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## G = C2×D5×C3⋊C8order 480 = 25·3·5

### Direct product of C2, D5 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C2×D5×C3⋊C8
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D5×C3⋊C8 — C2×D5×C3⋊C8
 Lower central C15 — C2×D5×C3⋊C8
 Upper central C1 — C2×C4

Generators and relations for C2×D5×C3⋊C8
G = < a,b,c,d,e | a2=b5=c2=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 476 in 152 conjugacy classes, 84 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×6], C5, C6, C6 [×2], C6 [×4], C8 [×4], C2×C4, C2×C4 [×5], C23, D5 [×4], C10, C10 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×6], C15, C2×C8 [×6], C22×C4, Dic5 [×2], C20 [×2], D10 [×6], C2×C10, C3⋊C8 [×2], C3⋊C8 [×2], C2×C12, C2×C12 [×5], C22×C6, C3×D5 [×4], C30, C30 [×2], C22×C8, C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C2×C3⋊C8 [×5], C22×C12, C3×Dic5 [×2], C60 [×2], C6×D5 [×6], C2×C30, C8×D5 [×4], C2×C52C8, C2×C40, C2×C4×D5, C22×C3⋊C8, C5×C3⋊C8 [×2], C153C8 [×2], D5×C12 [×4], C6×Dic5, C2×C60, D5×C2×C6, D5×C2×C8, D5×C3⋊C8 [×4], C10×C3⋊C8, C2×C153C8, D5×C2×C12, C2×D5×C3⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C8 [×4], C2×C4 [×6], C23, D5, Dic3 [×4], D6 [×3], C2×C8 [×6], C22×C4, D10 [×3], C3⋊C8 [×4], C2×Dic3 [×6], C22×S3, C22×C8, C4×D5 [×2], C22×D5, C2×C3⋊C8 [×6], C22×Dic3, S3×D5, C8×D5 [×2], C2×C4×D5, C22×C3⋊C8, D5×Dic3 [×2], C2×S3×D5, D5×C2×C8, D5×C3⋊C8 [×2], C2×D5×Dic3, C2×D5×C3⋊C8

