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G = C2×C10×C3⋊D4  order 480 = 25·3·5

Direct product of C2×C10 and C3⋊D4

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

Aliases: C2×C10×C3⋊D4, C30.98C24, C6⋊3(D4×C10), C30⋊18(C2×D4), (C2×C30)⋊35D4, C24⋊6(C5×S3), C23⋊5(S3×C10), (C23×C6)⋊6C10, (C23×C10)⋊8S3, C15⋊19(C22×D4), (S3×C23)⋊5C10, (C2×C30)⋊14C23, (C23×C30)⋊10C2, D6⋊3(C22×C10), (C22×C10)⋊16D6, (S3×C10)⋊12C23, C6.15(C23×C10), C10.83(S3×C23), (C22×C30)⋊23C22, (C22×Dic3)⋊9C10, (C5×Dic3)⋊10C23, Dic3⋊2(C22×C10), (C10×Dic3)⋊38C22, C3⋊3(D4×C2×C10), (C2×C6)⋊9(C5×D4), C22⋊3(S3×C2×C10), (S3×C2×C10)⋊23C22, (S3×C22×C10)⋊11C2, (C2×C6)⋊3(C22×C10), (C22×C6)⋊7(C2×C10), (Dic3×C2×C10)⋊20C2, C2.15(S3×C22×C10), (C22×S3)⋊7(C2×C10), (C2×C10)⋊11(C22×S3), (C2×Dic3)⋊11(C2×C10), SmallGroup(480,1164)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C2×C10×C3⋊D4
Chief series
C1 — C3 — C6 — C30 — S3×C10 — S3×C2×C10 — S3×C22×C10 — C2×C10×C3⋊D4
Lower central
C3 — C6 — C2×C10×C3⋊D4

Subgroups: 996 in 472 conjugacy classes, 210 normal (22 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×11], C22 [×28], C5, S3 [×4], C6, C6 [×6], C6 [×4], C2×C4 [×6], D4 [×16], C23, C23 [×6], C23 [×14], C10, C10 [×6], C10 [×8], Dic3 [×4], D6 [×4], D6 [×12], C2×C6 [×11], C2×C6 [×12], C15, C22×C4, C2×D4 [×12], C24, C24, C20 [×4], C2×C10 [×11], C2×C10 [×28], C2×Dic3 [×6], C3⋊D4 [×16], C22×S3 [×6], C22×S3 [×4], C22×C6, C22×C6 [×6], C22×C6 [×4], C5×S3 [×4], C30, C30 [×6], C30 [×4], C22×D4, C2×C20 [×6], C5×D4 [×16], C22×C10, C22×C10 [×6], C22×C10 [×14], C22×Dic3, C2×C3⋊D4 [×12], S3×C23, C23×C6, C5×Dic3 [×4], S3×C10 [×4], S3×C10 [×12], C2×C30 [×11], C2×C30 [×12], C22×C20, D4×C10 [×12], C23×C10, C23×C10, C22×C3⋊D4, C10×Dic3 [×6], C5×C3⋊D4 [×16], S3×C2×C10 [×6], S3×C2×C10 [×4], C22×C30, C22×C30 [×6], C22×C30 [×4], D4×C2×C10, Dic3×C2×C10, C10×C3⋊D4 [×12], S3×C22×C10, C23×C30, C2×C10×C3⋊D4

Quotients:
C1, C2 [×15], C22 [×35], C5, S3, D4 [×4], C23 [×15], C10 [×15], D6 [×7], C2×D4 [×6], C24, C2×C10 [×35], C3⋊D4 [×4], C22×S3 [×7], C5×S3, C22×D4, C5×D4 [×4], C22×C10 [×15], C2×C3⋊D4 [×6], S3×C23, S3×C10 [×7], D4×C10 [×6], C23×C10, C22×C3⋊D4, C5×C3⋊D4 [×4], S3×C2×C10 [×7], D4×C2×C10, C10×C3⋊D4 [×6], S3×C22×C10, C2×C10×C3⋊D4

Generators and relations
 G = < a,b,c,d,e | a2=b10=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

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

(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)

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

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

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

Matrix representation ►G ⊆ GL5(𝔽61)

600000
060000
006000
000600
000060
,
200000
03000
00300
000200
000020
,
10000
01000
00100
000060
000160
,
10000
006000
01000
000439
0005218
,
600000
01000
006000
000060
000600

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[20,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,20,0,0,0,0,0,20],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,60,60],[1,0,0,0,0,0,0,1,0,0,0,60,0,0,0,0,0,0,43,52,0,0,0,9,18],[60,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,60,0] >;
 

180 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O 3 4A4B4C4D5A5B5C5D6A···6O10A···10AB10AC···10AR10AS···10BH15A15B15C15D20A···20P30A···30BH
order12···2222222223444455556···610···1010···1010···101515151520···2030···30
size11···1222266662666611112···21···12···26···622226···62···2

180 irreducible representations

dim111111111122222222
type++++++++
imageC1C2C2C2C2C5C10C10C10C10S3D4D6C3⋊D4C5×S3C5×D4S3×C10C5×C3⋊D4
kernelC2×C10×C3⋊D4Dic3×C2×C10C10×C3⋊D4S3×C22×C10C23×C30C22×C3⋊D4C22×Dic3C2×C3⋊D4S3×C23C23×C6C23×C10C2×C30C22×C10C2×C10C24C2×C6C23C22
# reps11121144484414784162832

In GAP, Magma, Sage, TeX 

C_2\times C_{10}\times C_3\rtimes D_4
 
% in TeX
 
G:=Group("C2xC10xC3:D4");
 
// GroupNames label
 
G:=SmallGroup(480,1164);
 
// by ID
 
G=gap.SmallGroup(480,1164);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,2467,15686]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^10=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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