direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C15⋊7D4, C24⋊6D15, C23⋊5D30, D30⋊8C23, C30.68C24, Dic15⋊5C23, (C2×C30)⋊31D4, C30⋊16(C2×D4), (C23×C6)⋊6D5, (C23×C30)⋊4C2, C15⋊17(C22×D4), (C2×C30)⋊10C23, (C23×C10)⋊10S3, (C23×D15)⋊5C2, (C22×C10)⋊17D6, (C22×C6)⋊14D10, C6.68(C23×D5), C10.68(S3×C23), C2.15(C23×D15), C22⋊3(C22×D15), (C22×C30)⋊19C22, (C22×Dic15)⋊9C2, (C2×Dic15)⋊28C22, (C22×D15)⋊19C22, C6⋊4(C2×C5⋊D4), C10⋊4(C2×C3⋊D4), C3⋊4(C22×C5⋊D4), C5⋊4(C22×C3⋊D4), (C2×C6)⋊9(C22×D5), (C2×C6)⋊15(C5⋊D4), (C2×C10)⋊19(C3⋊D4), (C2×C10)⋊12(C22×S3), SmallGroup(480,1179)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2452 in 472 conjugacy classes, 159 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×11], C22 [×28], C5, S3 [×4], C6, C6 [×6], C6 [×4], C2×C4 [×6], D4 [×16], C23, C23 [×6], C23 [×14], D5 [×4], C10, C10 [×6], C10 [×4], Dic3 [×4], D6 [×16], C2×C6 [×11], C2×C6 [×12], C15, C22×C4, C2×D4 [×12], C24, C24, Dic5 [×4], D10 [×16], C2×C10 [×11], C2×C10 [×12], C2×Dic3 [×6], C3⋊D4 [×16], C22×S3 [×10], C22×C6, C22×C6 [×6], C22×C6 [×4], D15 [×4], C30, C30 [×6], C30 [×4], C22×D4, C2×Dic5 [×6], C5⋊D4 [×16], C22×D5 [×10], C22×C10, C22×C10 [×6], C22×C10 [×4], C22×Dic3, C2×C3⋊D4 [×12], S3×C23, C23×C6, Dic15 [×4], D30 [×4], D30 [×12], C2×C30 [×11], C2×C30 [×12], C22×Dic5, C2×C5⋊D4 [×12], C23×D5, C23×C10, C22×C3⋊D4, C2×Dic15 [×6], C15⋊7D4 [×16], C22×D15 [×6], C22×D15 [×4], C22×C30, C22×C30 [×6], C22×C30 [×4], C22×C5⋊D4, C22×Dic15, C2×C15⋊7D4 [×12], C23×D15, C23×C30, C22×C15⋊7D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], C3⋊D4 [×4], C22×S3 [×7], D15, C22×D4, C5⋊D4 [×4], C22×D5 [×7], C2×C3⋊D4 [×6], S3×C23, D30 [×7], C2×C5⋊D4 [×6], C23×D5, C22×C3⋊D4, C15⋊7D4 [×4], C22×D15 [×7], C22×C5⋊D4, C2×C15⋊7D4 [×6], C23×D15, C22×C15⋊7D4
Generators and relations
G = < a,b,c,d,e | a2=b2=c15=d4=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
(1 149)(2 150)(3 136)(4 137)(5 138)(6 139)(7 140)(8 141)(9 142)(10 143)(11 144)(12 145)(13 146)(14 147)(15 148)(16 123)(17 124)(18 125)(19 126)(20 127)(21 128)(22 129)(23 130)(24 131)(25 132)(26 133)(27 134)(28 135)(29 121)(30 122)(31 167)(32 168)(33 169)(34 170)(35 171)(36 172)(37 173)(38 174)(39 175)(40 176)(41 177)(42 178)(43 179)(44 180)(45 166)(46 163)(47 164)(48 165)(49 151)(50 152)(51 153)(52 154)(53 155)(54 156)(55 157)(56 158)(57 159)(58 160)(59 161)(60 162)(61 197)(62 198)(63 199)(64 200)(65 201)(66 202)(67 203)(68 204)(69 205)(70 206)(71 207)(72 208)(73 209)(74 210)(75 196)(76 191)(77 192)(78 193)(79 194)(80 195)(81 181)(82 182)(83 183)(84 184)(85 185)(86 186)(87 187)(88 188)(89 189)(90 190)(91 229)(92 230)(93 231)(94 232)(95 233)(96 234)(97 235)(98 236)(99 237)(100 238)(101 239)(102 240)(103 226)(104 227)(105 228)(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 84)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 76)(9 77)(10 78)(11 79)(12 80)(13 81)(14 82)(15 83)(16 73)(17 74)(18 75)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 113)(32 114)(33 115)(34 