direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×D30.C2, C30.45C24, D30.43C23, C15⋊7(C23×C4), D30⋊31(C2×C4), C30⋊7(C22×C4), (C2×Dic5)⋊26D6, D15⋊3(C22×C4), C6.45(C23×D5), C23.69(S3×D5), (C2×Dic3)⋊26D10, (C22×D15)⋊11C4, C10.45(S3×C23), (C5×Dic3)⋊7C23, Dic3⋊7(C22×D5), Dic5⋊7(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, C6⋊2(C2×C4×D5), C10⋊3(S3×C2×C4), C5⋊3(S3×C22×C4), C3⋊2(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
| C15 — C22×D30.C2 |
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, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C22×C4, C24, Dic5, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, D15, C30, C30, C23×C4, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C10, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C5×Dic3, C3×Dic5, D30, C2×C30, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, S3×C22×C4, D30.C2, C6×Dic5, C10×Dic3, C22×D15, C22×C30, D5×C22×C4, C2×D30.C2, C2×C6×Dic5, Dic3×C2×C10, C23×D15, C22×D30.C2
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, C24, D10, C4×S3, C22×S3, C23×C4, C4×D5, C22×D5, S3×C2×C4, S3×C23, S3×D5, C2×C4×D5, C23×D5, S3×C22×C4, D30.C2, C2×S3×D5, D5×C22×C4, C2×D30.C2, C22×S3×D5, C22×D30.C2
►On 240 points
Generators in S240
(1 117)(2 118)(3 119)(4 120)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 113)(28 114)(29 115)(30 116)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)(41 78)(42 79)(43 80)(44 81)(45 82)(46 83)(47 84)(48 85)(49 86)(50 87)(51 88)(52 89)(53 90)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)(121 226)(122 227)(123 228)(124 229)(125 230)(126 231)(127 232)(128 233)(129 234)(130 235)(131 236)(132 237)(133 238)(134 239)(135 240)(136 211)(137 212)(138 213)(139 214)(140 215)(141 216)(142 217)(143 218)(144 219)(145 220)(146 221)(147 222)(148 223)(149 224)(150 225)(151 181)(152 182)(153 183)(154 184)(155 185)(156 186)(157 187)(158 188)(159 189)(160 190)(161 191)(162 192)(163 193)(164 194)(165 195)(166 196)(167 197)(168 198)(169 199)(170 200)(171 201)(172 202)(173 203)(174 204)(175 205)(176 206)(177 207)(178 208)(179 209)(180 210)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 113)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(121 151)(122 152)(123 153)(124 154)(125 155)(126 156)(127 157)(128 158)(129 159)(130 160)(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)(181 226)(182 227)(183 228)(184 229)(185 230)(186 231)(187 232)(188 233)(189 234)(190 235)(191 236)(192 237)(193 238)(194 239)(195 240)(196 211)(197 212)(198 213)(199 214)(200 215)(201 216)(202 217)(203 218)(204 219)(205 220)(206 221)(207 222)(208 223)(209 224)(210 225)
(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)
(1 116)(2 115)(3 114)(4 113)(5 112)(6 111)(7 110)(8 109)(9 108)(10 107)(11 106)(12 105)(13 104)(14 103)(15 102)(16 101)(17 100)(18 99)(19 98)(20 97)(21 96)(22 95)(23 94)(24 93)(25 92)(26 91)(27 120)(28 119)(29 118)(30 117)(31 63)(32 62)(33 61)(34 90)(35 89)(36 88)(37 87)(38 86)(39 85)(40 84)(41 83)(42 82)(43 81)(44 80)(45 79)(46 78)(47 77)(48 76)(49 75)(50 74)(51 73)(52 72)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)(121 239)(122 238)(123 237)(124 236)(125 235)(126 234)(127 233)(128 232)(129 231)(130 230)(131 229)(132 228)(133 227)(134 226)(135 225)(136 224)(137 223)(138 222)(139 221)(140 220)(141 219)(142 218)(143 217)(144 216)(145 215)(146 214)(147 213)(148 212)(149 211)(150 240)(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)