Smallest permutation representation of C2×D5×C3⋊C8
On 240 points
Generators in S240
(1 192)(2 185)(3 186)(4 187)(5 188)(6 189)(7 190)(8 191)(9 146)(10 147)(11 148)(12 149)(13 150)(14 151)(15 152)(16 145)(17 140)(18 141)(19 142)(20 143)(21 144)(22 137)(23 138)(24 139)(25 162)(26 163)(27 164)(28 165)(29 166)(30 167)(31 168)(32 161)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 89)(40 90)(41 88)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 73)(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)(97 178)(98 179)(99 180)(100 181)(101 182)(102 183)(103 184)(104 177)(113 194)(114 195)(115 196)(116 197)(117 198)(118 199)(119 200)(120 193)(121 224)(122 217)(123 218)(124 219)(125 220)(126 221)(127 222)(128 223)(129 210)(130 211)(131 212)(132 213)(133 214)(134 215)(135 216)(136 209)(153 203)(154 204)(155 205)(156 206)(157 207)(158 208)(159 201)(160 202)(169 228)(170 229)(171 230)(172 231)(173 232)(174 225)(175 226)(176 227)
(1 216 233 76 155)(2 209 234 77 156)(3 210 235 78 157)(4 211 236 79 158)(5 212 237 80 159)(6 213 238 73 160)(7 214 239 74 153)(8 215 240 75 154)(9 138 125 94 193)(10 139 126 95 194)(11 140 127 96 195)(12 141 128 89 196)(13 142 121 90 197)(14 143 122 91 198)(15 144 123 92 199)(16 137 124 93 200)(17 222 38 114 148)(18 223 39 115 149)(19 224 40 116 150)(20 217 33 117 151)(21 218 34 118 152)(22 219 35 119 145)(23 220 36 120 146)(24 221 37 113 147)(25 179 111 230 46)(26 180 112 231 47)(27 181 105 232 48)(28 182 106 225 41)(29 183 107 226 42)(30 184 108 227 43)(31 177 109 228 44)(32 178 110 229 45)(49 203 190 133 69)(50 204 191 134 70)(51 205 192 135 71)(52 206 185 136 72)(53 207 186 129 65)(54 208 187 130 66)(55 201 188 131 67)(56 202 189 132 68)(57 172 86 163 99)(58 173 87 164 100)(59 174 88 165 101)(60 175 81 166 102)(61 176 82 167 103)(62 169 83 168 104)(63 170 84 161 97)(64 171 85 162 98)
(1 159)(2 160)(3 153)(4 154)(5 155)(6 156)(7 157)(8 158)(9 197)(10 198)(11 199)(12 200)(13 193)(14 194)(15 195)(16 196)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 33)(25 226)(26 227)(27 228)(28 229)(29 230)(30 231)(31 232)(32 225)(41 45)(42 46)(43 47)(44 48)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 103)(58 104)(59 97)(60 98)(61 99)(62 100)(63 101)(64 102)(65 69)(66 70)(67 71)(68 72)(73 209)(74 210)(75 211)(76 212)(77 213)(78 214)(79 215)(80 216)(81 85)(82 86)(83 87)(84 88)(89 137)(90 138)(91 139)(92 140)(93 141)(94 142)(95 143)(96 144)(105 177)(106 178)(107 179)(108 180)(109 181)(110 182)(111 183)(112 184)(113 151)(114 152)(115 145)(116 146)(117 147)(118 148)(119 149)(120 150)(121 125)(122 126)(123 127)(124 128)(161 174)(162 175)(163 176)(164 169)(165 170)(166 171)(167 172)(168 173)(185 202)(186 203)(187 204)(188 205)(189 206)(190 207)(191 208)(192 201)(217 221)(218 222)(219 223)(220 224)(233 237)(234 238)(235 239)(236 240)
(1 11 106)(2 107 12)(3 13 108)(4 109 14)(5 15 110)(6 111 16)(7 9 112)(8 105 10)(17 174 135)(18 136 175)(19 176 129)(20 130 169)(21 170 131)(22 132 171)(23 172 133)(24 134 173)(25 93 73)(26 74 94)(27 95 75)(28 76 96)(29 89 77)(30 78 90)(31 91 79)(32 80 92)(33 54 168)(34 161 55)(35 56 162)(36 163 49)(37 50 164)(38 165 51)(39 52 166)(40 167 53)(41 233 127)(42 128 234)(43 235 121)(44 122 236)(45 237 123)(46 124 238)(47 239 125)(48 126 240)(57 190 146)(58 147 191)(59 192 148)(60 149 185)(61 186 150)(62 151 187)(63 188 152)(64 145 189)(65 224 82)(66 83 217)(67 218 84)(68 85 219)(69 220 86)(70 87 221)(71 222 88)(72 81 223)(97 201 118)(98 119 202)(99 203 120)(100 113 204)(101 205 114)(102 115 206)(103 207 116)(104 117 208)(137 213 230)(138 231 214)(139 215 232)(140 225 216)(141 209 226)(142 227 210)(143 211 228)(144 229 212)(153 193 180)(154 181 194)(155 195 182)(156 183 196)(157 197 184)(158 177 198)(159 199 178)(160 179 200)
