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## G = C2×C10×C3⋊D4order 480 = 25·3·5

### Direct product of C2×C10 and C3⋊D4

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
 Upper central C1 — C22×C10 — C23×C10

Generators and relations for C2×C10×C3⋊D4
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 >

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

Smallest permutation representation of C2×C10×C3⋊D4
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 94)(12 95)(13 96)(14 97)(15 98)(16 99)(17 100)(18 91)(19 92)(20 93)(21 46)(22 47)(23 48)(24 49)(25 50)(26 41)(27 42)(28 43)(29 44)(30 45)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 71)(40 72)(51 233)(52 234)(53 235)(54 236)(55 237)(56 238)(57 239)(58 240)(59 231)(60 232)(61 109)(62 110)(63 101)(64 102)(65 103)(66 104)(67 105)(68 106)(69 107)(70 108)(81 115)(82 116)(83 117)(84 118)(85 119)(86 120)(87 111)(88 112)(89 113)(90 114)(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 93 189)(2 94 190)(3 95 181)(4 96 182)(5 97 183)(6 98 184)(7 99 185)(8 100 186)(9 91 187)(10 92 188)(11 215 203)(12 216 204)(13 217 205)(14 218 206)(15 219 207)(16 220 208)(17 211 209)(18 212 210)(19 213 201)(20 214 202)(21 152 225)(22 153 226)(23 154 227)(24 155 228)(25 156 229)(26 157 230)(27 158 221)(28 159 222)(29 160 223)(30 151 224)(31 117 140)(32 118 131)(33 119 132)(34 120 133)(35 111 134)(36 112 135)(37 113 136)(38 114 137)(39 115 138)(40 116 139)(41 122 195)(42 123 196)(43 124 197)(44 125 198)(45 126 199)(46 127 200)(47 128 191)(48 129 192)(49 130 193)(50 121 194)(51 144 107)(52 145 108)(53 146 109)(54 147 110)(55 148 101)(56 149 102)(57 150 103)(58 141 104)(59 142 105)(60 143 106)(61 235 171)(62 236 172)(63 237 173)(64 238 174)(65 239 175)(66 240 176)(67 231 177)(68 232 178)(69 233 179)(70 234 180)(71 81 163)(72 82 164)(73 83 165)(74 84 166)(75 85 167)(76 86 168)(77 87 169)(78 88 170)(79 89 161)(80 90 162)
(1 195 55 113)(2 196 56 114)(3 197 57 115)(4 198 58 116)(5 199 59 117)(6 200 60 118)(7 191 51 119)(8 192 52 120)(9 193 53 111)(10 194 54 112)(11 158 174 80)(12 159 175 71)(13 160 176 72)(14 151 177 73)(15 152 178 74)(16 153 179 75)(17 154 180 76)(18 155 171 77)(19 156 172 78)(20 157 173 79)(21 68 166 219)(22 69 167 220)(23 70 168 211)(24 61 169 212)(25 62 170 213)(26 63 161 214)(27 64 162 215)(28 65 163 216)(29 66 164 217)(30 67 165 218)(31 97 126 142)(32 98 127 143)(33 99 128 144)(34 100 129 145)(35 91 130 146)(36 92 121 147)(37 93 122 148)(38 94 123 149)(39 95 124 150)(40 96 125 141)(41 101 136 189)(42 102 137 190)(43 103 138 181)(44 104 139 182)(45 105 140 183)(46 106 131 184)(47 107 132 185)(48 108 133 186)(49 109 134 187)(50 110 135 188)(81 204 222 239)(82 205 223 240)(83 206 224 231)(84 207 225 232)(85 208 226 233)(86 209 227 234)(87 210 228 235)(88 201 229 236)(89 202 230 237)(90 203 221 238)
(1 60)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 69)(12 70)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 157)(22 158)(23 159)(24 160)(25 151)(26 152)(27 153)(28 154)(29 155)(30 156)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 131)(38 132)(39 133)(40 134)(41 127)(42 128)(43 129)(44 130)(45 121)(46 122)(47 123)(48 124)(49 125)(50 126)(71 168)(72 169)(73 170)(74 161)(75 162)(76 163)(77 164)(78 165)(79 166)(80 167)(81 86)(82 87)(83 88)(84 89)(85 90)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)(97 110)(98 101)(99 102)(100 103)(111 116)(112 117)(113 118)(114 119)(115 120)(141 187)(142 188)(143 189)(144 190)(145 181)(146 182)(147 183)(148 184)(149 185)(150 186)(171 217)(172 218)(173 219)(174 220)(175 211)(176 212)(177 213)(178 214)(179 215)(180 216)(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)(11,94)(12,95)(13,96)(14,97)(15,98)(16,99)(17,100)(18,91)(19,92)(20,93)(21,46)(22,47)(23,48)(24,49)(25,50)(26,41)(27,42)(28,43)(29,44)(30,45)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,71)(40,72)(51,233)(52,234)(53,235)(54,236)(55,237)(56,238)(57,239)(58,240)(59,231)(60,232)(61,109)(62,110)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,108)(81,115)(82,116)(83,117)(84,118)(85,119)(86,120)(87,111)(88,112)(89,113)(90,114)(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), (1,93,189)(2,94,190)(3,95,181)(4,96,182)(5,97,183)(6,98,184)(7,99,185)(8,100,186)(9,91,187)(10,92,188)(11,215,203)(12,216,204)(13,217,205)(14,218,206)(15,219,207)(16,220,208)(17,211,209)(18,212,210)(19,213,201)(20,214,202)(21,152,225)(22,153,226)(23,154,227)(24,155,228)(25,156,229)(26,157,230)(27,158,221)(28,159,222)(29,160,223)(30,151,224)