(1 203 16 188)(2 192 17 207)(3 181 18 196)(4 200 19 185)(5 189 20 204)(6 208 21 193)(7 197 22 182)(8 186 23 201)(9 205 24 190)(10 194 25 209)(11 183 26 198)(12 202 27 187)(13 191 28 206)(14 210 29 195)(15 199 30 184)(31 211 46 226)(32 230 47 215)(33 219 48 234)(34 238 49 223)(35 227 50 212)(36 216 51 231)(37 235 52 220)(38 224 53 239)(39 213 54 228)(40 232 55 217)(41 221 56 236)(42 240 57 225)(43 229 58 214)(44 218 59 233)(45 237 60 222)(61 123 76 138)(62 142 77 127)(63 131 78 146)(64 150 79 135)(65 139 80 124)(66 128 81 143)(67 147 82 132)(68 136 83 121)(69 125 84 140)(70 144 85 129)(71 133 86 148)(72 122 87 137)(73 141 88 126)(74 130 89 145)(75 149 90 134)(91 159 106 174)(92 178 107 163)(93 167 108 152)(94 156 109 171)(95 175 110 160)(96 164 111 179)(97 153 112 168)(98 172 113 157)(99 161 114 176)(100 180 115 165)(101 169 116 154)(102 158 117 173)(103 177 118 162)(104 166 119 151)(105 155 120 170)
G:=sub<Sym(240)| (1,117)(2,118)(3,119)(4,120)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(41,78)(42,79)(43,80)(44,81)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67)(121,226)(122,227)(123,228)(124,229)(125,230)(126,231)(127,232)(128,233)(129,234)(130,235)(131,236)(132,237)(133,238)(134,239)(135,240)(136,211)(137,212)(138,213)(139,214)(140,215)(141,216)(142,217)(143,218)(144,219)(145,220)(146,221)(147,222)(148,223)(149,224)(150,225)(151,181)(152,182)(153,183)(154,184)(155,185)(156,186)(157,187)(158,188)(159,189)(160,190)(161,191)(162,192)(163,193)(164,194)(165,195)(166,196)(167,197)(168,198)(169,199)(170,200)(171,201)(172,202)(173,203)(174,204)(175,205)(176,206)(177,207)(178,208)(179,209)(180,210), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(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)(181,226)(182,227)(183,228)(184,229)(185,230)(186,231)(187,232)(188,233)(189,234)(190,235)(191,236)(192,237)(193,238)(194,239)(195,240)(196,211)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,218)(204,219)(205,220)(206,221)(207,222)(208,223)(209,224)(210,225), (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), (1,116)(2,115)(3,114)(4,113)(5,112)(6,111)(7,110)(8,109)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,100)(18,99)(19,98)(20,97)(21,96)(22,95)(23,94)(24,93)(25,92)(26,91)(27,120)(28,119)(29,118)(30,117)(31,63)(32,62)(33,61)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,84)(41,83)(42,82)(43,81)(44,80)(45,79)(46,78)(47,77)(48,76)(49,75)(50,74)(51,73)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(121,239)(122,238)(123,237)(124,236)(125,235)(126,234)(127,233)(128,232)(129,231)(130,230)(131,229)(132,228)(133,227)(134,226)(135,225)(136,224)(137,223)(138,222)(139,221)(140,220)(141,219)(142,218)(143,217)(144,216)(145,215)(146,214)(147,213)(148,212)(149,211)(150,240)(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), (1,203,16,188)(2,192,17,207)(3,181,18,196)(4,200,19,185)(5,189,20,204)(6,208,21,193)(7,197,22,182)(8,186,23,201)(9,205,24,190)(10,194,25,209)(11,183,26,198)(12,202,27,187)(13,191,28,206)(14,210,29,195)(15,199,30,184)(31,211,46,226)(32,230,47,215)(33,219,48,234)(34,238,49,223)(35,227,50,212)(36,216,51,231)(37,235,52,220)(38,224,53,239)(39,213,54,228)(40,232,55,217)(41,221,56,236)(42,240,57,225)(43,229,58,214)(44,218,59,233)(45,237,60,222)(61,123,76,138)(62,142,77,127)(63,131,78,146)(64,150,79,135)(65,139,80,124)(66,128,81,143)(67,147,82,132)(68,136,83,121)(69,125,84,140)(70,144,85,129)(71,133,86,148)(72,122,87,137)(73,141,88,126)(74,130,89,145)(75,149,90,134)(91,159,106,174)(92,178,107,163)(93,167,108,152)(94,156,109,171)(95,175,110,160)(96,164,111,179)(97,153,112,168)(98,172,113,157)(99,161,114,176)(100,180,115,165)(101,169,116,154)(102,158,117,173)(103,177,118,162)(104,166,119,151)(105,155,120,170)>;
