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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, C6.21(C22×F5), C22.20(S3×F5), C30.21(C22×C4), (C2×Dic15).9C4, Dic5.27(C4×S3), 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×C30).16(C2×C4), (C2×C10).18(C4×S3), (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
Upper central
C1 — C22

Generators and relations for C2×S3×C5⋊C8
 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 >

Subgroups: 564 in 152 conjugacy classes, 70 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C23, C10, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, C22×C4, Dic5, Dic5, C2×C10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C30, C22×C8, C5⋊C8, C5⋊C8, C2×Dic5, C2×Dic5, C22×C10, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, C3×Dic5, Dic15, S3×C10, C2×C30, C2×C5⋊C8, C2×C5⋊C8, C22×Dic5, S3×C2×C8, C3×C5⋊C8, C15⋊C8, S3×Dic5, C6×Dic5, C2×Dic15, S3×C2×C10, C22×C5⋊C8, S3×C5⋊C8, C6×C5⋊C8, C2×C15⋊C8, C2×S3×Dic5, C2×S3×C5⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C2×C8, C22×C4, F5, C4×S3, C22×S3, C22×C8, C5⋊C8, C2×F5, S3×C8, S3×C2×C4, C2×C5⋊C8, C22×F5, S3×C2×C8, S3×F5, C22×C5⋊C8, S3×C5⋊C8, C2×S3×F5, C2×S3×C5⋊C8

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

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

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

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

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

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

Matrix representation of C2×S3×C5⋊C8 ►in 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] >;
 

C2×S3×C5⋊C8 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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