116)(35 117)(36 118)(37 119)(38 120)(39 106)(40 107)(41 108)(42 109)(43 110)(44 111)(45 112)(46 101)(47 102)(48 103)(49 104)(50 105)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 97)(58 98)(59 99)(60 100)(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)(136 186)(137 187)(138 188)(139 189)(140 190)(141 191)(142 192)(143 193)(144 194)(145 195)(146 181)(147 182)(148 183)(149 184)(150 185)(151 227)(152 228)(153 229)(154 230)(155 231)(156 232)(157 233)(158 234)(159 235)(160 236)(161 237)(162 238)(163 239)(164 240)(165 226)(166 224)(167 225)(168 211)(169 212)(170 213)(171 214)(172 215)(173 216)(174 217)(175 218)(176 219)(177 220)(178 221)(179 222)(180 223)
(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)
(1 154 27 174)(2 153 28 173)(3 152 29 172)(4 151 30 171)(5 165 16 170)(6 164 17 169)(7 163 18 168)(8 162 19 167)(9 161 20 166)(10 160 21 180)(11 159 22 179)(12 158 23 178)(13 157 24 177)(14 156 25 176)(15 155 26 175)(31 141 60 126)(32 140 46 125)(33 139 47 124)(34 138 48 123)(35 137 49 122)(36 136 50 121)(37 150 51 135)(38 149 52 134)(39 148 53 133)(40 147 54 132)(41 146 55 131)(42 145 56 130)(43 144 57 129)(44 143 58 128)(45 142 59 127)(61 225 76 238)(62 224 77 237)(63 223 78 236)(64 222 79 235)(65 221 80 234)(66 220 81 233)(67 219 82 232)(68 218 83 231)(69 217 84 230)(70 216 85 229)(71 215 86 228)(72 214 87 227)(73 213 88 226)(74 212 89 240)(75 211 90 239)(91 206 119 185)(92 205 120 184)(93 204 106 183)(94 203 107 182)(95 202 108 181)(96 201 109 195)(97 200 110 194)(98 199 111 193)(99 198 112 192)(100 197 113 191)(101 196 114 190)(102 210 115 189)(103 209 116 188)(104 208 117 187)(105 207 118 186)
(1 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 75)(11 74)(12 73)(13 72)(14 71)(15 70)(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)(31 112)(32 111)(33 110)(34 109)(35 108)(36 107)(37 106)(38 120)(39 119)(40 118)(41 117)(42 116)(43 115)(44 114)(45 113)(46 98)(47 97)(48 96)(49 95)(50 94)(51 93)(52 92)(53 91)(54 105)(55 104)(56 103)(57 102)(58 101)(59 100)(60 99)(121 182)(122 181)(123 195)(124 194)(125 193)(126 192)(127 191)(128 190)(129 189)(130 188)(131 187)(132 186)(133 185)(134 184)(135 183)(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)(151 233)(152 232)(153 231)(154 230)(155 229)(156 228)(157 227)(158 226)(159 240)(160 239)(161 238)(162 237)(163 236)(164 235)(165 234)(166 225)(167 224)(168 223)(169 222)(170 221)(171 220)(172 219)(173 218)(174 217)(175 216)(176 215)(177 214)(178 213)(179 212)(180 211)
G:=sub<Sym(240)| (1,149)(2,150)(3,136)(4,137)(5,138)(6,139)(7,140)(8,141)(9,142)(10,143)(11,144)(12,145)(13,146)(14,147)(15,148)(16,123)(17,124)(18,125)(19,126)(20,127)(21,128)(22,129)(23,130)(24,131)(25,132)(26,133)(27,134)(28,135)(29,121)(30,122)(31,167)(32,168)(33,169)(34,170)(35,171)(36,172)(37,173)(38,174)(39,175)(40,176)(41,177)(42,178)(43,179)(44,180)(45,166)(46,163)(47,164)(48,165)(49,151)(50,152)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,159)(58,160)(59,161)(60,162)(61