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G = C2×S3×C5⋊C8  order 480 = 25·3·5

Direct product of C2, S3 and C5⋊C8

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

Aliases: C2×S3×C5⋊C8, C10⋊2(S3×C8), C30⋊1(C2×C8), (S3×C10)⋊2C8, C15⋊2(C22×C8), D6.15(C2×F5), C15⋊C8⋊7C22, (S3×Dic5).4C4, (C22×S3).5F5, C22.20(S3×F5), C6.21(C22×F5), C30.21(C22×C4), Dic5.27(C4×S3), (C2×Dic15).9C4, Dic15.14(C2×C4), (C2×Dic5).147D6, Dic5.33(C22×S3), (S3×Dic5).16C22, (C3×Dic5).31C23, (C6×Dic5).142C22, C5⋊3(S3×C2×C8), C6⋊1(C2×C5⋊C8), (C6×C5⋊C8)⋊3C2, C3⋊1(C22×C5⋊C8), C2.4(C2×S3×F5), (C5×S3)⋊2(C2×C8), (C3×C5⋊C8)⋊7C22, (S3×C2×C10).4C4, C10.21(S3×C2×C4), (C2×C15⋊C8)⋊3C2, (C2×C6).21(C2×F5), (C2×C10).18(C4×S3), (C2×C30).16(C2×C4), (C2×S3×Dic5).11C2, (S3×C10).13(C2×C4), (C3×Dic5).23(C2×C4), SmallGroup(480,1002)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C15 — C2×S3×C5⋊C8
Chief series
C1 — C5 — C15 — C30 — C3×Dic5 — C3×C5⋊C8 — S3×C5⋊C8 — C2×S3×C5⋊C8
Lower central
C15 — C2×S3×C5⋊C8

Subgroups: 564 in 152 conjugacy classes, 70 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×4], C22, C22 [×6], C5, S3 [×4], C6, C6 [×2], C8 [×4], C2×C4 [×6], C23, C10, C10 [×2], C10 [×4], Dic3 [×2], C12 [×2], D6 [×6], C2×C6, C15, C2×C8 [×6], C22×C4, Dic5 [×2], Dic5 [×2], C2×C10, C2×C10 [×6], C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C5×S3 [×4], C30, C30 [×2], C22×C8, C5⋊C8 [×2], C5⋊C8 [×2], C2×Dic5, C2×Dic5 [×5], C22×C10, S3×C8 [×4], C2×C3⋊C8, C2×C24, S3×C2×C4, C3×Dic5 [×2], Dic15 [×2], S3×C10 [×6], C2×C30, C2×C5⋊C8, C2×C5⋊C8 [×5], C22×Dic5, S3×C2×C8, C3×C5⋊C8 [×2], C15⋊C8 [×2], S3×Dic5 [×4], C6×Dic5, C2×Dic15, S3×C2×C10, C22×C5⋊C8, S3×C5⋊C8 [×4], C6×C5⋊C8, C2×C15⋊C8, C2×S3×Dic5, C2×S3×C5⋊C8

Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], S3, C8 [×4], C2×C4 [×6], C23, D6 [×3], C2×C8 [×6], C22×C4, F5, C4×S3 [×2], C22×S3, C22×C8, C5⋊C8 [×4], C2×F5 [×3], S3×C8 [×2], S3×C2×C4, C2×C5⋊C8 [×6], C22×F5, S3×C2×C8, S3×F5, C22×C5⋊C8, S3×C5⋊C8 [×2], C2×S3×F5, C2×S3×C5⋊C8

Generators and relations
 G = < a,b,c,d,e | a2=b3=c2=d5=e8=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

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

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

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

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

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

24000000
02400000
00240000
00024000
00002400
00000240
,
2402400000
100000
001000
000100
000010
000001
,
100000
2402400000
00240000
00024000
00002400
00000240
,
100000
010000
00240100
00240010
00240001
00240000
,
24000000
02400000
00147521693
00755539150
001862029178
0023813094189

G:=sub<GL(6,GF(241))| [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],[240,1,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,240,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,147,75,186,238,0,0,52,55,202,130,0,0,169,39,91,94,0,0,3,150,78,189] >;
 

60 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H 5 6A6B6C8A···8H8I···8P10A10B10C10D10E10F10G12A12B12C12D 15 24A···24H30A30B30C
order1222222234444444456668···88···810101010101010121212121524···24303030
size11113333255551515151542225···515···154441212121210101010810···10888

60 irreducible representations

dim1111111112222224444888
type+++++++++-+++-+
imageC1C2C2C2C2C4C4C4C8S3D6D6C4×S3C4×S3S3×C8F5C5⋊C8C2×F5C2×F5S3×F5S3×C5⋊C8C2×S3×F5
kernelC2×S3×C5⋊C8S3×C5⋊C8C6×C5⋊C8C2×C15⋊C8C2×S3×Dic5S3×Dic5C2×Dic15S3×C2×C10S3×C10C2×C5⋊C8C5⋊C8C2×Dic5Dic5C2×C10C10C22×S3D6D6C2×C6C22C2C2
# reps14111422161212281421121

In GAP, Magma, Sage, TeX 

C_2\times S_3\times C_5\rtimes C_8
 
% in TeX
 
G:=Group("C2xS3xC5:C8");
 
// GroupNames label
 
G:=SmallGroup(480,1002);
 
// by ID
 
G=gap.SmallGroup(480,1002);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,80,1356,9414,2379]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^5=e^8=1,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,c*e=e*c,e*d*e^-1=d^3>;
 
// generators/relations
 

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