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## G = C22×C4×D15order 480 = 25·3·5

### Direct product of C22×C4 and D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C22×C4×D15
 Chief series C1 — C5 — C15 — C30 — D30 — C22×D15 — C23×D15 — C22×C4×D15
 Lower central C15 — C22×C4×D15
 Upper central C1 — C22×C4

Generators and relations for C22×C4×D15
G = < a,b,c,d,e | a2=b2=c4=d15=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 2292 in 472 conjugacy classes, 199 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, D15, C30, C30, C23×C4, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C10, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, Dic15, C60, D30, C2×C30, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, S3×C22×C4, C4×D15, C2×Dic15, C2×C60, C22×D15, C22×C30, D5×C22×C4, C2×C4×D15, C22×Dic15, C22×C60, C23×D15, C22×C4×D15
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, C24, D10, C4×S3, C22×S3, D15, C23×C4, C4×D5, C22×D5, S3×C2×C4, S3×C23, D30, C2×C4×D5, C23×D5, S3×C22×C4, C4×D15, C22×D15, D5×C22×C4, C2×C4×D15, C23×D15, C22×C4×D15

Smallest permutation representation of C22×C4×D15
On 240 points
Generators in S240
(1 144)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 136)(9 137)(10 138)(11 139)(12 140)(13 141)(14 142)(15 143)(16 125)(17 126)(18 127)(19 128)(20 129)(21 130)(22 131)(23 132)(24 133)(25 134)(26 135)(27 121)(28 122)(29 123)(30 124)(31 167)(32 168)(33 169)(34 170)(35 171)(36 172)(37 173)(38 174)(39 175)(40 176)(41 177)(42 178)(43 179)(44 180)(45 166)(46 151)(47 152)(48 153)(49 154)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 161)(57 162)(58 163)(59 164)(60 165)(61 199)(62 200)(63 201)(64 202)(65 203)(66 204)(67 205)(68 206)(69 207)(70 208)(71 209)(72 210)(73 196)(74 197)(75 198)(76 181)(77 182)(78 183)(79 184)(80 185)(81 186)(82 187)(83 188)(84 189)(85 190)(86 191)(87 192)(88 193)(89 194)(90 195)(91 227)(92 228)(93 229)(94 230)(95 231)(96 232)(97 233)(98 234)(99 235)(100 236)(101 237)(102 238)(103 239)(104 240)(105 226)(106 215)(107 216)(108 217)(109 218)(110 219)(111 220)(112 221)(113 222)(114 223)(115 224)(116 225)(117 211)(118 212)(119 213)(120 214)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 61)(28 62)(29 63)(30 64)(31 114)(32 115)(33 116)(34 117)(35 118)(36 119)(37 120)(38 106)(39 107)(40 108)(41 109)(42 110)(43 111)(44 112)(45 113)(46 98)(47 99)(48 100)(49 101)(50 102)(51 103)(52 104)(53 105)(54 91)(55 92)(56 93)(57 94)(58 95)(59 96)(60 97)(121 199)(122 200)(123 201)(124 202)(125 203)(126 204)(127 205)(128 206)(129 207)(130 208)(131 209)(132 210)(133 196)(134 197)(135 198)(136 188)(137 189)(138 190)(139 191)(140 192)(141 193)(142 194)(143 195)(144 181)(145 182)(146 183)(147 184)(148 185)(149 186)(150 187)(151 234)(152 235)(153 236)(154 237)(155 238)(156 239)(157 240)(158 226)(159 227)(160 228)(161 229)(162 230)(163 231)(164 232)(165 233)(166 222)(167 223)(168 224)(169 225)(170 211)(171 212)(172 213)(173 214)(174 215)(175 216)(176 217)(177 218)(178 219)(179 220)(180 221)
(1 31 28 54)(2 32 29 55)(3 33 30 56)(4 34 16 57)(5 35 17 58)(6 36 18 59)(7 37 19 60)(8 38 20 46)(9 39 21 47)(10 40 22 48)(11 41 23 49)(12 42 24 50)(13 43 25 51)(14 44 26 52)(15 45 27 53)(61 105 90 113)(62 91 76 114)(63 92 77 115)(64 93 78 116)(65 94 79 117)(66 95 80 118)(67 96 81 119)(68 97 82 120)(69 98 83 106)(70 99 84 107)(71 100 85 108)(72 101 86 109)(73 102 87 110)(74 103 88 111)(75 104 89 112)(121 158 143 166)(122 159 144 167)(123 160 145 168)(124 161 146 169)(125 162 147 170)(126 163 148 171)(127 164 149 172)(128 165 150 173)(129 151 136 174)(130 152 137 175)(131 153 138 176)(132 154 139 177)(133 155 140 178)(134 156 141 179)(135 157 142 180)(181 223 200 227)(182 224 201 228)(183 225 202 229)(184 211 203 230)(185 212 204 231)(186 213 205 232)(187 214 206 233)(188 215 207 234)(189 216 208 235)(190 217 209 236)(191 218 210 237)(192 219 196 238)(193 220 197 239)(194 221 198 240)(195 222 199 226)
(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 61)(2 75)(3 74)(4 73)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 65)(13 64)(14 63)(15 62)(16 87)(17 86)(18 85)(19 84)(20 83)(21 82)(22 81)(23 80)(24 79)(25 78)(26 77)(27 76)(28 90)(29 89)(30 88)(31 105)(32 104)(33 103)(34 102)(35 101)(36 100)(37 99)(38 98)(39 97)(40 96)(41 95)(42 94)(43 93)(44 92)(45 91)(46 106)(47 120)(48 119)(49 118)(50 117)(51 116)(52 115)(53 114)(54 113)(55 112)(56 111)(57 110)(58 109)(59 108)(60 107)(121 181)(122 195)(123 194)(124 193)(125 192)(126 191)(127 190)(128 189)(129 188)(130 187)(131 186)(132 185)(133 184)(134 183)(135 182)(136 207)(137 206)(138 205)(139 204)(140 203)(141 202)(142 201)(143 200)(144 199)(145 198)(146 197)(147 196)(148 210)(149 209)(150 208)(151 215)(152 214)(153 213)(154 212)(155 211)(156 225)(157 224)(158 223)(159 222)(160 221)(161 220)(162 219)(163 218)(164 217)(165 216)(166 227)(167 226)(168 240)(169 239)(170 238)(171 237)(172 236)(173 235)(174 234)(175 233)(176 232)(177 231)(178 230)(179 229)(180 228)

