Copied to
clipboard

G = C22×D30.C2order 480 = 25·3·5

Direct product of C22 and D30.C2

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

Aliases: C22×D30.C2, C30.45C24, D30.43C23, C157(C23×C4), D3031(C2×C4), C307(C22×C4), (C2×Dic5)⋊26D6, D153(C22×C4), C6.45(C23×D5), C23.69(S3×D5), (C2×Dic3)⋊26D10, (C22×D15)⋊11C4, C10.45(S3×C23), (C5×Dic3)⋊7C23, Dic37(C22×D5), Dic57(C22×S3), (C3×Dic5)⋊7C23, (C23×D15).5C2, (C2×C30).248C23, (C22×Dic5)⋊13S3, (C6×Dic5)⋊30C22, (C22×Dic3)⋊10D5, (C22×C6).100D10, (C22×C10).117D6, (C10×Dic3)⋊30C22, (C22×C30).86C22, (C22×D15).123C22, C62(C2×C4×D5), C103(S3×C2×C4), C53(S3×C22×C4), C32(D5×C22×C4), (C2×C6)⋊9(C4×D5), (C2×C10)⋊18(C4×S3), (C2×C30)⋊25(C2×C4), C2.4(C22×S3×D5), (C2×C6×Dic5)⋊10C2, (Dic3×C2×C10)⋊10C2, C22.108(C2×S3×D5), (C2×C6).254(C22×D5), (C2×C10).252(C22×S3), SmallGroup(480,1117)

Series: Derived Chief Lower central Upper central

C1C15 — C22×D30.C2
C1C5C15C30C3×Dic5D30.C2C2×D30.C2 — C22×D30.C2
C15 — C22×D30.C2
C1C23

Generators and relations for C22×D30.C2
 G = < a,b,c,d,e | a2=b2=c30=d2=1, e2=c15, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ece-1=c19, ede-1=c18d >

Subgroups: 2204 in 472 conjugacy classes, 188 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], D5 [×8], C10, C10 [×6], Dic3 [×4], C12 [×4], D6 [×28], C2×C6 [×7], C15, C22×C4 [×14], 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, C5×Dic3 [×4], C3×Dic5 [×4], D30 [×28], C2×C30 [×7], C2×C4×D5 [×12], C22×Dic5, C22×C20, C23×D5, S3×C22×C4, D30.C2 [×16], C6×Dic5 [×6], C10×Dic3 [×6], C22×D15 [×14], C22×C30, D5×C22×C4, C2×D30.C2 [×12], C2×C6×Dic5, Dic3×C2×C10, C23×D15, C22×D30.C2
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], C23×C4, C4×D5 [×4], C22×D5 [×7], S3×C2×C4 [×6], S3×C23, S3×D5, C2×C4×D5 [×6], C23×D5, S3×C22×C4, D30.C2 [×4], C2×S3×D5 [×3], D5×C22×C4, C2×D30.C2 [×6], C22×S3×D5, C22×D30.C2

