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## G = C22×D5×Dic3order 480 = 25·3·5

### Direct product of C22, D5 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C22×D5×Dic3
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×Dic3 — C2×D5×Dic3 — C22×D5×Dic3
 Lower central C15 — C22×D5×Dic3
 Upper central C1 — C23

Generators and relations for C22×D5×Dic3
G = < a,b,c,d,e,f | a2=b2=c5=d2=e6=1, f2=e3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 1724 in 472 conjugacy classes, 228 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, C6, C6, C6, C2×C4, C23, C23, D5, C10, C10, Dic3, Dic3, C2×C6, C2×C6, C15, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×Dic3, C2×Dic3, C22×C6, C22×C6, C3×D5, C30, C30, C23×C4, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C10, C22×Dic3, C22×Dic3, C23×C6, C5×Dic3, Dic15, C6×D5, C2×C30, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, C23×Dic3, D5×Dic3, C10×Dic3, C2×Dic15, D5×C2×C6, C22×C30, D5×C22×C4, C2×D5×Dic3, Dic3×C2×C10, C22×Dic15, D5×C22×C6, C22×D5×Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, Dic3, D6, C22×C4, C24, D10, C2×Dic3, C22×S3, C23×C4, C4×D5, C22×D5, C22×Dic3, S3×C23, S3×D5, C2×C4×D5, C23×D5, C23×Dic3, D5×Dic3, C2×S3×D5, D5×C22×C4, C2×D5×Dic3, C22×S3×D5, C22×D5×Dic3

Smallest permutation representation of C22×D5×Dic3
On 240 points
Generators in S240
(1 88)(2 89)(3 90)(4 85)(5 86)(6 87)(7 202)(8 203)(9 204)(10 199)(11 200)(12 201)(13 198)(14 193)(15 194)(16 195)(17 196)(18 197)(19 188)(20 189)(21 190)(22 191)(23 192)(24 187)(25 184)(26 185)(27 186)(28 181)(29 182)(30 183)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 132)(62 127)(63 128)(64 129)(65 130)(66 131)(67 122)(68 123)(69 124)(70 125)(71 126)(72 121)(73 144)(74 139)(75 140)(76 141)(77 142)(78 143)(79 134)(80 135)(81 136)(82 137)(83 138)(84 133)(145 205)(146 206)(147 207)(148 208)(149 209)(150 210)(151 211)(152 212)(153 213)(154 214)(155 215)(156 216)(157 217)(158 218)(159 219)(160 220)(161 221)(162 222)(163 223)(164 224)(165 225)(166 226)(167 227)(168 228)(169 229)(170 230)(171 231)(172 232)(173 233)(174 234)(175 235)(176 236)(177 237)(178 238)(179 239)(180 240)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 229)(8 230)(9 231)(10 232)(11 233)(12 234)(13 225)(14 226)(15 227)(16 228)(17 223)(18 224)(19 221)(20 222)(21 217)(22 218)(23 219)(24 220)(25 211)(26 212)(27 213)(28 214)(29 215)(30 216)(31 64)(32 65)(33 66)(34 61)(35 62)(36 63)(37 70)(38 71)(39 72)(40 67)(41 68)(42 69)(43 76)(44 77)(45 78)(46 73)(47 74)(48 75)(49 82)(50 83)(51 84)(52 79)(53 80)(54 81)(85 118)(86 119)(87 120)(88 115)(89 116)(90 117)(91 129)(92 130)(93 131)(94 132)(95 127)(96 128)(97 125)(98 126)(99 121)(100 122)(101 123)(102 124)(103 141)(104 142)(105 143)(106 144)(107 139)(108 140)(109 137)(110 138)(111 133)(112 134)(113 135)(114 136)(145 178)(146 179)(147 180)(148 175)(149 176)(150 177)(151 184)(152 185)(153 186)(154 181)(155 182)(156 183)(157 190)(158 191)(159 192)(160 187)(161 188)(162 189)(163 196)(164 197)(165 198)(166 193)(167 194)(168 195)(169 202)(170 203)(171 204)(172 199)(173 200)(174 201)(205 238)(206 239)(207 240)(208 235)(209 236)(210 237)
(1 39 51 48 31)(2 40 52 43 32)(3 41 53 44 33)(4 42 54 45 34)(5 37 49 46 35)(6 38 50 47 36)(7 24 235 26 14)(8 19 236 27 15)(9 20 237 28 16)(10 21 238 29 17)(11 22 239 30 18)(12 23 240 25 13)(55 72 84 75 64)(56 67 79 76 65)(57 68 80 77 66)(58 69 81 78 61)(59 70 82 73 62)(60 71 83 74 63)(85 102 114 105 94)(86 97 109 106 95)(87 98 110 107 96)(88 99 111 108 91)(89 100 112 103 92)(90 101 113 104 93)(115 121 133 140 129)(116 122 134 141 130)(117 123 135 142 131)(118 124 136 143 132)(119 125 137 144 127)(120 126 138 139 128)(145 155 163 172 157)(146 156 164 173 158)(147 151 165 174 159)(148 152 166 169 160)(149 153 167 170 161)(150 154 168 171 162)(175 185 193 202 187)(176 186 194 203 188)(177 181 195 204 189)(178 182 196 199 190)(179 183 197 200 191)(180 184 198 201 192)(205 215 223 232 217)(206 216 224 233 218)(207 211 225 234 219)(208 212 226 229 220)(209 213 227 230 221)(210 214 228 231 222)
(1 94)(2 95)(3 96)(4 91)(5 92)(6 93)(7 182)(8 183)(9 184)(10 185)(11 186)(12 181)(13 195)(14 196)(15 197)(16 198)(17 193)(18 194)(19 179)(20 180)(21 175)(22 176)(23 177)(24 178)(25 204)(26 199)(27 200)(28 201)(29 202)(30 203)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 103)(38 104)(39 105)(40 106)(41 107)(42 108)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(49 112)(50 113)(51 114)(52 109)(53 110)(54 111)(55 132)(56 127)(57 128)(58 129)(59 130)(60 131)(61 115)(62 116)(63 117)(64 118)(65 119)(66 120)(67 144)(68 139)(69 140)(70 141)(71 142)(72 143)(73 122)(74 123)(75 124)(76 125)(77 126)(78 121)(79 137)(80 138)(81 133)(82 134)(83 135)(84 136)(145 220)(146 221)(147 222)(148 217)(149 218)(150 219)(151 231)(152 232)(153 233)(154 234)(155 229)(156 230)(157 208)(158 209)(159 210)(160 205)(161 206)(162 207)(163 226)(164 227)(165 228)(166 223)(167 224)(168 225)(169 215)(170 216)(171 211)(172 212)(173 213)(174 214)(187 238)(188 239)(189 240)(190 235)(191 236)(192 237)
(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)
(1 239 4 236)(2 238 5 235)(3 237 6 240)(7 43 10 46)(8 48 11 45)(9 47 12 44)(13 53 16 50)(14 52 17 49)(15 51 18 54)(19 31 22 34)(20 36 23 33)(21 35 24 32)(25 41 28 38)(26 40 29 37)(27 39 30 42)(55 206 58 209)(56 205 59 208)(57 210 60 207)(61 221 64 218)(62 220 65 217)(63 219 66 222)(67 215 70 212)(68 214 71 211)(69 213 72 216)(73 229 76 232)(74 234 77 231)(75 233 78 230)(79 223 82 226)(80 228 83 225)(81 227 84 224)(85 176 88 179)(86 175 89 178)(87 180 90 177)(91 191 94 188)(92 190 95 187)(93 189 96 192)(97 185 100 182)(98 184 101 181)(99 183 102 186)(103 199 106 202)(104 204 107 201)(105 203 108 200)(109 193 112 196)(110 198 113 195)(111 197 114 194)(115 146 118 149)(116 145 119 148)(117 150 120 147)(121 156 124 153)(122 155 125 152)(123 154 126 151)(127 160 130 157)(128 159 131 162)(129 158 132 161)(133 164 136 167)(134 163 137 166)(135 168 138 165)(139 174 142 171)(140 173 143 170)(141 172 144 169)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 4A ··· 4H 4I ··· 4P 5A 5B 6A ··· 6G 6H ··· 6O 10A ··· 10N 15A 15B 20A ··· 20P 30A ··· 30N order 1 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 5 ··· 5 2 3 ··· 3 15 ··· 15 2 2 2 ··· 2 10 ··· 10 2 ··· 2 4 4 6 ··· 6 4 ··· 4