G:=Group( (1,117)(2,118)(3,119)(4,120)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(41,78)(42,79)(43,80)(44,81)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67)(121,226)(122,227)(123,228)(124,229)(125,230)(126,231)(127,232)(128,233)(129,234)(130,235)(131,236)(132,237)(133,238)(134,239)(135,240)(136,211)(137,212)(138,213)(139,214)(140,215)(141,216)(142,217)(143,218)(144,219)(145,220)(146,221)(147,222)(148,223)(149,224)(150,225)(151,181)(152,182)(153,183)(154,184)(155,185)(156,186)(157,187)(158,188)(159,189)(160,190)(161,191)(162,192)(163,193)(164,194)(165,195)(166,196)(167,197)(168,198)(169,199)(170,200)(171,201)(172,202)(173,203)(174,204)(175,205)(176,206)(177,207)(178,208)(179,209)(180,210), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(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)(181,226)(182,227)(183,228)(184,229)(185,230)(186,231)(187,232)(188,233)(189,234)(190,235)(191,236)(192,237)(193,238)(194,239)(195,240)(196,211)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,218)(204,219)(205,220)(206,221)(207,222)(208,223)(209,224)(210,225), (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), (1,116)(2,115)(3,114)(4,113)(5,112)(6,111)(7,110)(8,109)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,100)(18,99)(19,98)(20,97)(21,96)(22,95)(23,94)(24,93)(25,92)(26,91)(27,120)(28,119)(29,118)(30,117)(31,63)(32,62)(33,61)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,84)(41,83)(42,82)(43,81)(44,80)(45,79)(46,78)(47,77)(48,76)(49,75)(50,74)(51,73)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(121,239)(122,238)(123,237)(124,236)(125,235)(126,234)(127,233)(128,232)(129,231)(130,230)(131,229)(132,228)(133,227)(134,226)(135,225)(136,224)(137,223)(138,222)(139,221)(140,220)(141,219)(142,218)(143,217)(144,216)(145,215)(146,214)(147,213)(148,212)(149,211)(150,240)(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), (1,203,16,188)(2,192,17,207)(3,181,18,196)(4,200,19,185)(5,189,20,204)(6,208,21,193)(7,197,22,182)(8,186,23,201)(9,205,24,190)(10,194,25,209)(11,183,26,198)(12,202,27,187)(13,191,28,206)(14,210,29,195)(15,199,30,184)(31,211,46,226)(32,230,47,215)(33,219,48,234)(34,238,49,223)(35,227,50,212)(36,216,51,231)(37,235,52,220)(38,224,53,239)(39,213,54,228)(40,232,55,217)(41,221,56,236)(42,240,57,225)(43,229,58,214)(44,218,59,233)(45,237,60,222)(61,123,76,138)(62,142,77,127)(63,131,78,146)(64,150,79,135)(65,139,80,124)(66,128,81,143)(67,147,82,132)(68,136,83,121)(69,125,84,140)(70,144,85,129)(71,133,86,148)(72,122,87,137)(73,141,88,126)(74,130,89,145)(75,149,90,134)(91,159,106,174)(92,178,107,163)(93,167,108,152)(94,156,109,171)(95,175,110,160)(96,164,111,179)(97,153,112,168)(98,172,113,157)(99,161,114,176)(100,180,115,165)(101,169,116,154)(102,158,117,173)(103,177,118,162)(104,166,119,151)(105,155,120,170) );
G=PermutationGroup([[(1,117),(2,118),(3,119),(4,120),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(17,103),(18,104),(19,105),(20,106),(21,107),(22,108),(23,109),(24,110),(25,111),(26,112),(27,113),(28,114),(29,115),(30,116),(31,68),(32,69),(33,70),(34,71),(35,72),(36,73),(37,74),(38,75),(39,76),(40,77),(41,78),(42,79),(43,80),(44,81),(45,82),(46,83),(47,84),(48,85),(49,86),(50,87),(51,88),(52,89),(53,90),(54,61),(55,62),(56,63),(57,64),(58,65),(59,66),(60,67),(121,226),(122,227),(123,228),(124,229),(125,230),(126,231),(127,232),(128,233),(129,234),(130,235),(131,236),(132,237),(133,238),(134,239),(135,240),(136,211),(137,212),(138,213),(139,214),(140,215),(141,216),(142,217),(143,218),(144,219),(145,220),(146,221),(147,222),(148,223),(149,224),(150,225),(151,181),(152,182),(153,183),(154,184),(155,185),(156,186),(157,187),(158,188),(159,189),(160,190),(161,191),(162,192),(163,193),(164,194),(165,195),(166,196),(167,197),(168,198),(169,199),(170,200),(171,201),(172,202),(173,203),(174,204),(175,205),(176,206),(177,207),(178,208),(179,209),(180,210)], [(1,44),(2,45),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,31),(19,32),(20,33),(21,34),(22,35),(23,36),(24,37),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(61,97),