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G = C3×Q8.F5  order 480 = 25·3·5

Direct product of C3 and Q8.F5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3×Q8.F5, D20.C12, D5⋊C8⋊3C6, C4.F5⋊4C6, C4.6(C6×F5), Q8.2(C3×F5), (C3×Q8).2F5, C15⋊20(C8○D4), C20.6(C2×C12), C60.45(C2×C4), (C3×D20).2C4, C12.45(C2×F5), (C5×Q8).3C12, (Q8×C15).2C4, D10.2(C2×C12), Q8⋊2D5.5C6, C6.53(C22×F5), C30.91(C22×C4), C10.9(C22×C12), (D5×C12).88C22, (C3×Dic5).73C23, Dic5.13(C22×C6), C5⋊2(C3×C8○D4), C5⋊C8.2(C2×C6), (C3×D5⋊C8)⋊8C2, C2.10(C2×C6×F5), (C3×C4.F5)⋊10C2, (C3×C5⋊C8).8C22, (C6×D5).29(C2×C4), (C4×D5).13(C2×C6), (C3×Q8⋊2D5).6C2, SmallGroup(480,1055)

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

Subgroups: 360 in 124 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2 [×3], C3, C4 [×3], C4, C22 [×3], C5, C6, C6 [×3], C8 [×4], C2×C4 [×3], D4 [×3], Q8, D5 [×3], C10, C12 [×3], C12, C2×C6 [×3], C15, C2×C8 [×3], M4(2) [×3], C4○D4, Dic5, C20 [×3], D10 [×3], C24 [×4], C2×C12 [×3], C3×D4 [×3], C3×Q8, C3×D5 [×3], C30, C8○D4, C5⋊C8, C5⋊C8 [×3], C4×D5 [×3], D20 [×3], C5×Q8, C2×C24 [×3], C3×M4(2) [×3], C3×C4○D4, C3×Dic5, C60 [×3], C6×D5 [×3], D5⋊C8 [×3], C4.F5 [×3], Q8⋊2D5, C3×C8○D4, C3×C5⋊C8, C3×C5⋊C8 [×3], D5×C12 [×3], C3×D20 [×3], Q8×C15, Q8.F5, C3×D5⋊C8 [×3], C3×C4.F5 [×3], C3×Q8⋊2D5, C3×Q8.F5

Quotients:
C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, C12 [×4], C2×C6 [×7], C22×C4, F5, C2×C12 [×6], C22×C6, C8○D4, C2×F5 [×3], C22×C12, C3×F5, C22×F5, C3×C8○D4, C6×F5 [×3], Q8.F5, C2×C6×F5, C3×Q8.F5

Generators and relations
 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 >

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽241)

1500000
0150000
0015000
0001500
0000150
0000015
,
511830000
2361900000
00240000
00024000
00002400
00000240
,
177440000
0640000
00240000
00024000
00002400
00000240
,
100000
010000
000100
000010
000001
00240240240240
,
800000
080000
00095795
001530146146
001461460153
00957950

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] >;
 

75 conjugacy classes

class 1 2A2B2C2D3A3B4A4B4C4D4E 5 6A6B6C···6H8A8B8C8D8E···8J 10 12A···12F12G12H12I12J15A15B20A20B20C24A···24H24I···24T30A30B60A···60F
order1222233444445666···688888···81012···1212121212151520202024···2424···24303060···60
size11101010112225541110···10555510···1042···25555448885···510···10448···8

75 irreducible representations

dim11111111111122444488
type+++++++
imageC1C2C2C2C3C4C4C6C6C6C12C12C8○D4C3×C8○D4F5C2×F5C3×F5C6×F5Q8.F5C3×Q8.F5
kernelC3×Q8.F5C3×D5⋊C8C3×C4.F5C3×Q8⋊2D5Q8.F5C3×D20Q8×C15D5⋊C8C4.F5Q8⋊2D5D20C5×Q8C15C5C3×Q8C12Q8C4C3C1
# reps133126266212448132612

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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