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## G = C22×S3×Dic5order 480 = 25·3·5

### Direct product of C22, S3 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C22×S3×Dic5
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — S3×Dic5 — C2×S3×Dic5 — C22×S3×Dic5
 Lower central C15 — C22×S3×Dic5
 Upper central C1 — C23

Generators and relations for C22×S3×Dic5
G = < a,b,c,d,e,f | a2=b2=c3=d2=e10=1, f2=e5, 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: 1564 in 472 conjugacy classes, 228 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×8], C22 [×7], C22 [×28], C5, S3 [×8], C6, C6 [×6], C2×C4 [×28], C23, C23 [×14], C10, C10 [×6], C10 [×8], Dic3 [×4], C12 [×4], D6 [×28], C2×C6 [×7], C15, C22×C4 [×14], C24, Dic5 [×4], Dic5 [×4], C2×C10 [×7], C2×C10 [×28], C4×S3 [×16], C2×Dic3 [×6], C2×C12 [×6], C22×S3 [×14], C22×C6, C5×S3 [×8], C30, C30 [×6], C23×C4, C2×Dic5 [×6], C2×Dic5 [×22], C22×C10, C22×C10 [×14], S3×C2×C4 [×12], C22×Dic3, C22×C12, S3×C23, C3×Dic5 [×4], Dic15 [×4], S3×C10 [×28], C2×C30 [×7], C22×Dic5, C22×Dic5 [×13], C23×C10, S3×C22×C4, S3×Dic5 [×16], C6×Dic5 [×6], C2×Dic15 [×6], S3×C2×C10 [×14], C22×C30, C23×Dic5, C2×S3×Dic5 [×12], C2×C6×Dic5, C22×Dic15, S3×C22×C10, C22×S3×Dic5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D5, D6 [×7], C22×C4 [×14], C24, Dic5 [×8], D10 [×7], C4×S3 [×4], C22×S3 [×7], C23×C4, C2×Dic5 [×28], C22×D5 [×7], S3×C2×C4 [×6], S3×C23, S3×D5, C22×Dic5 [×14], C23×D5, S3×C22×C4, S3×Dic5 [×4], C2×S3×D5 [×3], C23×Dic5, C2×S3×Dic5 [×6], C22×S3×D5, C22×S3×Dic5