,197)(62,198)(63,199)(64,200)(65,201)(66,202)(67,203)(68,204)(69,205)(70,206)(71,207)(72,208)(73,209)(74,210)(75,196)(76,191)(77,192)(78,193)(79,194)(80,195)(81,181)(82,182)(83,183)(84,184)(85,185)(86,186)(87,187)(88,188)(89,189)(90,190)(91,229)(92,230)(93,231)(94,232)(95,233)(96,234)(97,235)(98,236)(99,237)(100,238)(101,239)(102,240)(103,226)(104,227)(105,228)(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), (1,84)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,73)(17,74)(18,75)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,106)(40,107)(41,108)(42,109)(43,110)(44,111)(45,112)(46,101)(47,102)(48,103)(49,104)(50,105)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,97)(58,98)(59,99)(60,100)(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)(136,186)(137,187)(138,188)(139,189)(140,190)(141,191)(142,192)(143,193)(144,194)(145,195)(146,181)(147,182)(148,183)(149,184)(150,185)(151,227)(152,228)(153,229)(154,230)(155,231)(156,232)(157,233)(158,234)(159,235)(160,236)(161,237)(162,238)(163,239)(164,240)(165,226)(166,224)(167,225)(168,211)(169,212)(170,213)(171,214)(172,215)(173,216)(174,217)(175,218)(176,219)(177,220)(178,221)(179,222)(180,223), (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), (1,154,27,174)(2,153,28,173)(3,152,29,172)(4,151,30,171)(5,165,16,170)(6,164,17,169)(7,163,18,168)(8,162,19,167)(9,161,20,166)(10,160,21,180)(11,159,22,179)(12,158,23,178)(13,157,24,177)(14,156,25,176)(15,155,26,175)(31,141,60,126)(32,140,46,125)(33,139,47,124)(34,138,48,123)(35,137,49,122)(36,136,50,121)(37,150,51,135)(38,149,52,134)(39,148,53,133)(40,147,54,132)(41,146,55,131)(42,145,56,130)(43,144,57,129)(44,143,58,128)(45,142,59,127)(61,225,76,238)(62,224,77,237)(63,223,78,236)(64,222,79,235)(65,221,80,234)(66,220,81,233)(67,219,82,232)(68,218,83,231)(69,217,84,230)(70,216,85,229)(71,215,86,228)(72,214,87,227)(73,213,88,226)(74,212,89,240)(75,211,90,239)(91,206,119,185)(92,205,120,184)(93,204,106,183)(94,203,107,182)(95,202,108,181)(96,201,109,195)(97,200,110,194)(98,199,111,193)(99,198,112,192)(100,197,113,191)(101,196,114,190)(102,210,115,189)(103,209,116,188)(104,208,117,187)(105,207,118,186), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(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)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,120)(39,119)(40,118)(41,117)(42,116)(43,115)(44,114)(45,113)(46,98)(47,97)(48,96)(49,95)(50,94)(51,93)(52,92)(53,91)(54,105)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)(121,182)(122,181)(123,195)(124,194)(125,193)(126,192)(127,191)(128,190)(129,189)(130,188)(131,187)(132,186)(133,185)(134,184)(135,183)(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)(151,233)(152,232)(153,231)(154,230)(155,229)(156,228)(157,227)(158,226)(159,240)(160,239)(161,238)(162,237)(163,236)(164,235)(165,234)(166,225)(167,224)(168,223)(169,222)(170,221)(171,220)(172,219)(173,218)(174,217)(175,216)(176,215)(177,214)(178,213)(179,212)(180,211)>;
