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## G = C22×C5⋊D12order 480 = 25·3·5

### Direct product of C22 and C5⋊D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C22×C5⋊D12
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C5⋊D12 — C2×C5⋊D12 — C22×C5⋊D12
 Lower central C15 — C30 — C22×C5⋊D12
 Upper central C1 — C23

Generators and relations for C22×C5⋊D12
G = < a,b,c,d,e | a2=b2=c5=d12=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: 2524 in 472 conjugacy classes, 148 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C10, C12, D6, D6, C2×C6, C15, C22×C4, C2×D4, C24, Dic5, D10, C2×C10, C2×C10, D12, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, D15, C30, C30, C22×D4, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C22×C10, C2×D12, C22×C12, S3×C23, S3×C23, C3×Dic5, S3×C10, S3×C10, D30, D30, C2×C30, C22×Dic5, C2×C5⋊D4, C23×D5, C23×C10, C22×D12, C5⋊D12, C6×Dic5, S3×C2×C10, S3×C2×C10, C22×D15, C22×D15, C22×C30, C22×C5⋊D4, C2×C5⋊D12, C2×C6×Dic5, S3×C22×C10, C23×D15, C22×C5⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C24, D10, D12, C22×S3, C22×D4, C5⋊D4, C22×D5, C2×D12, S3×C23, S3×D5, C2×C5⋊D4, C23×D5, C22×D12, C5⋊D12, C2×S3×D5, C22×C5⋊D4, C2×C5⋊D12, C22×S3×D5, C22×C5⋊D12

Smallest permutation representation of C22×C5⋊D12
On 240 points
Generators in S240
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 61)(10 62)(11 63)(12 64)(13 153)(14 154)(15 155)(16 156)(17 145)(18 146)(19 147)(20 148)(21 149)(22 150)(23 151)(24 152)(25 228)(26 217)(27 218)(28 219)(29 220)(30 221)(31 222)(32 223)(33 224)(34 225)(35 226)(36 227)(37 175)(38 176)(39 177)(40 178)(41 179)(42 180)(43 169)(44 170)(45 171)(46 172)(47 173)(48 174)(49 125)(50 126)(51 127)(52 128)(53 129)(54 130)(55 131)(56 132)(57 121)(58 122)(59 123)(60 124)(73 235)(74 236)(75 237)(76 238)(77 239)(78 240)(79 229)(80 230)(81 231)(82 232)(83 233)(84 234)(85 168)(86 157)(87 158)(88 159)(89 160)(90 161)(91 162)(92 163)(93 164)(94 165)(95 166)(96 167)(97 139)(98 140)(99 141)(100 142)(101 143)(102 144)(103 133)(104 134)(105 135)(106 136)(107 137)(108 138)(109 200)(110 201)(111 202)(112 203)(113 204)(114 193)(115 194)(116 195)(117 196)(118 197)(119 198)(120 199)(181 207)(182 208)(183 209)(184 210)(185 211)(186 212)(187 213)(188 214)(189 215)(190 216)(191 205)(192 206)
(1 175)(2 176)(3 177)(4 178)(5 179)(6 180)(7 169)(8 170)(9 171)(10 172)(11 173)(12 174)(13 144)(14 133)(15 134)(16 135)(17 136)(18 137)(19 138)(20 139)(21 140)(22 141)(23 142)(24 143)(25 162)(26 163)(27 164)(28 165)(29 166)(30 167)(31 168)(32 157)(33 158)(34 159)(35 160)(36 161)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 61)(46 62)(47 63)(48 64)(49 231)(50 232)(51 233)(52 234)(53 235)(54 236)(55 237)(56 238)(57 239)(58 240)(59 229)(60 230)(73 129)(74 130)(75 131)(76 132)(77 121)(78 122)(79 123)(80 124)(81 125)(82 126)(83 127)(84 128)(85 222)(86 223)(87 224)(88 225)(89 226)(90 227)(91 228)(92 217)(93 218)(94 219)(95 220)(96 221)(97 148)(98 149)(99 150)(100 151)(101 152)(102 153)(103 154)(104 155)(105 156)(106 145)(107 146)(108 147)(109 214)(110 215)(111 216)(112 205)(113 206)(114 207)(115 208)(116 209)(117 210)(118 211)(119 212)(120 213)(181 193)(182 194)(183 195)(184 196)(185 197)(186 198)(187 199)(188 200)(189 201)(190 202)(191 203)(192 204)
(1 53 161 186 14)(2 15 187 162 54)(3 55 163 188 16)(4 17 189 164 56)(5 57 165 190 18)(6 19 191 166 58)(7 59 167 192 20)(8 21 181 168 60)(9 49 157 182 22)(10 23 183 158 50)(11 51 159 184 24)(12 13 185 160 52)(25 236 176 134 199)(26 200 135 177 237)(27 238 178 136 201)(28 202 137 179 239)(29 240 180 138 203)(30 204 139 169 229)(31 230 170 140 193)(32 194 141 171 231)(33 232 172 142 195)(34 196 143 173 233)(35 234 174 144 197)(36 198 133 175 235)(37 73 227 119 103)(38 104 120 228 74)(39 75 217 109 105)(40 106 110 218 76)(41 77 219 111 107)(42 108 112 220 78)(43 79 221 113 97)(44 98 114 222 80)(45 81 223 115 99)(46 100 116 224 82)(47 83 225 117 101)(48 102 118 226 84)(61 125 86 208 150)(62 151 209 87 126)(63 127 88 210 152)(64 153 211 89 128)(65 129 90 212 154)(66 155 213 91 130)(67 131 92 214 156)(68 145 215 93 132)(69 121 94 216 146)(70 147 205 95 122)(71 123 96 206 148)(72 149 207 85 124)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 34)(26 33)(27 32)(28 31)(29 30)(35 36)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 68)(62 67)(63 66)(64 65)(69 72)(70 71)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 94)(86 93)(87 92)(88 91)(89 90)(95 96)(97 108)(98 107)(99 106)(100 105)(101 104)(102 103)(109 116)(110 115)(111 114)(112 113)(117 120)(118 119)(121 124)(122 123)(125 132)(126 131)(127 130)(128 129)(133 144)(134 143)(135 142)(136 141)(137 140)(138 139)(145 150)(146 149)(147 148)(151 156)(152 155)(153 154)(157 164)(158 163)(159 162)(160 161)(165 168)(166 167)(169 180)(170 179)(171 178)(172 177)(173 176)(174 175)(181 190)(182 189)(183 188)(184 187)(185 186)(191 192)(193 202)(194 201)(195 200)(196 199)(197 198)(203 204)(205 206)(207 216)(208 215)(209 214)(210 213)(211 212)(217 224)(218 223)(219 222)(220 221)(225 228)(226 227)(229 240)(230 239)(231 238)(232 237)(233 236)(234 235)