(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)| (1,192)(2,185)(3,186)(4,187)(5,188)(6,189)(7,190)(8,191)(9,146)(10,147)(11,148)(12,149)(13,150)(14,151)(15,152)(16,145)(17,140)(18,141)(19,142)(20,143)(21,144)(22,137)(23,138)(24,139)(25,162)(26,163)(27,164)(28,165)(29,166)(30,167)(31,168)(32,161)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,89)(40,90)(41,88)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,73)(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)(97,178)(98,179)(99,180)(100,181)(101,182)(102,183)(103,184)(104,177)(113,194)(114,195)(115,196)(116,197)(117,198)(118,199)(119,200)(120,193)(121,224)(122,217)(123,218)(124,219)(125,220)(126,221)(127,222)(128,223)(129,210)(130,211)(131,212)(132,213)(133,214)(134,215)(135,216)(136,209)(153,203)(154,204)(155,205)(156,206)(157,207)(158,208)(159,201)(160,202)(169,228)(170,229)(171,230)(172,231)(173,232)(174,225)(175,226)(176,227), (1,216,233,76,155)(2,209,234,77,156)(3,210,235,78,157)(4,211,236,79,158)(5,212,237,80,159)(6,213,238,73,160)(7,214,239,74,153)(8,215,240,75,154)(9,138,125,94,193)(10,139,126,95,194)(11,140,127,96,195)(12,141,128,89,196)(13,142,121,90,197)(14,143,122,91,198)(15,144,123,92,199)(16,137,124,93,200)(17,222,38,114,148)(18,223,39,115,149)(19,224,40,116,150)(20,217,33,117,151)(21,218,34,118,152)(22,219,35,119,145)(23,220,36,120,146)(24,221,37,113,147)(25,179,111,230,46)(26,180,112,231,47)(27,181,105,232,48)(28,182,106,225,41)(29,183,107,226,42)(30,184,108,227,43)(31,177,109,228,44)(32,178,110,229,45)(49,203,190,133,69)(50,204,191,134,70)(51,205,192,135,71)(52,206,185,136,72)(53,207,186,129,65)(54,208,187,130,66)(55,201,188,131,67)(56,202,189,132,68)(57,172,86,163,99)(58,173,87,164,100)(59,174,88,165,101)(60,175,81,166,102)(61,176,82,167,103)(62,169,83,168,104)(63,170,84,161,97)(64,171,85,162,98), (1,159)(2,160)(3,153)(4,154)(5,155)(6,156)(7,157)(8,158)(9,197)(10,198)(11,199)(12,200)(13,193)(14,194)(15,195)(16,196)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,33)(25,226)(26,227)(27,228)(28,229)(29,230)(30,231)(31,232)(32,225)(41,45)(42,46)(43,47)(44,48)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,103)(58,104)(59,97)(60,98)(61,99)(62,100)(63,101)(64,102)(65,69)(66,70)(67,71)(68,72)(73,209)(74,210)(75,211)(76,212)(77,213)(78,214)(79,215)(80,216)(81,85)(82,86)(83,87)(84,88)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144)(105,177)(106,178)(107,179)(108,180)(109,181)(110,182)(111,183)(112,184)(113,151)(114,152)(115,145)(116,146)(117,147)(118,148)(119,149)(120,150)(121,125)(122,126)(123,127)(124,128)(161,174)(162,175)(163,176)(164,169)(165,170)(166,171)(167,172)(168,173)(185,202)(186,203)(187,204)(188,205)(189,206)(190,207)(191,208)(192,201)(217,221)(218,222)(219,223)(220,224)(233,237)(234,238)(235,239)(236,240), (1,11,106)(2,107,12)(3,13,108)(4,109,14)(5,15,110)(6,111,16)(7,9,112)(8,105,10)(17,174,135)(18,136,175)(19,176,129)(20,130,169)(21,170,131)(22,132,171)(23,172,133)(24,134,173)(25,93,73)(26,74,94)(27,95,75)(28,76,96)(29,89,77)(30,78,90)(31,91,79)(32,80,92)(33,54,168)(34,161,55)(35,56,162)(36,163,49)(37,50,164)(38,165,51)(39,52,166)(40,167,53)(41,233,127)(42,128,234)(43,235,121)(44,122,236)(45,237,123)(46,124,238)(47,239,125)(48,126,240)(57,190,146)(58,147,191)(59,192,148)(60,149,185)(61,186,150)(62,151,187)(63,188,152)(64,145,189)(65,224,82)(66,83,217)(67,218,84)(68,85,219)(69,220,86)(70,87,221)(71,222,88)(72,81,223)(97,201,118)(98,119,202)(99,203,120)(100,113,204)(101,205,114)(102,115,206)(103,207,116)(104,117,208)(137,213,230)(138,231,214)(139,215,232)(140,225,216)(141,209,226)(142,227,210)(143,211,228)(144,229,212)(153,193,180)(154,181,194)(155,195,182)(156,183,196)(157,197,184)(158,177,198)(159,199,178)(160,179,200), (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( (1,192)(2,185)(3,186)(4,187)(5,188)(6,189)(7,190)(8,191)(9,146)(10,147)(11,148)(12,149)(13,150)(14,151)(15,152)(16,145)(17,140