G:=Group( (1,149)(2,150)(3,136)(4,137)(5,138)(6,139)(7,140)(8,141)(9,142)(10,143)(11,144)(12,145)(13,146)(14,147)(15,148)(16,123)(17,124)(18,125)(19,126)(20,127)(21,128)(22,129)(23,130)(24,131)(25,132)(26,133)(27,134)(28,135)(29,121)(30,122)(31,167)(32,168)(33,169)(34,170)(35,171)(36,172)(37,173)(38,174)(39,175)(40,176)(41,177)(42,178)(43,179)(44,180)(45,166)(46,163)(47,164)(48,165)(49,151)(50,152)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,159)(58,160)(59,161)(60,162)(61,197)(62,198)(63,199)(64,200)(65,201)(66,202)(67,203)(68,204)(69,205)(70,206)(71,207)(72,208)(73,209)(74,210)(75,196)(76,191)(77,192)(78,193)(79,194)(80,195)(81,181)(82,182)(83,183)(84,184)(85,185)(86,186)(87,187)(88,188)(89,189)(90,190)(91,229)(92,230)(93,231)(94,232)(95,233)(96,234)(97,235)(98,236)(99,237)(100,238)(101,239)(102,240)(103,226)(104,227)(105,228)(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), (1,84)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,73)(17,74)(18,75)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,106)(40,107)(41,108)(42,109)(43,110)(44,111)(45,112)(46,101)(47,102)(48,103)(49,104)(50,105)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,97)(58,98)(59,99)(60,100)(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)(136,186)(137,187)(138,188)(139,189)(140,190)(141,191)(142,192)(143,193)(144,194)(145,195)(146,181)(147,182)(148,183)(149,184)(150,185)(151,227)(152,228)(153,229)(154,230)(155,231)(156,232)(157,233)(158,234)(159,235)(160,236)(161,237)(162,238)(163,239)(164,240)(165,226)(166,224)(167,225)(168,211)(169,212)(170,213)(171,214)(172,215)(173,216)(174,217)(175,218)(176,219)(177,220)(178,221)(179,222)(180,223), (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), (1,154,27,174)(2,153,28,173)(3,152,29,172)(4,151,30,171)(5,165,16,170)(6,164,17,169)(7,163,18,168)(8,162,19,167)(9,161,20,166)(10,160,21,180)(11,159,22,179)(12,158,23,178)(13,157,24,177)(14,156,25,176)(15,155,26,175)(31,141,60,126)(32,140,46,125)(33,139,47,124)(34,138,48,123)(35,137,49,122)(36,136,50,121)(37,150,51,135)(38,149,52,134)(39,148,53,133)(40,147,54,132)(41,146,55,131)(42,145,56,130)(43,144,57,129)(44,143,58,128)(45,142,59,127)(61,225,76,238)(62,224,77,237)(63,223,78,236)(64,222,79,235)(65,221,80,234)(66,220,81,233)(67,219,82,232)(68,218,83,231)(69,217,84,230)(70,216,85,229)(71,215,86,228)(72,214,87,227)(73,213,88,226)(74,212,89,240)(75,211,90,239)(91,206,119,185)(92,205,120,184)(93,204,106,183)(94,203,107,182)(95,202,108,181)(96,201,109,195)(97,200,110,194)(98,199,111,193)(99,198,112,192)(100,197,113,191)(101,196,114,190)(102,210,115,189)(103,209,116,188)(104,208,117,187)(105,207,118,186), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(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)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,120)(39,119)(40,118)(41,117)(42,116)(43,115)(44,114)(45,113)(46,98)(47,97)(48,96)(49,95)(50,94)(51,93)(52,92)(53,91)(54,105)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)(121