,231)(130,180,232)(131,181,225)(132,182,226)(133,183,227)(134,184,228)(135,177,229)(136,178,230), (1,55,5,51)(2,56,6,52)(3,49,7,53)(4,50,8,54)(9,155,13,159)(10,156,14,160)(11,157,15,153)(12,158,16,154)(17,107,21,111)(18,108,22,112)(19,109,23,105)(20,110,24,106)(25,115,29,119)(26,116,30,120)(27,117,31,113)(28,118,32,114)(33,217,37,221)(34,218,38,222)(35,219,39,223)(36,220,40,224)(41,131,45,135)(42,132,46,136)(43,133,47,129)(44,134,48,130)(57,207,61,203)(58,208,62,204)(59,201,63,205)(60,202,64,206)(65,215,69,211)(66,216,70,212)(67,209,71,213)(68,210,72,214)(73,167,77,163)(74,168,78,164)(75,161,79,165)(76,162,80,166)(81,193,85,197)(82,194,86,198)(83,195,87,199)(84,196,88,200)(89,183,93,179)(90,184,94,180)(91,177,95,181)(92,178,96,182)(97,185,101,189)(98,186,102,190)(99,187,103,191)(100,188,104,192)(121,173,125,169)(122,174,126,170)(123,175,127,171)(124,176,128,172)(137,225,141,229)(138,226,142,230)(139,227,143,231)(140,228,144,232)(145,235,149,239)(146,236,150,240)(147,237,151,233)(148,238,152,234), (1,7,5,3)(2,8,6,4)(9,135,13,131)(10,136,14,132)(11,129,15,133)(12,130,16,134)(17,141,21,137)(18,142,22,138)(19,143,23,139)(20,144,24,140)(25,87,29,83)(26,88,30,84)(27,81,31,85)(28,82,32,86)(33,73,37,77)(34,74,38,78)(35,75,39,79)(36,76,40,80)(41,155,45,159)(42,156,46,160)(43,157,47,153)(44,158,48,154)(49,51,53,55)(50,52,54,56)(57,177,61,181)(58,178,62,182)(59,179,63,183)(60,180,64,184)(65,123,69,127)(66,124,70,128)(67,125,71,121)(68,126,72,122)(89,201,93,205)(90,202,94,206)(91,203,95,207)(92,204,96,208)(97,103,101,99)(98,104,102,100)(105,231,109,227)(106,232,110,228)(107,225,111,229)(108,226,112,230)(113,193,117,197)(114,194,118,198)(115,195,119,199)(116,196,120,200)(145,151,149,147)(146,152,150,148)(161,219,165,223)(162,220,166,224)(163,221,167,217)(164,222,168,218)(169,209,173,213)(170,210,174,214)(171,211,175,215)(172,212,176,216)(185,187,189,191)(186,188,190,192)(233,235,237,239)(234,236,238,240), (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( (1,101,145)(2,102,146)(3,103,147)(4,104,148)(5,97,149)(6,98,150)(7,99,151)(8,100,152)(9,57,111)(10,58,112)(11,59,105)(12,60,106)(13,61,107)(14,62,108)(15,63,109)(16,64,110)(17,155,207)(18,156,208)(19,157,201)(20,158,202)(21,159,203)(22,160,204)(23,153,205)(24,154,206)(25,79,215)(26,80,216)(27,73,209)(28,74,210)(29,75,211)(30,76,212)(31,77,213)(32,78,214)(33,169,85)(34,170,86)(35,171,87)(36,172,88)(37,173,81)(38,174,82)(39,175,83)(40,176,84)(41,95,137)(42,96,138)(43,89,139)(44,90,140)(45,91,141)(46,92,142)(47,93,143)(48,94,144)(49,191,237)(50,192,238)(51,185,239)(52,186,240)(53,187,233)(54,188,234)(55,189,235)(56,190,236)(65,119,161)(66,120,162)(67,113,163)(68,114,164)(69,115,165)(70,116,166)(71,117,167)(72,118,168)(121,197,217)(122,198,218)(123,199,219)(124,200,220)(125,193,221)(126,194,222)(127,195,223)(128,196,224)(129,179,231)(130,180,232)(131,181,225)(132,182,226)(133,183,227)(134,184,228)(135,177,229)(136,178,230), (1,55,5,51)(2,56,6,52)(3,49,7,53)(4,50,8,54)(9,155,13,159)(10,156,14,160)(11,157,15,153)(12,158,16,154)(17,107,21,111)(18,108,22,112)(19,109,23,105)(20,110,24,106)(25,115,29,119)(26,116,30,120)(27,117,31,113)(28,118,32,114)(33,217,37,221)(34,218,38,222)(35,219,39,223)(36,220,40,224)(41,131,45,135)(42,132,46,136)(43,133,47,129)(44,134,48,130)(57,207,61,203)(58,208,62,204)(59,201,63,205)(60,202,64,206)(65,215,69,211)(66,216,70,212)(67,209,71,213)(68,210,72,214)(73,167,77,163)(74,168,78,164)(75,161,79,165)(76,162,80,166)(81,193,85,197)(82,194,86,198)(83,195,87,199)(84,196,88,200)(89,183,93,179)(90,184,94,180)(91,177,95,181)(92,178,96,182)(97,185,101,189)(98,186,102,190)(99,187,103,191)(100,188,104,192)(121,173,125,169)(122,174,126,170)(123,175,127,171)(124,176,128,172)(137,225,141,229)(138,226,142,230)(139,227,143,231)(140,228,144,232)(145,235,149,239)(146,236,150,240)(147,237,151,233)(148,238,152,234), (1,7,5,3)(2,8,6,4)(9,135,13,131)(10,136,14,132)(11,129,15,133)(12,130,16,134)(17,141