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## G = C22×C3⋊D20order 480 = 25·3·5

### Direct product of C22 and C3⋊D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C22×C3⋊D20
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C3⋊D20 — C2×C3⋊D20 — C22×C3⋊D20
 Lower central C15 — C30 — C22×C3⋊D20
 Upper central C1 — C23

Generators and relations for C22×C3⋊D20
G = < a,b,c,d,e | a2=b2=c3=d20=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: 2684 in 472 conjugacy classes, 148 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×7], C22 [×32], C5, S3 [×4], C6, C6 [×6], C6 [×4], C2×C4 [×6], D4 [×16], C23, C23 [×20], D5 [×8], C10, C10 [×6], Dic3 [×4], D6 [×16], C2×C6 [×7], C2×C6 [×16], C15, C22×C4, C2×D4 [×12], C24 [×2], C20 [×4], D10 [×4], D10 [×28], C2×C10 [×7], C2×Dic3 [×6], C3⋊D4 [×16], C22×S3 [×10], C22×C6, C22×C6 [×10], C3×D5 [×4], D15 [×4], C30, C30 [×6], C22×D4, D20 [×16], C2×C20 [×6], C22×D5 [×6], C22×D5 [×14], C22×C10, C22×Dic3, C2×C3⋊D4 [×12], S3×C23, C23×C6, C5×Dic3 [×4], C6×D5 [×4], C6×D5 [×12], D30 [×4], D30 [×12], C2×C30 [×7], C2×D20 [×12], C22×C20, C23×D5, C23×D5, C22×C3⋊D4, C3⋊D20 [×16], C10×Dic3 [×6], D5×C2×C6 [×6], D5×C2×C6 [×4], C22×D15 [×6], C22×D15 [×4], C22×C30, C22×D20, C2×C3⋊D20 [×12], Dic3×C2×C10, D5×C22×C6, C23×D15, C22×C3⋊D20
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], C3⋊D4 [×4], C22×S3 [×7], C22×D4, D20 [×4], C22×D5 [×7], C2×C3⋊D4 [×6], S3×C23, S3×D5, C2×D20 [×6], C23×D5, C22×C3⋊D4, C3⋊D20 [×4], C2×S3×D5 [×3], C22×D20, C2×C3⋊D20 [×6], C22×S3×D5, C22×C3⋊D20