,182)(122,181)(123,195)(124,194)(125,193)(126,192)(127,191)(128,190)(129,189)(130,188)(131,187)(132,186)(133,185)(134,184)(135,183)(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)(151,233)(152,232)(153,231)(154,230)(155,229)(156,228)(157,227)(158,226)(159,240)(160,239)(161,238)(162,237)(163,236)(164,235)(165,234)(166,225)(167,224)(168,223)(169,222)(170,221)(171,220)(172,219)(173,218)(174,217)(175,216)(176,215)(177,214)(178,213)(179,212)(180,211) );
G=PermutationGroup([(1,149),(2,150),(3,136),(4,137),(5,138),(6,139),(7,140),(8,141),(9,142),(10,143),(11,144),(12,145),(13,146),(14,147),(15,148),(16,123),(17,124),(18,125),(19,126),(20,127),(21,128),(22,129),(23,130),(24,131),(25,132),(26,133),(27,134),(28,135),(29,121),(30,122),(31,167),(32,168),(33,169),(34,170),(35,171),(36,172),(37,173),(38,174),(39,175),(40,176),(41,177),(42,178),(43,179),(44,180),(45,166),(46,163),(47,164),(48,165),(49,151),(50,152),(51,153),(52,154),(53,155),(54,156),(55,157),(56,158),(57,159),(58,160),(59,161),(60,162),(61,197),(62,198),(63,199),(64,200),(65,201),(66,202),(67,203),(68,204),(69,205),(70,206),(71,207),(72,208),(73,209),(74,210),(75,196),(76,191),(77,192),(78,193),(79,194),(80,195),(81,181),(82,182),(83,183),(84,184),(85,185),(86,186),(87,187),(88,188),(89,189),(90,190),(91,229),(92,230),(93,231),(94,232),(95,233),(96,234),(97,235),(98,236),(99,237),(100,238),(101,239),(102,240),(103,226),(104,227),(105,228),(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)], [(1,84),(2,85),(3,86),(4,87),(5,88),(6,89),(7,90),(8,76),(9,77),(10,78),(11,79),(12,80),(13,81),(14,82),(15,83),(16,73),(17,74),(18,75),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,113),(32,114),(33,115),(34,116),(35,117),(36,118),(37,119),(38,120),(39,106),(40,107),(41,108),(42,109),(43,110),(44,111),(45,112),(46,101),(47,102),(48,103),(49,104),(50,105),(51,91),(52,92),(53,93),(54,94),(55,95),(56,96),(57,97),(58,98),(59,99),(60,100),(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),(136,186),(137,187),(138,188),(139,189),(140,190),(141,191),(142,192),(143,193),(144,194),(145,195),(146,181),(147,182),(148,183),(149,184),(150,185),(151,227),(152,228),(153,229),(154,230),(155,231),(156,232),(157,233),(158,234),(159,235),(160,236),(161,237),(162,238),(163,239),(164,240),(165,226),(166,224),(167,225),(168,211),(169,212),(170,213),(171,214),(172,215),(173,216),(174,217),(175,218),(176,219),(177,220),(178,221),(179,222),(180,223)], [(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)], [(1,154,27,174),(2,153,28,173),(3,152,29,172),(4,151,30,171),(5,165,16,170),(6,164,17,169),(7,163,18,168),(8,162,19,167),(9,161,20,166),(10,160,21,180),(11,159,22,179),(12,158,23,178),(13,157,24,177),(14,156,25,176),(15,155,26,175),(31,141,60,126),(32,140,46,125),(33,139,47,124),(34,138,48,123),(35,137,49,122),(36,136,50,121),(37,150,51,135),(38,149,52,134),(39,148,53,133),(40,147,54,132),(41,146,55,131),(42,145,56,130),(43,144,57,129),(44,143,58,128),(45,142,59,127),(61,