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

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

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

144 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 4A ··· 4H 4I ··· 4P 5A 5B 6A ··· 6G 10A ··· 10N 12A ··· 12H 15A 15B 15C 15D 20A ··· 20P 30A ··· 30AB 60A ··· 60AF order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 4 ··· 4 5 5 6 ··· 6 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 ··· 1 15 ··· 15 2 1 ··· 1 15 ··· 15 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

144 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D5 D6 D6 D10 D10 C4×S3 D15 C4×D5 D30 D30 C4×D15 kernel C22×C4×D15 C2×C4×D15 C22×Dic15 C22×C60 C23×D15 C22×D15 C22×C20 C22×C12 C2×C20 C22×C10 C2×C12 C22×C6 C2×C10 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 12 1 1 1 16 1 2 6 1 12 2 8 4 16 24 4 32

Matrix representation of C22×C4×D15 in GL4(𝔽61) generated by

 1 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 11 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 23 14 0 0 39 45
,
 60 0 0 0 0 1 0 0 0 0 1 60 0 0 0 60
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[11,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,23,39,0,0,14,45],[60,0,0,0,0,1,0,0,0,0,1,0,0,0,60,60] >;

C22×C4×D15 in GAP, Magma, Sage, TeX

C_2^2\times C_4\times D_{15}
% in TeX

G:=Group("C2^2xC4xD15");
// GroupNames label

G:=SmallGroup(480,1166);
// by ID

G=gap.SmallGroup(480,1166);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^15=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,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
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𝔽