direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C5⋊D12, D30⋊10C23, C30.48C24, C30⋊6(C2×D4), C10⋊3(C2×D12), (C2×C30)⋊15D4, C15⋊7(C22×D4), C5⋊3(C22×D12), (C2×C10)⋊12D12, (S3×C23)⋊4D5, D6⋊7(C22×D5), (S3×C10)⋊7C23, (C2×Dic5)⋊24D6, (C23×D15)⋊9C2, C6.48(C23×D5), C23.72(S3×D5), (C22×S3)⋊14D10, C10.48(S3×C23), Dic5⋊5(C22×S3), (C3×Dic5)⋊8C23, (C2×C30).251C23, (C6×Dic5)⋊31C22, (C22×Dic5)⋊12S3, (C22×C10).120D6, (C22×C6).103D10, (C22×D15)⋊21C22, (C22×C30).89C22, C6⋊1(C2×C5⋊D4), C3⋊1(C22×C5⋊D4), (C2×C6)⋊9(C5⋊D4), (S3×C22×C10)⋊4C2, (C2×C6×Dic5)⋊11C2, C2.48(C22×S3×D5), (S3×C2×C10)⋊18C22, C22.111(C2×S3×D5), (C2×C6).257(C22×D5), (C2×C10).255(C22×S3), SmallGroup(480,1120)
Series: Derived ►Chief ►Lower central ►Upper central
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
Generators and relations
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 >
►On 240 points
Generators in S240
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 109)(9 110)(10 111)(11 112)(12 113)(13 96)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 146)(26 147)(27 148)(28 149)(29 150)(30 151)(31 152)(32 153)(33 154)(34 155)(35 156)(36 145)(37 186)(38 187)(39 188)(40 189)(41 190)(42 191)(43 192)(44 181)(45 182)(46 183)(47 184)(48 185)(49 81)(50 82)(51 83)(52 84)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 158)(62 159)(63 160)(64 161)(65 162)(66 163)(67 164)(68 165)(69 166)(70 167)(71 168)(72 157)(97 204)(98 193)(99 194)(100 195)(101 196)(102 197)(103 198)(104 199)(105 200)(106 201)(107 202)(108 203)(121 235)(122 236)(123 237)(124 238)(125 239)(126 240)(127 229)(128 230)(129 231)(130 232)(131 233)(132 234)(133 225)(134 226)(135 227)(136 228)(137 217)(138 218)(139 219)(140 220)(141 221)(142 222)(143 223)(144 224)(169 212)(170 213)(171 214)(172 215)(173 216)(174 205)(175 206)(176 207)(177 208)(178 209)(179 210)(180 211)
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(25 186)(26 187)(27 188)(28 189)(29 190)(30 191)(31 192)(32 181)(33 182)(34 183)(35 184)(36 185)(37 146)(38 147)(39 148)(40 149)(41 150)(42 151)(43 152)(44 153)(45 154)(46 155)(47 156)(48 145)(49 162)(50 163)(51 164)(52 165)(53 166)(54 167)(55 168)(56 157)(57 158)(58 159)(59 160)(60 161)(61 77)(62 78)(63 79)(64 80)(65 81)(66 82)(67 83)(68 84)(69 73)(70 74)(71 75)(72 76)(85 118)(86 119)(87 120)(88 109)(89 110)(90 111)(91 112)(92 113)(93 114)(94 115)(95 116)(96 117)(97 135)(98 136)(99 137)(100 138)(101 139)(102 140)(103 141)(104 142)(105 143)(106 144)(107 133)(108 134)(121 211)(122 212)(123 213)(124 214)(125 215)(126 216)(127 205)(128 206)(129 207)(130 208)(131 209)(132 210)(169 236)(170 237)(171 238)(172 239)(173 240)(174 229)(175 230)(176 231)(177 232)(178 233)(179 234)(180 235)(193 228)(194 217)(195 218)(196 219)(197 220)(198 221)(199 222)(200 223)(201 224)(202 225)(203 226)(204 227)
(1 36 84 144 232)(2 233 133 73 25)(3 26 74 134 234)(4 235 135 75 27)(5 28 76 136 236)(6 237 137 77 29)(7 30 78 138 238)(8 239 139 79 31)(9 32 80 140 240)(10 229 141 81 33)(11 34 82 142 230)(12 231 143 83 35)(13 180 97 71 188)(14 189 72 98 169)(15 170 99 61 190)(16 191 62 100 171)(17 172 101 63 192)(18 181 64 102 173)(19 174 103 65 182)(20 183 66 104 175)(21 176 105 67 184)(22 185 68 106 177)(23 178 107 69 186)(24 187 70 108 179)(37 94 209 202 166)(38 167 203 210 95)(39 96 211 204 168)(40 157 193 212 85)(41 86 213 194 158)(42 159 195 214 87)(43 88 215 196 160)(44 161 197 216 89)(45 90 205 198 162)(46 163 199 206 91)(47 92 207 200 164)(48 165 201 208 93)(49 154 111 127 221)(50 222 128 112 155)(51 156 113 129 223)(52 224 130 114 145)(53 146 115 131 225)(54 226 132 116 147)(55 148 117 121 227)(56 228 122 118 149)(57 150 119 123 217)(58 218 124 120 151)(59 152 109 125 219)(60 220 126 110 153)