,21,137)(18,142,22,138)(19,143,23,139)(20,144,24,140)(25,87,29,83)(26,88,30,84)(27,81,31,85)(28,82,32,86)(33,73,37,77)(34,74,38,78)(35,75,39,79)(36,76,40,80)(41,155,45,159)(42,156,46,160)(43,157,47,153)(44,158,48,154)(49,51,53,55)(50,52,54,56)(57,177,61,181)(58,178,62,182)(59,179,63,183)(60,180,64,184)(65,123,69,127)(66,124,70,128)(67,125,71,121)(68,126,72,122)(89,201,93,205)(90,202,94,206)(91,203,95,207)(92,204,96,208)(97,103,101,99)(98,104,102,100)(105,231,109,227)(106,232,110,228)(107,225,111,229)(108,226,112,230)(113,193,117,197)(114,194,118,198)(115,195,119,199)(116,196,120,200)(145,151,149,147)(146,152,150,148)(161,219,165,223)(162,220,166,224)(163,221,167,217)(164,222,168,218)(169,209,173,213)(170,210,174,214)(171,211,175,215)(172,212,176,216)(185,187,189,191)(186,188,190,192)(233,235,237,239)(234,236,238,240), (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([[(1,101,145),(2,102,146),(3,103,147),(4,104,148),(5,97,149),(6,98,150),(7,99,151),(8,100,152),(9,57,111),(10,58,112),(11,59,105),(12,60,106),(13,61,107),(14,62,108),(15,63,109),(16,64,110),(17,155,207),(18,156,208),(19,157,201),(20,158,202),(21,159,203),(22,160,204),(23,153,205),(24,154,206),(25,79,215),(26,80,216),(27,73,209),(28,74,210),(29,75,211),(30,76,212),(31,77,213),(32,78,214),(33,169,85),(34,170,86),(35,171,87),(36,172,88),(37,173,81),(38,174,82),(39,175,83),(40,176,84),(41,95,137),(42,96,138),(43,89,139),(44,90,140),(45,91,141),(46,92,142),(47,93,143),(48,94,144),(49,191,237),(50,192,238),(51,185,239),(52,186,240),(53,187,233),(54,188,234),(55,189,235),(56,190,236),(65,119,161),(66,120,162),(67,113,163),(68,114,164),(69,115,165),(70,116,166),(71,117,167),(72,118,168),(121,197,217),(122,198,218),(123,199,219),(124,200,220),(125,193,221),(126,194,222),(127,195,223),(128,196,224),(129,179,231),(130,180,232),(131,181,225),(132,182,226),(133,183,227),(134,184,228),(135,177,229),(136,178,230)], [(1,55,5,51),(2,56,6,52),(3,49,7,53),(4,50,8,54),(9,155,13,159),(10,156,14,160),(11,157,15,153),(12,158,16,154),(17,107,21,111),(18,108,22,112),(19,109,23,105),(20,110,24,106),(25,115,29,119),(26,116,30,120),(27,117,31,113),(28,118,32,114),(33,217,37,221),(34,218,38,222),(35,219,39,223),(36,220,40,224),(41,131,45,135),(42,132,46,136),(43,133,47,129),(44,134,48,130),(57,207,61,203),(58,208,62,204),(59,201,63,205),(60,202,64,206),(65,215,69,211),(66,216,70,212),(67,209,71,213),(68,210,72,214),(73,167,77,163),(74,168,78,164),(75,161,79,165),(76,162,80,166),(81,193,85,197),(82,194,86,198),(83,195,87,199),(84,196,88,200),(89,183,93,179),(90,184,94,180),(91,177,95,181),(92,178,96,182),(97,185,101,189),(98,186,102,190),(99,187,103,191),(100,188,104,192),(121,173,125,169),(122,174,126,170),(123,175,127,171),(124,176,128,172),(137,225,141,229),(138,226,142,230),(139,227,143,231),(140,228,144,232),(145,235,149,239),(146,236,150,240),(147,237,151,233),(148,238,152,234)], [(1,7,5,3),(2,8,6,4),(9,135,13,131),(10,136,14,132),(11,129,15,133),(12,130,16,134),(17,141,21,137),(18,142,22,138),(19,143,23,139),(20,144,24,140),(25,87,29,83),(26,88,30,84),(27,81,31,85),(28,82,32,86),(33,73,37,77),(34,74,38,78),(35,75,39,79),(36,76,40,80),(41,155,45,159),(42,156,46,160),(43,157,47,153),(44,158,48,154),(49,51,53,55),(50,52,54,56),(57,177,61,181),(58,178,62,182),(59,179,63,183),(60,180,64,184),(65,123,69,127),(66,124,70,128),(67,125,71,121),(68,126,72,122),(89,201,93,205),(90,202,94,206),(91,203,95,207),(92,204,96,208),(97,103,101,99),(98,104,102,100),(105,231,109,227),(106,232,110,228),(107,225,111,229),(108,226,112,230),(113,193,117,197),(114,194,118,198),(115,195,119,199),(116,196,120,200),(145,151,149,147),(146,152,150,148),(161,219,165,223),(162,220,166,224),(163,221,167,217),(164,222,168,218),(169,209,173,213),(170,210,174,214),(171,211,175,215),(172,212,176,216),(185,187,189,191),(186,188,190,192),(233,235,237,239),(234,236,238,240)], [(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)]])