225,76,238),(62,224,77,237),(63,223,78,236),(64,222,79,235),(65,221,80,234),(66,220,81,233),(67,219,82,232),(68,218,83,231),(69,217,84,230),(70,216,85,229),(71,215,86,228),(72,214,87,227),(73,213,88,226),(74,212,89,240),(75,211,90,239),(91,206,119,185),(92,205,120,184),(93,204,106,183),(94,203,107,182),(95,202,108,181),(96,201,109,195),(97,200,110,194),(98,199,111,193),(99,198,112,192),(100,197,113,191),(101,196,114,190),(102,210,115,189),(103,209,116,188),(104,208,117,187),(105,207,118,186)], [(1,69),(2,68),(3,67),(4,66),(5,65),(6,64),(7,63),(8,62),(9,61),(10,75),(11,74),(12,73),(13,72),(14,71),(15,70),(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),(31,112),(32,111),(33,110),(34,109),(35,108),(36,107),(37,106),(38,120),(39,119),(40,118),(41,117),(42,116),(43,115),(44,114),(45,113),(46,98),(47,97),(48,96),(49,95),(50,94),(51,93),(52,92),(53,91),(54,105),(55,104),(56,103),(57,102),(58,101),(59,100),(60,99),(121,182),(122,181),(123,195),(124,194),(125,193),(126,192),(127,191),(128,190),(129,189),(130,188),(131,187),(132,186),(133,185),(134,184),(135,183),(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),(151,233),(152,232),(153,231),(154,230),(155,229),(156,228),(157,227),(158,226),(159,240),(160,239),(161,238),(162,237),(163,236),(164,235),(165,234),(166,225),(167,224),(168,223),(169,222),(170,221),(171,220),(172,219),(173,218),(174,217),(175,216),(176,215),(177,214),(178,213),(179,212),(180,211)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 5 | 38 | 0 | 0 | 0 | 0 |
| 23 | 53 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 47 |
| 0 | 0 | 0 | 0 | 22 | 23 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 14 |
| 0 | 0 | 0 | 0 | 52 | 53 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 43 | 1 |
| 0 | 0 | 0 | 0 | 43 | 18 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[5,23,0,0,0,0,38,53,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,45,22,0,0,0,0,47,23],[60,18,0,0,0,0,0,1,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,8,52,0,0,0,0,14,53],[60,18,0,0,0,0,0,1,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,43,43,0,0,0,0,1,18] >;
132 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | ··· | 6O | 10A | ··· | 10AD | 15A | 15B | 15C | 15D | 30A | ··· | 30BH |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D10 | C3⋊D4 | D15 | C5⋊D4 | D30 | C15⋊7D4 |
| kernel | C22×C15⋊7D4 | C22×Dic15 | C2×C15⋊7D4 | C23×D15 | C23×C30 | C23×C10 | C2×C30 | C23×C6 | C22×C10 | C22×C6 | C2×C10 | C24 | C2×C6 | C23 | C22 |
| # reps | 1 | 1 | 12 | 1 | 1 | 1 | 4 | 2 | 7 | 14 | 8 | 4 | 16 | 28 | 32 |
In GAP, Magma, Sage, TeX
C_2^2\times C_{15}\rtimes_7D_4 % in TeX
G:=Group("C2^2xC15:7D4"); // GroupNames label
G:=SmallGroup(480,1179);
// by ID
G=gap.SmallGroup(480,1179);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,675,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^15=d^4=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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