Smallest permutation representation of C22×C3⋊D20
On 240 points
Generators in S240
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 120)(16 101)(17 102)(18 103)(19 104)(20 105)(21 145)(22 146)(23 147)(24 148)(25 149)(26 150)(27 151)(28 152)(29 153)(30 154)(31 155)(32 156)(33 157)(34 158)(35 159)(36 160)(37 141)(38 142)(39 143)(40 144)(41 97)(42 98)(43 99)(44 100)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 235)(62 236)(63 237)(64 238)(65 239)(66 240)(67 221)(68 222)(69 223)(70 224)(71 225)(72 226)(73 227)(74 228)(75 229)(76 230)(77 231)(78 232)(79 233)(80 234)(121 209)(122 210)(123 211)(124 212)(125 213)(126 214)(127 215)(128 216)(129 217)(130 218)(131 219)(132 220)(133 201)(134 202)(135 203)(136 204)(137 205)(138 206)(139 207)(140 208)(161 191)(162 192)(163 193)(164 194)(165 195)(166 196)(167 197)(168 198)(169 199)(170 200)(171 181)(172 182)(173 183)(174 184)(175 185)(176 186)(177 187)(178 188)(179 189)(180 190)
(1 99)(2 100)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 223)(22 224)(23 225)(24 226)(25 227)(26 228)(27 229)(28 230)(29 231)(30 232)(31 233)(32 234)(33 235)(34 236)(35 237)(36 238)(37 239)(38 240)(39 221)(40 222)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 101)(59 102)(60 103)(61 157)(62 158)(63 159)(64 160)(65 141)(66 142)(67 143)(68 144)(69 145)(70 146)(71 147)(72 148)(73 149)(74 150)(75 151)(76 152)(77 153)(78 154)(79 155)(80 156)(121 173)(122 174)(123 175)(124 176)(125 177)(126 178)(127 179)(128 180)(129 161)(130 162)(131 163)(132 164)(133 165)(134 166)(135 167)(136 168)(137 169)(138 170)(139 171)(140 172)(181 207)(182 208)(183 209)(184 210)(185 211)(186 212)(187 213)(188 214)(189 215)(190 216)(191 217)(192 218)(193 219)(194 220)(195 201)(196 202)(197 203)(198 204)(199 205)(200 206)
(1 37 210)(2 211 38)(3 39 212)(4 213 40)(5 21 214)(6 215 22)(7 23 216)(8 217 24)(9 25 218)(10 219 26)(11 27 220)(12 201 28)(13 29 202)(14 203 30)(15 31 204)(16 205 32)(17 33 206)(18 207 34)(19 35 208)(20 209 36)(41 63 172)(42 173 64)(43 65 174)(44 175 66)(45 67 176)(46 177 68)(47 69 178)(48 179 70)(49 71 180)(50 161 72)(51 73 162)(52 163 74)(53 75 164)(54 165 76)(55 77 166)(56 167 78)(57 79 168)(58 169 80)(59 61 170)(60 171 62)(81 221 186)(82 187 222)(83 223 188)(84 189 224)(85 225 190)(86 191 226)(87 227 192)(88 193 228)(89 229 194)(90 195 230)(91 231 196)(92 197 232)(93 233 198)(94 199 234)(95 235 200)(96 181 236)(97 237 182)(98 183 238)(99 239 184)(100 185 240)(101 137 156)(102 157 138)(103 139 158)(104 159 140)(105 121 160)(106 141 122)(107 123 142)(108 143 124)(109 125 144)(110 145 126)(111 127 146)(112 147 128)(113 129 148)(114 149 130)(115 131 150)(116 151 132)(117 133 152)(118 153 134)(119 135 154)(120 155 136)
(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)
(1 115)(2 114)(3 113)(4 112)(5 111)(6 110)(7 109)(8 108)(9 107)(10 106)(11 105)(12 104)(13 103)(14 102)(15 101)(16 120)(17 119)(18 118)(19 117)(20 116)(21 146)(22 145)(23 144)(24 143)(25 142)(26 141)(27 160)(28 159)(29 158)(30 157)(31 156)(32 155)(33 154)(34 153)(35 152)(36 151)(37 150)(38 149)(39 148)(40 147)(41 90)(42 89)(43 88)(44 87)(45 86)(46 85)(47 84)(48 83)(49 82)(50 81)(51 100)(52 99)(53 98)(54 97)(55 96)(56 95)(57 94)(58 93)(59 92)(60 91)(61 232)(62 231)(63 230)(64 229)(65 228)(66 227)(67 226)(68 225)(69 224)(70 223)(71 222)(72 221)(73 240)(74 239)(75 238)(76 237)(77 236)(78 235)(79 234)(80 233)(121 220)(122 219)(123 218)(124 217)(125 216)(126 215)(127 214)(128 213)(129 212)(130 211)(131 210)(132 209)(133 208)(134 207)(135 206)(136 205)(137 204)(138 203)(139 202)(140 201)(161 186)(162 185)(163 184)(164 183)(165 182)(166 181)(167 200)(168 199)(169 198)(170 197)(171 196)(172 195)(173 194)(174 193)(175 192)(176 191)(177 190)(178 189)(179 188)(180 187)

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D5 D6 D6 D10 D10 C3⋊D4 D20 S3×D5 C3⋊D20 C2×S3×D5 kernel C22×C3⋊D20 C2×C3⋊D20 Dic3×C2×C10 D5×C22×C6 C23×D15 C23×D5 C2×C30 C22×Dic3 C22×D5 C22×C10 C2×Dic3 C22×C6 C2×C10 C2×C6 C23 C22 C22 # reps 1 12 1 1 1 1 4 2 6 1 12 2 8 16 2 8 6

Matrix representation of C22×C3⋊D20 in GL5(𝔽61)

 60 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 60
,
 1 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 47 0 0 0 0 0 13
,
 1 0 0 0 0 0 0 60 0 0 0 1 17 0 0 0 0 0 0 1 0 0 0 60 0
,
 1 0 0 0 0 0 44 60 0 0 0 44 17 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,47,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,60,17,0,0,0,0,0,0,60,0,0,0,1,0],[1,0,0,0,0,0,44,44,0,0,0,60,17,0,0,0,0,0,0,1,0,0,0,1,0] >;

C22×C3⋊D20 in GAP, Magma, Sage, TeX

C_2^2\times C_3\rtimes D_{20}
% in TeX

G:=Group("C2^2xC3:D20");
// GroupNames label

G:=SmallGroup(480,1119);
// by ID

G=gap.SmallGroup(480,1119);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^20=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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