Copied to
clipboard

G = C22×C4×D15order 480 = 25·3·5

Direct product of C22×C4 and D15

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C22×C4×D15, C6010C23, C30.55C24, C23.39D30, D30.44C23, Dic1511C23, (C2×C20)⋊35D6, C158(C23×C4), (C2×C12)⋊35D10, C209(C22×S3), C308(C22×C4), C129(C22×D5), (C22×C60)⋊14C2, (C22×C20)⋊14S3, (C2×C60)⋊46C22, (C22×C12)⋊10D5, C2.1(C23×D15), C6.55(C23×D5), C10.55(S3×C23), (C23×D15).6C2, (C2×C30).319C23, (C22×C6).125D10, (C22×C10).143D6, (C22×Dic15)⋊21C2, (C2×Dic15)⋊39C22, C22.29(C22×D15), (C22×C30).148C22, (C22×D15).127C22, C63(C2×C4×D5), C104(S3×C2×C4), C54(S3×C22×C4), C33(D5×C22×C4), (C2×C6)⋊12(C4×D5), (C2×C10)⋊21(C4×S3), (C2×C30)⋊34(C2×C4), (C2×C6).315(C22×D5), (C2×C10).314(C22×S3), SmallGroup(480,1166)

Series: Derived Chief Lower central Upper central

C1C15 — C22×C4×D15
C1C5C15C30D30C22×D15C23×D15 — C22×C4×D15
C15 — C22×C4×D15
C1C22×C4

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

Subgroups: 2292 in 472 conjugacy classes, 199 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C4 [×4], C22 [×7], C22 [×28], C5, S3 [×8], C6, C6 [×6], C2×C4 [×6], C2×C4 [×22], C23, C23 [×14], D5 [×8], C10, C10 [×6], Dic3 [×4], C12 [×4], D6 [×28], C2×C6 [×7], C15, C22×C4, C22×C4 [×13], C24, Dic5 [×4], C20 [×4], D10 [×28], C2×C10 [×7], C4×S3 [×16], C2×Dic3 [×6], C2×C12 [×6], C22×S3 [×14], C22×C6, D15 [×8], C30, C30 [×6], C23×C4, C4×D5 [×16], C2×Dic5 [×6], C2×C20 [×6], C22×D5 [×14], C22×C10, S3×C2×C4 [×12], C22×Dic3, C22×C12, S3×C23, Dic15 [×4], C60 [×4], D30 [×28], C2×C30 [×7], C2×C4×D5 [×12], C22×Dic5, C22×C20, C23×D5, S3×C22×C4, C4×D15 [×16], C2×Dic15 [×6], C2×C60 [×6], C22×D15 [×14], C22×C30, D5×C22×C4, C2×C4×D15 [×12], C22×Dic15, C22×C60, C23×D15, C22×C4×D15
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D5, D6 [×7], C22×C4 [×14], C24, D10 [×7], C4×S3 [×4], C22×S3 [×7], D15, C23×C4, C4×D5 [×4], C22×D5 [×7], S3×C2×C4 [×6], S3×C23, D30 [×7], C2×C4×D5 [×6], C23×D5, S3×C22×C4, C4×D15 [×4], C22×D15 [×7], D5×C22×C4, C2×C4×D15 [×6], C23×D15, C22×C4×D15

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

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

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

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

144 conjugacy classes

class 1 2A···2G2H···2O 3 4A···4H4I···4P5A5B6A···6G10A···10N12A···12H15A15B15C15D20A···20P30A···30AB60A···60AF
order12···22···234···44···4556···610···1012···121515151520···2030···3060···60
size11···115···1521···115···15222···22···22···222222···22···22···2

144 irreducible representations

dim111111222222222222
type++++++++++++++
imageC1C2C2C2C2C4S3D5D6D6D10D10C4×S3D15C4×D5D30D30C4×D15
kernelC22×C4×D15C2×C4×D15C22×Dic15C22×C60C23×D15C22×D15C22×C20C22×C12C2×C20C22×C10C2×C12C22×C6C2×C10C22×C4C2×C6C2×C4C23C22
# reps112111161261122841624432

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

1000
06000
0010
0001
,
60000
06000
00600
00060
,
11000
0100
0010
0001
,
1000
0100
002314
003945
,
60000
0100
00160
00060
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[11,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,23,39,0,0,14,45],[60,0,0,0,0,1,0,0,0,0,1,0,0,0,60,60] >;

C22×C4×D15 in GAP, Magma, Sage, TeX

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

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

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

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

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

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

׿
×
𝔽