Smallest permutation representation of C22×S3×Dic5
On 240 points
Generators in S240
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 111)(9 112)(10 113)(11 157)(12 158)(13 159)(14 160)(15 151)(16 152)(17 153)(18 154)(19 155)(20 156)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 129)(38 130)(39 121)(40 122)(41 76)(42 77)(43 78)(44 79)(45 80)(46 71)(47 72)(48 73)(49 74)(50 75)(51 86)(52 87)(53 88)(54 89)(55 90)(56 81)(57 82)(58 83)(59 84)(60 85)(61 96)(62 97)(63 98)(64 99)(65 100)(66 91)(67 92)(68 93)(69 94)(70 95)(131 226)(132 227)(133 228)(134 229)(135 230)(136 221)(137 222)(138 223)(139 224)(140 225)(141 236)(142 237)(143 238)(144 239)(145 240)(146 231)(147 232)(148 233)(149 234)(150 235)(161 196)(162 197)(163 198)(164 199)(165 200)(166 191)(167 192)(168 193)(169 194)(170 195)(171 206)(172 207)(173 208)(174 209)(175 210)(176 201)(177 202)(178 203)(179 204)(180 205)(181 216)(182 217)(183 218)(184 219)(185 220)(186 211)(187 212)(188 213)(189 214)(190 215)
(1 59)(2 60)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 212)(12 213)(13 214)(14 215)(15 216)(16 217)(17 218)(18 219)(19 220)(20 211)(21 46)(22 47)(23 48)(24 49)(25 50)(26 41)(27 42)(28 43)(29 44)(30 45)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(91 128)(92 129)(93 130)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 127)(131 161)(132 162)(133 163)(134 164)(135 165)(136 166)(137 167)(138 168)(139 169)(140 170)(141 171)(142 172)(143 173)(144 174)(145 175)(146 176)(147 177)(148 178)(149 179)(150 180)(151 181)(152 182)(153 183)(154 184)(155 185)(156 186)(157 187)(158 188)(159 189)(160 190)(191 221)(192 222)(193 223)(194 224)(195 225)(196 226)(197 227)(198 228)(199 229)(200 230)(201 231)(202 232)(203 233)(204 234)(205 235)(206 236)(207 237)(208 238)(209 239)(210 240)
(1 23 33)(2 24 34)(3 25 35)(4 26 36)(5 27 37)(6 28 38)(7 29 39)(8 30 40)(9 21 31)(10 22 32)(11 231 227)(12 232 228)(13 233 229)(14 234 230)(15 235 221)(16 236 222)(17 237 223)(18 238 224)(19 239 225)(20 240 226)(41 66 52)(42 67 53)(43 68 54)(44 69 55)(45 70 56)(46 61 57)(47 62 58)(48 63 59)(49 64 60)(50 65 51)(71 96 82)(72 97 83)(73 98 84)(74 99 85)(75 100 86)(76 91 87)(77 92 88)(78 93 89)(79 94 90)(80 95 81)(101 123 112)(102 124 113)(103 125 114)(104 126 115)(105 127 116)(106 128 117)(107 129 118)(108 130 119)(109 121 120)(110 122 111)(131 156 145)(132 157 146)(133 158 147)(134 159 148)(135 160 149)(136 151 150)(137 152 141)(138 153 142)(139 154 143)(140 155 144)(161 186 175)(162 187 176)(163 188 177)(164 189 178)(165 190 179)(166 181 180)(167 182 171)(168 183 172)(169 184 173)(170 185 174)(191 216 205)(192 217 206)(193 218 207)(194 219 208)(195 220 209)(196 211 210)(197 212 201)(198 213 202)(199 214 203)(200 215 204)
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 111)(9 112)(10 113)(11 132)(12 133)(13 134)(14 135)(15 136)(16 137)(17 138)(18 139)(19 140)(20 131)(21 123)(22 124)(23 125)(24 126)(25 127)(26 128)(27 129)(28 130)(29 121)(30 122)(31 101)(32 102)(33 103)(34 104)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 86)(52 87)(53 88)(54 89)(55 90)(56 81)(57 82)(58 83)(59 84)(60 85)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(141 236)(142 237)(143 238)(144 239)(145 240)(146 231)(147 232)(148 233)(149 234)(150 235)(151 221)(152 222)(153 223)(154 224)(155 225)(156 226)(157 227)(158 228)(159 229)(160 230)(161 211)(162 212)(163 213)(164 214)(165 215)(166 216)(167 217)(168 218)(169 219)(170 220)(171 206)(172 207)(173 208)(174 209)(175 210)(176 201)(177 202)(178 203)(179 204)(180 205)(181 191)(182 192)(183 193)(184 194)(185 195)(186 196)(187 197)(188 198)(189 199)(190 200)
(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)
(1 206 6 201)(2 205 7 210)(3 204 8 209)(4 203 9 208)(5 202 10 207)(11 63 16 68)(12 62 17 67)(13 61 18 66)(14 70 19 65)(15 69 20 64)(21 194 26 199)(22 193 27 198)(23 192 28 197)(24 191 29 196)(25 200 30 195)(31 219 36 214)(32 218 37 213)(33 217 38 212)(34 216 39 211)(35 215 40 220)(41 229 46 224)(42 228 47 223)(43 227 48 222)(44 226 49 221)(45 225 50 230)(51 234 56 239)(52 233 57 238)(53 232 58 237)(54 231 59 236)(55 240 60 235)(71 139 76 134)(72 138 77 133)(73 137 78 132)(74 136 79 131)(75 135 80 140)(81 144 86 149)(82 143 87 148)(83 142 88 147)(84 141 89 146)(85 150 90 145)(91 159 96 154)(92 158 97 153)(93 157 98 152)(94 156 99 151)(95 155 100 160)(101 169 106 164)(102 168 107 163)(103 167 108 162)(104 166 109 161)(105 165 110 170)(111 174 116 179)(112 173 117 178)(113 172 118 177)(114 171 119 176)(115 180 120 175)(121 186 126 181)(122 185 127 190)(123 184 128 189)(124 183 129 188)(125 182 130 187)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 4A ··· 4H 4I ··· 4P 5A 5B 6A ··· 6G 10A ··· 10N 10O ··· 10AD 12A ··· 12H 15A 15B 30A ··· 30N order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 4 ··· 4 5 5 6 ··· 6 10 ··· 10 10 ··· 10 12 ··· 12 15 15 30 ··· 30 size 1 1 ··· 1 3 ··· 3 2 5 ··· 5 15 ··· 15 2 2 2 ··· 2 2 ··· 2 6 ··· 6 10 ··· 10 4 4 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 D6 D6 Dic5 D10 D10 C4×S3 S3×D5 S3×Dic5 C2×S3×D5 kernel C22×S3×Dic5 C2×S3×Dic5 C2×C6×Dic5 C22×Dic15 S3×C22×C10 S3×C2×C10 C22×Dic5 S3×C23 C2×Dic5 C22×C10 C22×S3 C22×S3 C22×C6 C2×C10 C23 C22 C22 # reps 1 12 1 1 1 16 1 2 6 1 16 12 2 8 2 8 6

Matrix representation of C22×S3×Dic5 in GL5(𝔽61)

 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
,
 60 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 0 60 0 0 0 1 60 0 0 0 0 0 1 0 0 0 0 0 1
,
 60 0 0 0 0 0 0 60 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 1 60 0 0 0 45 17
,
 1 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 25 9 0 0 0 5 36

G:=sub<GL(5,GF(61))| [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],[60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,60,60,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,45,0,0,0,60,17],[1,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,25,5,0,0,0,9,36] >;

C22×S3×Dic5 in GAP, Magma, Sage, TeX

C_2^2\times S_3\times {\rm Dic}_5
% in TeX

G:=Group("C2^2xS3xDic5");
// GroupNames label

G:=SmallGroup(480,1115);
// by ID

G=gap.SmallGroup(480,1115);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^2=e^10=1,f^2=e^5,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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𝔽