)(18,141)(19,142)(20,143)(21,144)(22,137)(23,138)(24,139)(25,162)(26,163)(27,164)(28,165)(29,166)(30,167)(31,168)(32,161)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,89)(40,90)(41,88)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,73)(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)(97,178)(98,179)(99,180)(100,181)(101,182)(102,183)(103,184)(104,177)(113,194)(114,195)(115,196)(116,197)(117,198)(118,199)(119,200)(120,193)(121,224)(122,217)(123,218)(124,219)(125,220)(126,221)(127,222)(128,223)(129,210)(130,211)(131,212)(132,213)(133,214)(134,215)(135,216)(136,209)(153,203)(154,204)(155,205)(156,206)(157,207)(158,208)(159,201)(160,202)(169,228)(170,229)(171,230)(172,231)(173,232)(174,225)(175,226)(176,227), (1,216,233,76,155)(2,209,234,77,156)(3,210,235,78,157)(4,211,236,79,158)(5,212,237,80,159)(6,213,238,73,160)(7,214,239,74,153)(8,215,240,75,154)(9,138,125,94,193)(10,139,126,95,194)(11,140,127,96,195)(12,141,128,89,196)(13,142,121,90,197)(14,143,122,91,198)(15,144,123,92,199)(16,137,124,93,200)(17,222,38,114,148)(18,223,39,115,149)(19,224,40,116,150)(20,217,33,117,151)(21,218,34,118,152)(22,219,35,119,145)(23,220,36,120,146)(24,221,37,113,147)(25,179,111,230,46)(26,180,112,231,47)(27,181,105,232,48)(28,182,106,225,41)(29,183,107,226,42)(30,184,108,227,43)(31,177,109,228,44)(32,178,110,229,45)(49,203,190,133,69)(50,204,191,134,70)(51,205,192,135,71)(52,206,185,136,72)(53,207,186,129,65)(54,208,187,130,66)(55,201,188,131,67)(56,202,189,132,68)(57,172,86,163,99)(58,173,87,164,100)(59,174,88,165,101)(60,175,81,166,102)(61,176,82,167,103)(62,169,83,168,104)(63,170,84,161,97)(64,171,85,162,98), (1,159)(2,160)(3,153)(4,154)(5,155)(6,156)(7,157)(8,158)(9,197)(10,198)(11,199)(12,200)(13,193)(14,194)(15,195)(16,196)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,33)(25,226)(26,227)(27,228)(28,229)(29,230)(30,231)(31,232)(32,225)(41,45)(42,46)(43,47)(44,48)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,103)(58,104)(59,97)(60,98)(61,99)(62,100)(63,101)(64,102)(65,69)(66,70)(67,71)(68,72)(73,209)(74,210)(75,211)(76,212)(77,213)(78,214)(79,215)(80,216)(81,85)(82,86)(83,87)(84,88)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144)(105,177)(106,178)(107,179)(108,180)(109,181)(110,182)(111,183)(112,184)(113,151)(114,152)(115,145)(116,146)(117,147)(118,148)(119,149)(120,150)(121,125)(122,126)(123,127)(124,128)(161,174)(162,175)(163,176)(164,169)(165,170)(166,171)(167,172)(168,173)(185,202)(186,203)(187,204)(188,205)(189,206)(190,207)(191,208)(192,201)(217,221)(218,222)(219,223)(220,224)(233,237)(234,238)(235,239)(236,240), (1,11,106)(2,107,12)(3,13,108)(4,109,14)(5,15,110)(6,111,16)(7,9,112)(8,105,10)(17,174,135)(18,136,175)(19,176,129)(20,130,169)(21,170,131)(22,132,171)(23,172,133)(24,134,173)(25,93,73)(26,74,94)(27,95,75)(28,76,96)(29,89,77)(30,78,90)(31,91,79)(32,80,92)(33,54,168)(34,161,55)(35,56,162)(36,163,49)(37,50,164)(38,165,51)(39,52,166)(40,167,53)(41,233,127)(42,128,234)(43,235,121)(44,122,236)(45,237,123)(46,124,238)(47,239,125)(48,126,240)(57,190,146)(58,147,191)(59,192,148)(60,149,185)(61,186,150)(62,151,187)(63,188,152)(64,145,189)(65,224,82)(66,83,217)(67,218,84)(68,85,219)(69,220,86)(70,87,221)(71,222,88)(72,81,223)(97,201,118)(98,119,202)(99,203,120)(100,113,204)(101,205,114)(102,115,206)(103,207,116)(104,117,208)(137,213,230)(138,231,214)(139,215,232)(140,225,216)(141,209,226)(142,227,210)(143,211,228)(144,229,212)(153,193,180)(154,181,194)(155,195,182)(156,183,196)(157,197,184)(158,177,198)(159,199,178)(160,179,200), (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([(1,192),(2,185),(3,186),(4,187),(5,188),(6,189),(7,190),(8,191),(9,146),(10,147),(11,148),(12,149),(13,150),(14,151),(15,152),(16,145),(17,140),(18,141),(19,142),(20,143),(21,144),(22,137),(23,138),(24,139),(25,162),(26,163),(27,164