(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 18)(14 17)(15 16)(19 24)(20 23)(21 22)(25 34)(26 33)(27 32)(28 31)(29 30)(35 36)(37 46)(38 45)(39 44)(40 43)(41 42)(47 48)(49 54)(50 53)(51 52)(55 60)(56 59)(57 58)(61 62)(63 72)(64 71)(65 70)(66 69)(67 68)(73 82)(74 81)(75 80)(76 79)(77 78)(83 84)(85 88)(86 87)(89 96)(90 95)(91 94)(92 93)(97 102)(98 101)(99 100)(103 108)(104 107)(105 106)(109 118)(110 117)(111 116)(112 115)(113 114)(119 120)(121 126)(122 125)(123 124)(127 132)(128 131)(129 130)(133 142)(134 141)(135 140)(136 139)(137 138)(143 144)(145 156)(146 155)(147 154)(148 153)(149 152)(150 151)(157 160)(158 159)(161 168)(162 167)(163 166)(164 165)(169 172)(170 171)(173 180)(174 179)(175 178)(176 177)(181 188)(182 187)(183 186)(184 185)(189 192)(190 191)(193 196)(194 195)(197 204)(198 203)(199 202)(200 201)(205 210)(206 209)(207 208)(211 216)(212 215)(213 214)(217 218)(219 228)(220 227)(221 226)(222 225)(223 224)(229 234)(230 233)(231 232)(235 240)(236 239)(237 238)
G:=sub<Sym(240)| (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,109)(9,110)(10,111)(11,112)(12,113)(13,96)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,146)(26,147)(27,148)(28,149)(29,150)(30,151)(31,152)(32,153)(33,154)(34,155)(35,156)(36,145)(37,186)(38,187)(39,188)(40,189)(41,190)(42,191)(43,192)(44,181)(45,182)(46,183)(47,184)(48,185)(49,81)(50,82)(51,83)(52,84)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,158)(62,159)(63,160)(64,161)(65,162)(66,163)(67,164)(68,165)(69,166)(70,167)(71,168)(72,157)(97,204)(98,193)(99,194)(100,195)(101,196)(102,197)(103,198)(104,199)(105,200)(106,201)(107,202)(108,203)(121,235)(122,236)(123,237)(124,238)(125,239)(126,240)(127,229)(128,230)(129,231)(130,232)(131,233)(132,234)(133,225)(134,226)(135,227)(136,228)(137,217)(138,218)(139,219)(140,220)(141,221)(142,222)(143,223)(144,224)(169,212)(170,213)(171,214)(172,215)(173,216)(174,205)(175,206)(176,207)(177,208)(178,209)(179,210)(180,211), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,186)(26,187)(27,188)(28,189)(29,190)(30,191)(31,192)(32,181)(33,182)(34,183)(35,184)(36,185)(37,146)(38,147)(39,148)(40,149)(41,150)(42,151)(43,152)(44,153)(45,154)(46,155)(47,156)(48,145)(49,162)(50,163)(51,164)(52,165)(53,166)(54,167)(55,168)(56,157)(57,158)(58,159)(59,160)(60,161)(61,77)(62,78)(63,79)(64,80)(65,81)(66,82)(67,83)(68,84)(69,73)(70,74)(71,75)(72,76)(85,118)(86,119)(87,120)(88,109)(89,110)(90,111)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(97,135)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,133)(108,134)(121,211)(122,212)(123,213)(124,214)(125,215)(126,216)(127,205)(128,206)(129,207)(130,208)(131,209)(132,210)(169,236)(170,237)(171,238)(172,239)(173,240)(174,229)(175,230)(176,231)(177,232)(178,233)(179,234)(180,235)(193,228)(194,217)(195,218)(196,219)(197,220)(198,221)(199,222)(200,223)(201,224)(202,225)(203,226)(204,227), (1,36,84,144,232)(2,233,133,73,25)(3,26,74,134,234)(4,235,135,75,27)(5,28,76,136,236)(6,237,137,77,29)(7,30,78,138,238)(8,239,139,79,31)(9,32