(31,117,140)(32,118,131)(33,119,132)(34,120,133)(35,111,134)(36,112,135)(37,113,136)(38,114,137)(39,115,138)(40,116,139)(41,122,195)(42,123,196)(43,124,197)(44,125,198)(45,126,199)(46,127,200)(47,128,191)(48,129,192)(49,130,193)(50,121,194)(51,144,107)(52,145,108)(53,146,109)(54,147,110)(55,148,101)(56,149,102)(57,150,103)(58,141,104)(59,142,105)(60,143,106)(61,235,171)(62,236,172)(63,237,173)(64,238,174)(65,239,175)(66,240,176)(67,231,177)(68,232,178)(69,233,179)(70,234,180)(71,81,163)(72,82,164)(73,83,165)(74,84,166)(75,85,167)(76,86,168)(77,87,169)(78,88,170)(79,89,161)(80,90,162), (1,195,55,113)(2,196,56,114)(3,197,57,115)(4,198,58,116)(5,199,59,117)(6,200,60,118)(7,191,51,119)(8,192,52,120)(9,193,53,111)(10,194,54,112)(11,158,174,80)(12,159,175,71)(13,160,176,72)(14,151,177,73)(15,152,178,74)(16,153,179,75)(17,154,180,76)(18,155,171,77)(19,156,172,78)(20,157,173,79)(21,68,166,219)(22,69,167,220)(23,70,168,211)(24,61,169,212)(25,62,170,213)(26,63,161,214)(27,64,162,215)(28,65,163,216)(29,66,164,217)(30,67,165,218)(31,97,126,142)(32,98,127,143)(33,99,128,144)(34,100,129,145)(35,91,130,146)(36,92,121,147)(37,93,122,148)(38,94,123,149)(39,95,124,150)(40,96,125,141)(41,101,136,189)(42,102,137,190)(43,103,138,181)(44,104,139,182)(45,105,140,183)(46,106,131,184)(47,107,132,185)(48,108,133,186)(49,109,134,187)(50,110,135,188)(81,204,222,239)(82,205,223,240)(83,206,224,231)(84,207,225,232)(85,208,226,233)(86,209,227,234)(87,210,228,235)(88,201,229,236)(89,202,230,237)(90,203,221,238), (1,60)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,69)(12,70)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,157)(22,158)(23,159)(24,160)(25,151)(26,152)(27,153)(28,154)(29,155)(30,156)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,131)(38,132)(39,133)(40,134)(41,127)(42,128)(43,129)(44,130)(45,121)(46,122)(47,123)(48,124)(49,125)(50,126)(71,168)(72,169)(73,170)(74,161)(75,162)(76,163)(77,164)(78,165)(79,166)(80,167)(81,86)(82,87)(83,88)(84,89)(85,90)(91,104)(92,105)(93,106)(94,107)(95,108)(96,109)(97,110)(98,101)(99,102)(100,103)(111,116)(112,117)(113,118)(114,119)(115,120)(141,187)(142,188)(143,189)(144,190)(145,181)(146,182)(147,183)(148,184)(149,185)(150,186)(171,217)(172,218)(173,219)(174,220)(175,211)(176,212)(177,213)(178,214)(179,215)(180,216)(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)(11,94)(12,95)(13,96)(14,97)(15,98)(16,99)(17,100)(18,91)(19,92)(20,93)(21,46)(22,47)(23,48)(24,49)(25,50)(26,41)(27,42)(28,43)(29,44)(30,45)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,71)(40,72)(51,233)(52,234)(53,235)(54,236)(55,237)(56,238)(57,239)(58,240)(59,231)(60,232)(61,109)(62,110)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,108)(81,115)(82,116)(83,117)(84,118)(85,119)(86,120)(87,111)(88,112)(89,113)(90,114)(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), (1,93,189)(2,94,190)(3,95,181)(4,96,182)(5,97,183)(6,98,184)(7,99,185)(8,100,186)(9,91,187)(10,92,188)(11,215,203)(12,216,204)(13,217,205)(14,218,206)(15,219,207)(16,220,208)(17,211,209)(18,212,210)(19,213,201)(20,214,202)(21,152,225)(22,153,226)(23,154,227)(24,155,228)(25,156,229)(26,157,230)(27,158,221)(28,159,222)(29,160,223)(30,151,224)(31,117,140)(32,118,131)(33,119,132)(34,120,133)(35,111,134)(36,112,135)(37,113,136)(38,114,137)(39,115,138)(40,116,139)(41,122,195)(42,123,196)(43,124,197)(44,125,198)(45,126,199)(46,127,200)(47,128,191)(48,129,192)(49,130,193)(50,121,194)(51,144,107)(52,145,108)(53,146,109)(54,147,110)(55,148,101)(56,149,102)(57,150,103)(58,141,104)(59,142,105)(60,143,106)(61,235,171)(62,236,172)(63,237,173)(64,238,174)(65,239,175)(66,240,176)(67,231,177)(68,232,178)(69,233,179)(70,234,180)(71,81,163)(72,82,164)(73,83,165)(74,84,166)(75,85,167)(76,86,168)(77,87,169)(78,88,170)(79,89,161)(80,90,162), (1,195,55,113)(2,196,56,114)(3,197,57,115)(4,198,58,116)(5,199,59,117)(6,200,60,118)(7,191,51,119)(8,192,52,120)(9,193,53,111)(10,194,54,112)(11,158,174,80)(12,159,175,71)(13,160,176,72)(14,151,177,73)(15,152,178,74)(16,153,179,75)(17,154,180,76)(18,155,171,77)(19,156,172,78)(20,157,173,79)(21,68,166,219)(22,69,167,220)(23,70,168,211)(24,61,169,212)(25,62,170,213)(26,63,161,214)(27,64,162,215)(28,65,163,216)(29,66,164,217)(30,67,165,218)(31,97,126,142)(32,98,127,143)(33,99,128,144)(34,100,129,145)(35,91,130,146)(36,92,121,147)(37,93,122,148)(38,94,123,149)(39,95,124,150)(40,96,125,141)(41,101,136,189)(42,102,137,190)(43,103,138,181)(44,104,139,182)(45,105,140,183)(46,106,131,184)(47,107,132,185)(48,108,133,186)(49,109,134,