Smallest permutation representation of C22×D30.C2
On 240 points
Generators in S240
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 115)(14 116)(15 117)(16 118)(17 119)(18 120)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 90)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 81)(53 82)(54 83)(55 84)(56 85)(57 86)(58 87)(59 88)(60 89)(121 218)(122 219)(123 220)(124 221)(125 222)(126 223)(127 224)(128 225)(129 226)(130 227)(131 228)(132 229)(133 230)(134 231)(135 232)(136 233)(137 234)(138 235)(139 236)(140 237)(141 238)(142 239)(143 240)(144 211)(145 212)(146 213)(147 214)(148 215)(149 216)(150 217)(151 195)(152 196)(153 197)(154 198)(155 199)(156 200)(157 201)(158 202)(159 203)(160 204)(161 205)(162 206)(163 207)(164 208)(165 209)(166 210)(167 181)(168 182)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 57)(27 58)(28 59)(29 60)(30 31)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 113)(72 114)(73 115)(74 116)(75 117)(76 118)(77 119)(78 120)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)(85 97)(86 98)(87 99)(88 100)(89 101)(90 102)(121 160)(122 161)(123 162)(124 163)(125 164)(126 165)(127 166)(128 167)(129 168)(130 169)(131 170)(132 171)(133 172)(134 173)(135 174)(136 175)(137 176)(138 177)(139 178)(140 179)(141 180)(142 151)(143 152)(144 153)(145 154)(146 155)(147 156)(148 157)(149 158)(150 159)(181 225)(182 226)(183 227)(184 228)(185 229)(186 230)(187 231)(188 232)(189 233)(190 234)(191 235)(192 236)(193 237)(194 238)(195 239)(196 240)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)
(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)
(1 102)(2 101)(3 100)(4 99)(5 98)(6 97)(7 96)(8 95)(9 94)(10 93)(11 92)(12 91)(13 120)(14 119)(15 118)(16 117)(17 116)(18 115)(19 114)(20 113)(21 112)(22 111)(23 110)(24 109)(25 108)(26 107)(27 106)(28 105)(29 104)(30 103)(31 61)(32 90)(33 89)(34 88)(35 87)(36 86)(37 85)(38 84)(39 83)(40 82)(41 81)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 71)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(121 229)(122 228)(123 227)(124 226)(125 225)(126 224)(127 223)(128 222)(129 221)(130 220)(131 219)(132 218)(133 217)(134 216)(135 215)(136 214)(137 213)(138 212)(139 211)(140 240)(141 239)(142 238)(143 237)(144 236)(145 235)(146 234)(147 233)(148 232)(149 231)(150 230)(151 194)(152 193)(153 192)(154 191)(155 190)(156 189)(157 188)(158 187)(159 186)(160 185)(161 184)(162 183)(163 182)(164 181)(165 210)(166 209)(167 208)(168 207)(169 206)(170 205)(171 204)(172 203)(173 202)(174 201)(175 200)(176 199)(177 198)(178 197)(179 196)(180 195)
(1 195 16 210)(2 184 17 199)(3 203 18 188)(4 192 19 207)(5 181 20 196)(6 200 21 185)(7 189 22 204)(8 208 23 193)(9 197 24 182)(10 186 25 201)(11 205 26 190)(12 194 27 209)(13 183 28 198)(14 202 29 187)(15 191 30 206)(31 220 46 235)(32 239 47 224)(33 228 48 213)(34 217 49 232)(35 236 50 221)(36 225 51 240)(37 214 52 229)(38 233 53 218)(39 222 54 237)(40 211 55 226)(41 230 56 215)(42 219 57 234)(43 238 58 223)(44 227 59 212)(45 216 60 231)(61 142 76 127)(62 131 77 146)(63 150 78 135)(64 139 79 124)(65 128 80 143)(66 147 81 132)(67 136 82 121)(68 125 83 140)(69 144 84 129)(70 133 85 148)(71 122 86 137)(72 141 87 126)(73 130 88 145)(74 149 89 134)(75 138 90 123)(91 163 106 178)(92 152 107 167)(93 171 108 156)(94 160 109 175)(95 179 110 164)(96 168 111 153)(97 157 112 172)(98 176 113 161)(99 165 114 180)(100 154 115 169)(101 173 116 158)(102 162 117 177)(103 151 118 166)(104 170 119 155)(105 159 120 174)

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

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

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

96 conjugacy classes

class 1 2A···2G2H···2O 3 4A···4H4I···4P5A5B6A···6G10A···10N12A···12H15A15B20A···20P30A···30N
order12···22···234···44···4556···610···1012···12151520···2030···30
size11···115···1523···35···5222···22···210···10446···64···4

96 irreducible representations

dim11111122222222444
type++++++++++++++
imageC1C2C2C2C2C4S3D5D6D6D10D10C4×S3C4×D5S3×D5D30.C2C2×S3×D5
kernelC22×D30.C2C2×D30.C2C2×C6×Dic5Dic3×C2×C10C23×D15C22×D15C22×Dic5C22×Dic3C2×Dic5C22×C10C2×Dic3C22×C6C2×C10C2×C6C23C22C22
# reps112111161261122816286

Matrix representation of C22×D30.C2 in GL5(𝔽61)

600000
060000
006000
00010
00001
,
10000
060000
006000
00010
00001
,
10000
0606000
01000
0001716
00011
,
10000
01100
006000
000145
000060
,
600000
01000
00100
0005757
000504

G:=sub<GL(5,GF(61))| [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],[1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,60,1,0,0,0,60,0,0,0,0,0,0,17,1,0,0,0,16,1],[1,0,0,0,0,0,1,0,0,0,0,1,60,0,0,0,0,0,1,0,0,0,0,45,60],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,57,50,0,0,0,57,4] >;

C22×D30.C2 in GAP, Magma, Sage, TeX

C_2^2\times D_{30}.C_2
% in TeX

G:=Group("C2^2xD30.C2");
// GroupNames label

G:=SmallGroup(480,1117);
// by ID

G=gap.SmallGroup(480,1117);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^30=d^2=1,e^2=c^15,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=c^-1,e*c*e^-1=c^19,e*d*e^-1=c^18*d>;
// generators/relations

׿
×
𝔽