96 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - + + + + + - + image C1 C2 C2 C2 C2 C4 S3 D5 Dic3 D6 D6 D10 D10 C4×D5 S3×D5 D5×Dic3 C2×S3×D5 kernel C22×D5×Dic3 C2×D5×Dic3 Dic3×C2×C10 C22×Dic15 D5×C22×C6 D5×C2×C6 C23×D5 C22×Dic3 C22×D5 C22×D5 C22×C10 C2×Dic3 C22×C6 C2×C6 C23 C22 C22 # reps 1 12 1 1 1 16 1 2 8 6 1 12 2 16 2 8 6

Matrix representation of C22×D5×Dic3 in GL5(𝔽61)

 1 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 60 1 0 0 0 42 18 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 60 0 0 0 0 42 1 0 0 0 0 0 60 0 0 0 0 0 60
,
 60 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 59 20 0 0 0 9 1
,
 11 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 53 5 0 0 0 24 8

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[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],[1,0,0,0,0,0,60,42,0,0,0,1,18,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,60,42,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,59,9,0,0,0,20,1],[11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,53,24,0,0,0,5,8] >;

C22×D5×Dic3 in GAP, Magma, Sage, TeX

C_2^2\times D_5\times {\rm Dic}_3
% in TeX

G:=Group("C2^2xD5xDic3");
// GroupNames label

G:=SmallGroup(480,1112);
// by ID

G=gap.SmallGroup(480,1112);
# by ID

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

G:=Group<a,b,c,d,e,f|a^2=b^2=c^5=d^2=e^6=1,f^2=e^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

׿
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