75 conjugacy classes

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

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 8 8 type + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 C8○D4 C3×C8○D4 F5 C2×F5 C3×F5 C6×F5 Q8.F5 C3×Q8.F5 kernel C3×Q8.F5 C3×D5⋊C8 C3×C4.F5 C3×Q8⋊2D5 Q8.F5 C3×D20 Q8×C15 D5⋊C8 C4.F5 Q8⋊2D5 D20 C5×Q8 C15 C5 C3×Q8 C12 Q8 C4 C3 C1 # reps 1 3 3 1 2 6 2 6 6 2 12 4 4 8 1 3 2 6 1 2

Matrix representation of C3×Q8.F5 in GL6(𝔽241)

 15 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 0 0 0 0 0 0 15
,
 51 183 0 0 0 0 236 190 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
,
 177 44 0 0 0 0 0 64 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 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 240 240 240 240
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 95 7 95 0 0 153 0 146 146 0 0 146 146 0 153 0 0 95 7 95 0

G:=sub<GL(6,GF(241))| [15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[51,236,0,0,0,0,183,190,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[177,0,0,0,0,0,44,64,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,153,146,95,0,0,95,0,146,7,0,0,7,146,0,95,0,0,95,146,153,0] >;

C3×Q8.F5 in GAP, Magma, Sage, TeX

C_3\times Q_8.F_5
% in TeX

G:=Group("C3xQ8.F5");
// GroupNames label

G:=SmallGroup(480,1055);
// by ID

G=gap.SmallGroup(480,1055);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,344,555,268,102,9414,818]);
// Polycyclic

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

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