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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 [×6], C2 [×8], C3, C4 [×4], C22 [×7], C22 [×32], C5, S3 [×8], C6, C6 [×6], C2×C4 [×6], D4 [×16], C23, C23 [×20], D5 [×4], C10, C10 [×6], C10 [×4], C12 [×4], D6 [×4], D6 [×28], C2×C6 [×7], C15, C22×C4, C2×D4 [×12], C24 [×2], Dic5 [×4], D10 [×16], C2×C10 [×7], C2×C10 [×16], D12 [×16], C2×C12 [×6], C22×S3 [×6], C22×S3 [×14], C22×C6, C5×S3 [×4], D15 [×4], C30, C30 [×6], C22×D4, C2×Dic5 [×6], C5⋊D4 [×16], C22×D5 [×10], C22×C10, C22×C10 [×10], C2×D12 [×12], C22×C12, S3×C23, S3×C23, C3×Dic5 [×4], S3×C10 [×4], S3×C10 [×12], D30 [×4], D30 [×12], C2×C30 [×7], C22×Dic5, C2×C5⋊D4 [×12], C23×D5, C23×C10, C22×D12, C5⋊D12 [×16], C6×Dic5 [×6], S3×C2×C10 [×6], S3×C2×C10 [×4], C22×D15 [×6], C22×D15 [×4], C22×C30, C22×C5⋊D4, C2×C5⋊D12 [×12], C2×C6×Dic5, S3×C22×C10, C23×D15, C22×C5⋊D12
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], D12 [×4], C22×S3 [×7], C22×D4, C5⋊D4 [×4], C22×D5 [×7], C2×D12 [×6], S3×C23, S3×D5, C2×C5⋊D4 [×6], C23×D5, C22×D12, C5⋊D12 [×4], C2×S3×D5 [×3], C22×C5⋊D4, C2×C5⋊D12 [×6], C22×S3×D5, C22×C5⋊D12

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

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

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

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

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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