),(28,165),(29,166),(30,167),(31,168),(32,161),(33,91),(34,92),(35,93),(36,94),(37,95),(38,96),(39,89),(40,90),(41,88),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,80),(56,73),(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),(97,178),(98,179),(99,180),(100,181),(101,182),(102,183),(103,184),(104,177),(113,194),(114,195),(115,196),(116,197),(117,198),(118,199),(119,200),(120,193),(121,224),(122,217),(123,218),(124,219),(125,220),(126,221),(127,222),(128,223),(129,210),(130,211),(131,212),(132,213),(133,214),(134,215),(135,216),(136,209),(153,203),(154,204),(155,205),(156,206),(157,207),(158,208),(159,201),(160,202),(169,228),(170,229),(171,230),(172,231),(173,232),(174,225),(175,226),(176,227)], [(1,216,233,76,155),(2,209,234,77,156),(3,210,235,78,157),(4,211,236,79,158),(5,212,237,80,159),(6,213,238,73,160),(7,214,239,74,153),(8,215,240,75,154),(9,138,125,94,193),(10,139,126,95,194),(11,140,127,96,195),(12,141,128,89,196),(13,142,121,90,197),(14,143,122,91,198),(15,144,123,92,199),(16,137,124,93,200),(17,222,38,114,148),(18,223,39,115,149),(19,224,40,116,150),(20,217,33,117,151),(21,218,34,118,152),(22,219,35,119,145),(23,220,36,120,146),(24,221,37,113,147),(25,179,111,230,46),(26,180,112,231,47),(27,181,105,232,48),(28,182,106,225,41),(29,183,107,226,42),(30,184,108,227,43),(31,177,109,228,44),(32,178,110,229,45),(49,203,190,133,69),(50,204,191,134,70),(51,205,192,135,71),(52,206,185,136,72),(53,207,186,129,65),(54,208,187,130,66),(55,201,188,131,67),(56,202,189,132,68),(57,172,86,163,99),(58,173,87,164,100),(59,174,88,165,101),(60,175,81,166,102),(61,176,82,167,103),(62,169,83,168,104),(63,170,84,161,97),(64,171,85,162,98)], [(1,159),(2,160),(3,153),(4,154),(5,155),(6,156),(7,157),(8,158),(9,197),(10,198),(11,199),(12,200),(13,193),(14,194),(15,195),(16,196),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,33),(25,226),(26,227),(27,228),(28,229),(29,230),(30,231),(31,232),(32,225),(41,45),(42,46),(43,47),(44,48),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,103),(58,104),(59,97),(60,98),(61,99),(62,100),(63,101),(64,102),(65,69),(66,70),(67,71),(68,72),(73,209),(74,210),(75,211),(76,212),(77,213),(78,214),(79,215),(80,216),(81,85),(82,86),(83,87),(84,88),(89,137),(90,138),(91,139),(92,140),(93,141),(94,142),(95,143),(96,144),(105,177),(106,178),(107,179),(108,180),(109,181),(110,182),(111,183),(112,184),(113,151),(114,152),(115,145),(116,146),(117,147),(118,148),(119,149),(120,150),(121,125),(122,126),(123,127),(124,128),(161,174),(162,175),(163,176),(164,169),(165,170),(166,171),(167,172),(168,173),(185,202),(186,203),(187,204),(188,205),(189,206),(190,207),(191,208),(192,201),(217,221),(218,222),(219,223),(220,224),(233,237),(234,238),(235,239),(236,240)], [(1,11,106),(2,107,12),(3,13,108),(4,109,14),(5,15,110),(6,111,16),(7,9,112),(8,105,10),(17,174,135),(18,136,175),(19,176,129),(20,130,169),(21,170,131),(22,132,171),(23,172,133),(24,134,173),(25,93,73),(26,74,94),(27,95,75),(28,76,96),(29,89,77),(30,78,90),(31,91,79),(32,80,92),(33,54,168),(34,161,55),(35,56,162),(36,163,49),(37,50,164),(38,165,51),(39,52,166),(40,167,53),(41,233,127),(42,128,234),(43,235,121),(44,122,236),(45,237,123),(46,124,238),(47,239,125),(48,126,240),(57,190,146),(58,147,191),(59,192,148),(60,149,185),(61,186,150),(62,151,187),(63,188,152),(64,145,189),(65,224,82),(66,83,217),(67,218,84),(68,85,219),(69,220,86),(70,87,221),(71,222,88),(72,81,223),(97,201,118),(98,119,202),(99,203,120),(100,113,204),(101,205,114),(102,115,206),(103,207,116),(104,117,208),(137,213,230),(138,231,214),(139,215,232),(140,225,216),(141,209,226),(142,227,210),(143,211,228),(144,229,212),(153,193,180),(154,181,194),(155,195,182),(156,183,196),(157,197,184),(158,177,198),(159,199,178),(160,179,200)], [(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)])