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 5A 5B 6A ··· 6G 10A ··· 10N 10O ··· 10AD 12A ··· 12H 15A 15B 30A ··· 30N order 1 2 ··· 2 2 2 2 2 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 6 6 6 6 30 30 30 30 2 10 10 10 10 2 2 2 ··· 2 2 ··· 2 6 ··· 6 10 ··· 10 4 4 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 D12 C5⋊D4 S3×D5 C5⋊D12 C2×S3×D5 kernel C22×C5⋊D12 C2×C5⋊D12 C2×C6×Dic5 S3×C22×C10 C23×D15 C22×Dic5 C2×C30 S3×C23 C2×Dic5 C22×C10 C22×S3 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×C5⋊D12 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
,
 60 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60
,
 1 0 0 0 0 0 9 16 0 0 0 0 34 0 0 0 0 0 1 0 0 0 0 0 1
,
 60 0 0 0 0 0 25 48 0 0 0 20 36 0 0 0 0 0 1 53 0 0 0 8 59
,
 1 0 0 0 0 0 36 19 0 0 0 41 25 0 0 0 0 0 1 53 0 0 0 0 60

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],[60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,9,0,0,0,0,16,34,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,25,20,0,0,0,48,36,0,0,0,0,0,1,8,0,0,0,53,59],[1,0,0,0,0,0,36,41,0,0,0,19,25,0,0,0,0,0,1,0,0,0,0,53,60] >;

C22×C5⋊D12 in GAP, Magma, Sage, TeX

C_2^2\times C_5\rtimes D_{12}
% in TeX

G:=Group("C2^2xC5:D12");
// GroupNames label

G:=SmallGroup(480,1120);
// by ID

G=gap.SmallGroup(480,1120);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,253,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^5=d^12=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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