,80,140,240)(10,229,141,81,33)(11,34,82,142,230)(12,231,143,83,35)(13,180,97,71,188)(14,189,72,98,169)(15,170,99,61,190)(16,191,62,100,171)(17,172,101,63,192)(18,181,64,102,173)(19,174,103,65,182)(20,183,66,104,175)(21,176,105,67,184)(22,185,68,106,177)(23,178,107,69,186)(24,187,70,108,179)(37,94,209,202,166)(38,167,203,210,95)(39,96,211,204,168)(40,157,193,212,85)(41,86,213,194,158)(42,159,195,214,87)(43,88,215,196,160)(44,161,197,216,89)(45,90,205,198,162)(46,163,199,206,91)(47,92,207,200,164)(48,165,201,208,93)(49,154,111,127,221)(50,222,128,112,155)(51,156,113,129,223)(52,224,130,114,145)(53,146,115,131,225)(54,226,132,116,147)(55,148,117,121,227)(56,228,122,118,149)(57,150,119,123,217)(58,218,124,120,151)(59,152,109,125,219)(60,220,126,110,153), (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)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,34)(26,33)(27,32)(28,31)(29,30)(35,36)(37,46)(38,45)(39,44)(40,43)(41,42)(47,48)(49,54)(50,53)(51,52)(55,60)(56,59)(57,58)(61,62)(63,72)(64,71)(65,70)(66,69)(67,68)(73,82)(74,81)(75,80)(76,79)(77,78)(83,84)(85,88)(86,87)(89,96)(90,95)(91,94)(92,93)(97,102)(98,101)(99,100)(103,108)(104,107)(105,106)(109,118)(110,117)(111,116)(112,115)(113,114)(119,120)(121,126)(122,125)(123,124)(127,132)(128,131)(129,130)(133,142)(134,141)(135,140)(136,139)(137,138)(143,144)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)(157,160)(158,159)(161,168)(162,167)(163,166)(164,165)(169,172)(170,171)(173,180)(174,179)(175,178)(176,177)(181,188)(182,187)(183,186)(184,185)(189,192)(190,191)(193,196)(194,195)(197,204)(198,203)(199,202)(200,201)(205,210)(206,209)(207,208)(211,216)(212,215)(213,214)(217,218)(219,228)(220,227)(221,226)(222,225)(223,224)(229,234)(230,233)(231,232)(235,240)(236,239)(237,238)>;
G:=Group( (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,109)(9,110)(10,111)(11,112)(12,113)(13,96)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,146)(26,147)(27,148)(28,149)(29,150)(30,151)(31,152)(32,153)(33,154)(34,155)(35,156)(36,145)(37,186)(38,187)(39,188)(40,189)(41,190)(42,191)(43,192)(44,181)(45,182)(46,183)(47,184)(48,185)(49,81)(50,82)(51,83)(52,84)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,158)(62,159)(63,160)(64,161)(65,162)(66,163)(67,164)(68,165)(69,166)(70,167)(71,168)(72,157)(97,204)(98,193)(99,194)(100,195)(101,196)(102,197)(103,198)(104,199)(105,200)(106,201)(107,202)(108,203)(121,235)(122,236)(123,237)(124,238)(125,239)(126,240)(127,229)(128,230)(129,231)(130,232)(131,233)(132,234)(133,225)(134,226)(135,227)(136,228)(137,217)(138,218)(139,219)(140,220)(141,221)(142,222)(143,223)(144,224)(169,212)(170,213)(171,214)(172,215)(173,216)(174,205)(175,206)(176,207)(177,208)(178,209)(179,210)(180,211), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,186)(26,187)(27,188)(28,189)(29,190)(30,191)(31,192)(32,181)(33,182)(34,183)(35,184)(36,185)(37,146)(38,147)(39,148)(40,149)(41,150)(42,151)(43,152)(44,153)(45,154)(46,155)(47,156)(48,145)(49,162)(50,163)(51,164)(52,165)(53,166)(54,167)(55,168)(56,157)(57,158)(58,159)(59,160)(60,161)(61,77)(62,78)(63,79)(64,80)(65,81)(66,82)(67,83)(68,84)(69,73)(70,74)(71,75)(72,76)(85,118)(86,119)(87,120)(88,109)(89,110)(90,111)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(97,135)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,133)(108,134)(121,211)(122,212)(123,213)(124,214