96 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 6D 6E 6F 6G 8A ··· 8H 8I ··· 8P 10A ··· 10F 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 20A ··· 20H 30A ··· 30F 40A ··· 40P 60A ··· 60H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 8 ··· 8 8 ··· 8 10 ··· 10 12 12 12 12 12 12 12 12 15 15 20 ··· 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 1 1 5 5 5 5 2 1 1 1 1 5 5 5 5 2 2 2 2 2 10 10 10 10 3 ··· 3 15 ··· 15 2 ··· 2 2 2 2 2 10 10 10 10 4 4 2 ··· 2 4 ··· 4 6 ··· 6 4 ··· 4

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + - + - + - + + + - + - image C1 C2 C2 C2 C2 C4 C4 C4 C8 S3 D5 Dic3 D6 Dic3 D6 Dic3 D10 D10 C3⋊C8 C4×D5 C4×D5 C8×D5 S3×D5 D5×Dic3 C2×S3×D5 D5×Dic3 D5×C3⋊C8 kernel C2×D5×C3⋊C8 D5×C3⋊C8 C10×C3⋊C8 C2×C15⋊3C8 D5×C2×C12 D5×C12 C6×Dic5 D5×C2×C6 C6×D5 C2×C4×D5 C2×C3⋊C8 C4×D5 C4×D5 C2×Dic5 C2×C20 C22×D5 C3⋊C8 C2×C12 D10 C12 C2×C6 C6 C2×C4 C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 16 1 2 2 2 1 1 1 4 2 8 4 4 16 2 2 2 2 8

Matrix representation of C2×D5×C3⋊C8 in GL4(𝔽241) generated by

 240 0 0 0 0 240 0 0 0 0 1 0 0 0 0 1
,
 240 1 0 0 188 52 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 53 240 0 0 0 0 240 0 0 0 0 240
,
 1 0 0 0 0 1 0 0 0 0 0 240 0 0 1 240
,
 211 0 0 0 0 211 0 0 0 0 170 16 0 0 186 71
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[240,188,0,0,1,52,0,0,0,0,1,0,0,0,0,1],[1,53,0,0,0,240,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,240,240],[211,0,0,0,0,211,0,0,0,0,170,186,0,0,16,71] >;

C2×D5×C3⋊C8 in GAP, Magma, Sage, TeX

C_2\times D_5\times C_3\rtimes C_8
% in TeX

G:=Group("C2xD5xC3:C8");
// GroupNames label

G:=SmallGroup(480,357);
// by ID

G=gap.SmallGroup(480,357);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,80,1356,18822]);
// Polycyclic

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

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