187)(50,110,135,188)(81,204,222,239)(82,205,223,240)(83,206,224,231)(84,207,225,232)(85,208,226,233)(86,209,227,234)(87,210,228,235)(88,201,229,236)(89,202,230,237)(90,203,221,238), (1,60)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,69)(12,70)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,157)(22,158)(23,159)(24,160)(25,151)(26,152)(27,153)(28,154)(29,155)(30,156)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,131)(38,132)(39,133)(40,134)(41,127)(42,128)(43,129)(44,130)(45,121)(46,122)(47,123)(48,124)(49,125)(50,126)(71,168)(72,169)(73,170)(74,161)(75,162)(76,163)(77,164)(78,165)(79,166)(80,167)(81,86)(82,87)(83,88)(84,89)(85,90)(91,104)(92,105)(93,106)(94,107)(95,108)(96,109)(97,110)(98,101)(99,102)(100,103)(111,116)(112,117)(113,118)(114,119)(115,120)(141,187)(142,188)(143,189)(144,190)(145,181)(146,182)(147,183)(148,184)(149,185)(150,186)(171,217)(172,218)(173,219)(174,220)(175,211)(176,212)(177,213)(178,214)(179,215)(180,216)(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),(11,94),(12,95),(13,96),(14,97),(15,98),(16,99),(17,100),(18,91),(19,92),(20,93),(21,46),(22,47),(23,48),(24,49),(25,50),(26,41),(27,42),(28,43),(29,44),(30,45),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,71),(40,72),(51,233),(52,234),(53,235),(54,236),(55,237),(56,238),(57,239),(58,240),(59,231),(60,232),(61,109),(62,110),(63,101),(64,102),(65,103),(66,104),(67,105),(68,106),(69,107),(70,108),(81,115),(82,116),(83,117),(84,118),(85,119),(86,120),(87,111),(88,112),(89,113),(90,114),(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)], [(1,93,189),(2,94,190),(3,95,181),(4,96,182),(5,97,183),(6,98,184),(7,99,185),(8,100,186),(9,91,187),(10,92,188),(11,215,203),(12,216,204),(13,217,205),(14,218,206),(15,219,207),(16,220,208),(17,211,209),(18,212,210),(19,213,201),(20,214,202),(21,152,225),(22,153,226),(23,154,227),(24,155,228),(25,156,229),(26,157,230),(27,158,221),(28,159,222),(29,160,223),(30,151,224),(31,117,140),(32,118,131),(33,119,132),(34,120,133),(35,111,134),(36,112,135),(37,113,136),(38,114,137),(39,115,138),(40,116,139),(41,122,195),(42,123,196),(43,124,197),(44,125,198),(45,126,199),(46,127,200),(47,128,191),(48,129,192),(49,130,193),(50,121,194),(51,144,107),(52,145,108),(53,146,109),(54,147,110),(55,148,101),(56,149,102),(57,150,103),(58,141,104),(59,142,105),(60,143,106),(61,235,171),(62,236,172),(63,237,173),(64,238,174),(65,239,175),(66,240,176),(67,231,177),(68,232,178),(69,233,179),(70,234,180),(71,81,163),(72,82,164),(73,83,165),(74,84,166),(75,85,167),(76,86,168),(77,87,169),(78,88,170),(79,89,161),(80,90,162)], [(1,195,55,113),(2,196,56,114),(3,197,57,115),(4,198,58,116),(5,199,59,117),(6,200,60,118),(7,191,51,119),(8,192,52,120),(9,193,53,111),(10,194,54,112),(11,158,174,80),(12,159,175,71),(13,160,176,72),(14,151,177,73),(15,152,178,74),(16,153,179,75),(17,154,180,76),(18,155,171,77),(19,156,172,78),(20,157,173,79),(21,68,166,219),(22,69,167,220),(23,70,168,211),(24,61,169,212),(25,62,170,213),(26,63,161,214),(27,64,162,215),(28,65,163,216),(29,66,164,217),(30,67,165,218),(31,97,126,142),(32,98,127,143),(33,99,128,144),(34,100,129,145),(35,91,130,146),(36,92,121,147),(37,93,122,148),(38,94,123,149),(39,95,124,150),(40,96,125,141),(41,101,136,189),(42,102,137,190),(43,103,138,181),(44,104,139,182),(45,105,140,183),(46,106,131,184),(47,107,132,185),(48,108,133,186),(49,109,134,187),(50,110,135,188),(81,204,222,239),(82,205,223,240),(83,206,224,231),(84,207,225,232),(85,208,226,233),(86,209,227,234),(87,210,228,235),(88,201,229,236),(89,202,230,237),(90,203,221,238)], [(1,60),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,58),(10,59),(11,69),(12,70),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,157),(22,158),(23,159),(24,160),(25,151),(26,152),(27,153),(28,154),(29,155),(30,156),(31,135),(32,136),(33,137),(34,138),(35,139),(36,140),(37,131),(38,132),(39,133),(40,134),(41,127),(42,128),(43,129),(44,130),(45,121),(46,122),(47,123),(48,124),(49,125),(50,126),(71,168),(72,169),(73,170),(74,161),(75,162),(76,163),(77,164),(78,165),(79,166),(80,167),(81,86),(82,87),(83,88),(84,89),(85,90),(91,104),(92,105),(93,106),(94,107),(95,108),(96,109),(97,110),(98,101),(99,102),(100,103),(111,116),(112,117),(113,118),(114,119),(115,120),(141,187),(142,188),(143,189),(144,190),(145,181),(146,182),(147,183),(148,184),(149,185),(150,186),(171,217),(172,218),(173,219),(174,220),(175,211),(176,212),(177,213),(178,214),(179,215),(180,216),(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)])