)(125,215)(126,216)(127,205)(128,206)(129,207)(130,208)(131,209)(132,210)(169,236)(170,237)(171,238)(172,239)(173,240)(174,229)(175,230)(176,231)(177,232)(178,233)(179,234)(180,235)(193,228)(194,217)(195,218)(196,219)(197,220)(198,221)(199,222)(200,223)(201,224)(202,225)(203,226)(204,227), (1,36,84,144,232)(2,233,133,73,25)(3,26,74,134,234)(4,235,135,75,27)(5,28,76,136,236)(6,237,137,77,29)(7,30,78,138,238)(8,239,139,79,31)(9,32,80,140,240)(10,229,141,81,33)(11,34,82,142,230)(12,231,143,83,35)(13,180,97,71,188)(14,189,72,98,169)(15,170,99,61,190)(16,191,62,100,171)(17,172,101,63,192)(18,181,64,102,173)(19,174,103,65,182)(20,183,66,104,175)(21,176,105,67,184)(22,185,68,106,177)(23,178,107,69,186)(24,187,70,108,179)(37,94,209,202,166)(38,167,203,210,95)(39,96,211,204,168)(40,157,193,212,85)(41,86,213,194,158)(42,159,195,214,87)(43,88,215,196,160)(44,161,197,216,89)(45,90,205,198,162)(46,163,199,206,91)(47,92,207,200,164)(48,165,201,208,93)(49,154,111,127,221)(50,222,128,112,155)(51,156,113,129,223)(52,224,130,114,145)(53,146,115,131,225)(54,226,132,116,147)(55,148,117,121,227)(56,228,122,118,149)(57,150,119,123,217)(58,218,124,120,151)(59,152,109,125,219)(60,220,126,110,153), (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)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,34)(26,33)(27,32)(28,31)(29,30)(35,36)(37,46)(38,45)(39,44)(40,43)(41,42)(47,48)(49,54)(50,53)(51,52)(55,60)(56,59)(57,58)(61,62)(63,72)(64,71)(65,70)(66,69)(67,68)(73,82)(74,81)(75,80)(76,79)(77,78)(83,84)(85,88)(86,87)(89,96)(90,95)(91,94)(92,93)(97,102)(98,101)(99,100)(103,108)(104,107)(105,106)(109,118)(110,117)(111,116)(112,115)(113,114)(119,120)(121,126)(122,125)(123,124)(127,132)(128,131)(129,130)(133,142)(134,141)(135,140)(136,139)(137,138)(143,144)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)(157,160)(158,159)(161,168)(162,167)(163,166)(164,165)(169,172)(170,171)(173,180)(174,179)(175,178)(176,177)(181,188)(182,187)(183,186)(184,185)(189,192)(190,191)(193,196)(194,195)(197,204)(198,203)(199,202)(200,201)(205,210)(206,209)(207,208)(211,216)(212,215)(213,214)(217,218)(219,228)(220,227)(221,226)(222,225)(223,224)(229,234)(230,233)(231,232)(235,240)(236,239)(237,238) );
G=PermutationGroup([(1,114),(2,115),(3,116),(4,117),(5,118),(6,119),(7,120),(8,109),(9,110),(10,111),(11,112),(12,113),(13,96),(14,85),(15,86),(16,87),(17,88),(18,89),(19,90),(20,91),(21,92),(22,93),(23,94),(24,95),(25,146),(26,147),(27,148),(28,149),(29,150),(30,151),(31,152),(32,153),(33,154),(34,155),(35,156),(36,145),(37,186),(38,187),(39,188),(40,189),(41,190),(42,191),(43,192),(44,181),(45,182),(46,183),(47,184),(48,185),(49,81),(50,82),(51,83),(52,84),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,158),(62,159),(63,160),(64,161),(65,162),(66,163),(67,164),(68,165),(69,166),(70,167),(71,168),(72,157),(97,204),(98,193),(99,194),(100,195),(101,196),(102,197),(103,198),(104,199),(105,200),(106,201),(107,202),(108,203),(121,235),(122,236),(123,237),(124,238),(125,239),(126,240),(127,229),(128,230),(129,231),(130,232),(131,233),(132,234),(133,225),(134,226),(135,227),(136,228),(137,217),(138,218),(139,219),(140,220),(141,221),(142,222),(143,223),(144,224),(169,212),(170,213),(171,214),(172,215),(173,216),(174,205),(175,206),(176,207),(177,208),(178,209),(179,210),(180,211)], [(1,22),(2,23),(3,24),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