(62,98),(63,99),(64,100),(65,101),(66,102),(67,103),(68,104),(69,105),(70,106),(71,107),(72,108),(73,109),(74,110),(75,111),(76,112),(77,113),(78,114),(79,115),(80,116),(81,117),(82,118),(83,119),(84,120),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96),(121,151),(122,152),(123,153),(124,154),(125,155),(126,156),(127,157),(128,158),(129,159),(130,160),(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),(181,226),(182,227),(183,228),(184,229),(185,230),(186,231),(187,232),(188,233),(189,234),(190,235),(191,236),(192,237),(193,238),(194,239),(195,240),(196,211),(197,212),(198,213),(199,214),(200,215),(201,216),(202,217),(203,218),(204,219),(205,220),(206,221),(207,222),(208,223),(209,224),(210,225)], [(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)], [(1,116),(2,115),(3,114),(4,113),(5,112),(6,111),(7,110),(8,109),(9,108),(10,107),(11,106),(12,105),(13,104),(14,103),(15,102),(16,101),(17,100),(18,99),(19,98),(20,97),(21,96),(22,95),(23,94),(24,93),(25,92),(26,91),(27,120),(28,119),(29,118),(30,117),(31,63),(32,62),(33,61),(34,90),(35,89),(36,88),(37,87),(38,86),(39,85),(40,84),(41,83),(42,82),(43,81),(44,80),(45,79),(46,78),(47,77),(48,76),(49,75),(50,74),(51,73),(52,72),(53,71),(54,70),(55,69),(56,68),(57,67),(58,66),(59,65),(60,64),(121,239),(122,238),(123,237),(124,236),(125,235),(126,234),(127,233),(128,232),(129,231),(130,230),(131,229),(132,228),(133,227),(134,226),(135,225),(136,224),(137,223),(138,222),(139,221),(140,220),(141,219),(142,218),(143,217),(144,216),(145,215),(146,214),(147,213),(148,212),(149,211),(150,240),(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)], [(1,203,16,188),(2,192,17,207),(3,181,18,196),(4,200,19,185),(5,189,20,204),(6,208,21,193),(7,197,22,182),(8,186,23,201),(9,205,24,190),(10,194,25,209),(11,183,26,198),(12,202,27,187),(13,191,28,206),(14,210,29,195),(15,199,30,184),(31,211,46,226),(32,230,47,215),(33,219,48,234),(34,238,49,223),(35,227,50,212),(36,216,51,231),(37,235,52,220),(38,224,53,239),(39,213,54,228),(40,232,55,217),(41,221,56,236),(42,240,57,225),(43,229,58,214),(44,218,59,233),(45,237,60,222),(61,123,76,138),(62,142,77,127),(63,131,78,146),(64,150,79,135),(65,139,80,124),(66,128,81,143),(67,147,82,132),(68,136,83,121),(69,125,84,140),(70,144,85,129),(71,133,86,148),(72,122,87,137),(73,141,88,126),(74,130,89,145),(75,149,90,134),(91,159,106,174),(92,178,107,163),(93,167,108,152),(94,156,109,171),(95,175,110,160),(96,164,111,179),(97,153,112,168),(98,172,113,157),(99,161,114,176),(100,180,115,165),(101,169,116,154),(102,158,117,173),(103,177,118,162),(104,166,119,151),(105,155,120,170)]])
96 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3 | 4A | ··· | 4H | 4I | ··· | 4P | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10N | 12A | ··· | 12H | 15A | 15B | 20A | ··· | 20P | 30A | ··· | 30N |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 15 | ··· | 15 | 2 | 3 | ··· | 3 | 5 | ··· | 5 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | 4 | 6 | ··· | 6 | 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 | D10 | D10 | C4×S3 | C4×D5 | S3×D5 | D30.C2 | C2×S3×D5 |
| kernel | C22×D30.C2 | C2×D30.C2 | C2×C6×Dic5 | Dic3×C2×C10 | C23×D15 | C22×D15 | C22×Dic5 | C22×Dic3 | C2×Dic5 | C22×C10 | C2×Dic3 | C22×C6 | C2×C10 | C2×C6 | C23 | C22 | C22 |
| # reps | 1 | 12 | 1 | 1 | 1 | 16 | 1 | 2 | 6 | 1 | 12 | 2 | 8 | 16 | 2 | 8 | 6 |
Matrix representation of C22×D30.C2 ►in GL5(𝔽61)
| 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 | 60 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 16 |
| 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 45 |
| 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 57 | 57 |
| 0 | 0 | 0 | 50 | 4 |
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