180 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 5A 5B 5C 5D 6A ··· 6O 10A ··· 10AB 10AC ··· 10AR 10AS ··· 10BH 15A 15B 15C 15D 20A ··· 20P 30A ··· 30BH order 1 2 ··· 2 2 2 2 2 2 2 2 2 3 4 4 4 4 5 5 5 5 6 ··· 6 10 ··· 10 10 ··· 10 10 ··· 10 15 15 15 15 20 ··· 20 30 ··· 30 size 1 1 ··· 1 2 2 2 2 6 6 6 6 2 6 6 6 6 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 6 ··· 6 2 2 2 2 6 ··· 6 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 S3 D4 D6 C3⋊D4 C5×S3 C5×D4 S3×C10 C5×C3⋊D4 kernel C2×C10×C3⋊D4 Dic3×C2×C10 C10×C3⋊D4 S3×C22×C10 C23×C30 C22×C3⋊D4 C22×Dic3 C2×C3⋊D4 S3×C23 C23×C6 C23×C10 C2×C30 C22×C10 C2×C10 C24 C2×C6 C23 C22 # reps 1 1 12 1 1 4 4 48 4 4 1 4 7 8 4 16 28 32

Matrix representation of C2×C10×C3⋊D4 in GL5(𝔽61)

 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 60 0 0 0 1 60
,
 1 0 0 0 0 0 0 60 0 0 0 1 0 0 0 0 0 0 43 9 0 0 0 52 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

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] >;

C2×C10×C3⋊D4 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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