21),(25,186),(26,187),(27,188),(28,189),(29,190),(30,191),(31,192),(32,181),(33,182),(34,183),(35,184),(36,185),(37,146),(38,147),(39,148),(40,149),(41,150),(42,151),(43,152),(44,153),(45,154),(46,155),(47,156),(48,145),(49,162),(50,163),(51,164),(52,165),(53,166),(54,167),(55,168),(56,157),(57,158),(58,159),(59,160),(60,161),(61,77),(62,78),(63,79),(64,80),(65,81),(66,82),(67,83),(68,84),(69,73),(70,74),(71,75),(72,76),(85,118),(86,119),(87,120),(88,109),(89,110),(90,111),(91,112),(92,113),(93,114),(94,115),(95,116),(96,117),(97,135),(98,136),(99,137),(100,138),(101,139),(102,140),(103,141),(104,142),(105,143),(106,144),(107,133),(108,134),(121,211),(122,212),(123,213),(124,214),(125,215),(126,216),(127,205),(128,206),(129,207),(130,208),(131,209),(132,210),(169,236),(170,237),(171,238),(172,239),(173,240),(174,229),(175,230),(176,231),(177,232),(178,233),(179,234),(180,235),(193,228),(194,217),(195,218),(196,219),(197,220),(198,221),(199,222),(200,223),(201,224),(202,225),(203,226),(204,227)], [(1,36,84,144,232),(2,233,133,73,25),(3,26,74,134,234),(4,235,135,75,27),(5,28,76,136,236),(6,237,137,77,29),(7,30,78,138,238),(8,239,139,79,31),(9,32,80,140,240),(10,229,141,81,33),(11,34,82,142,230),(12,231,143,83,35),(13,180,97,71,188),(14,189,72,98,169),(15,170,99,61,190),(16,191,62,100,171),(17,172,101,63,192),(18,181,64,102,173),(19,174,103,65,182),(20,183,66,104,175),(21,176,105,67,184),(22,185,68,106,177),(23,178,107,69,186),(24,187,70,108,179),(37,94,209,202,166),(38,167,203,210,95),(39,96,211,204,168),(40,157,193,212,85),(41,86,213,194,158),(42,159,195,214,87),(43,88,215,196,160),(44,161,197,216,89),(45,90,205,198,162),(46,163,199,206,91),(47,92,207,200,164),(48,165,201,208,93),(49,154,111,127,221),(50,222,128,112,155),(51,156,113,129,223),(52,224,130,114,145),(53,146,115,131,225),(54,226,132,116,147),(55,148,117,121,227),(56,228,122,118,149),(57,150,119,123,217),(58,218,124,120,151),(59,152,109,125,219),(60,220,126,110,153)], [(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),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22),(25,34),(26,33),(27,32),(28,31),(29,30),(35,36),(37,46),(38,45),(39,44),(40,43),(41,42),(47,48),(49,54),(50,53),(51,52),(55,60),(56,59),(57,58),(61,62),(63,72),(64,71),(65,70),(66,69),(67,68),(73,82),(74,81),(75,80),(76,79),(77,78),(83,84),(85,88),(86,87),(89,96),(90,95),(91,94),(92,93),(97,102),(98,101),(99,100),(103,108),(104,107),(105,106),(109,118),(110,117),(111,116),(112,115),(113,114),(119,120),(121,126),(122,125),(123,124),(127,132),(128,131),(129,130),(133,142),(134,141),(135,140),(136,139),(137,138),(143,144),(145,156),(146,155),(147,154),(148,153),(149,152),(150,151),(157,160),(158,159),(161,168),(162,167),(163,166),(164,165),(169,172),(170,171),(173,180),(174,179),(175,178),(176,177),(181,188),(182,187),(183,186),(184,185),(189,192),(190,191),(193,196),(194,195),(197,204),(198,203),(199,202),(200,201),(205,210),(206,209),(207,208),(211,216),(212,215),(213,214),(217,218),(219,228),(220,227),(221,226),(222,225),(223,224),(229,234),(230,233),(231,232),(235,240),(236,239),(237,238)])
Matrix representation ►G ⊆ 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] >;
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 |
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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