direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C15⋊D4, C30.46C24, Dic15⋊10C23, C30⋊4(C2×D4), (C2×C30)⋊13D4, C15⋊5(C22×D4), (S3×C23)⋊3D5, (C6×D5)⋊6C23, D6⋊6(C22×D5), (C23×D5)⋊6S3, (S3×C10)⋊6C23, (C22×D5)⋊14D6, D10⋊6(C22×S3), C6.46(C23×D5), C23.70(S3×D5), (C22×S3)⋊13D10, C10.46(S3×C23), (C2×C30).249C23, (C22×C10).118D6, (C22×C6).101D10, (C2×Dic15)⋊38C22, (C22×Dic15)⋊20C2, (C22×C30).87C22, C6⋊3(C2×C5⋊D4), C10⋊3(C2×C3⋊D4), (D5×C22×C6)⋊3C2, C3⋊3(C22×C5⋊D4), C5⋊3(C22×C3⋊D4), (S3×C22×C10)⋊3C2, (D5×C2×C6)⋊17C22, (C2×C6)⋊12(C5⋊D4), C2.46(C22×S3×D5), (S3×C2×C10)⋊17C22, (C2×C10)⋊16(C3⋊D4), C22.109(C2×S3×D5), (C2×C6).255(C22×D5), (C2×C10).253(C22×S3), SmallGroup(480,1118)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2044 in 472 conjugacy classes, 148 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×7], C22 [×32], C5, S3 [×4], C6, C6 [×6], C6 [×4], C2×C4 [×6], D4 [×16], C23, C23 [×20], D5 [×4], C10, C10 [×6], C10 [×4], Dic3 [×4], D6 [×4], D6 [×12], C2×C6 [×7], C2×C6 [×16], C15, C22×C4, C2×D4 [×12], C24 [×2], Dic5 [×4], D10 [×4], D10 [×12], C2×C10 [×7], C2×C10 [×16], C2×Dic3 [×6], C3⋊D4 [×16], C22×S3 [×6], C22×S3 [×4], C22×C6, C22×C6 [×10], C5×S3 [×4], C3×D5 [×4], C30, C30 [×6], C22×D4, C2×Dic5 [×6], C5⋊D4 [×16], C22×D5 [×6], C22×D5 [×4], C22×C10, C22×C10 [×10], C22×Dic3, C2×C3⋊D4 [×12], S3×C23, C23×C6, Dic15 [×4], C6×D5 [×4], C6×D5 [×12], S3×C10 [×4], S3×C10 [×12], C2×C30 [×7], C22×Dic5, C2×C5⋊D4 [×12], C23×D5, C23×C10, C22×C3⋊D4, C15⋊D4 [×16], C2×Dic15 [×6], D5×C2×C6 [×6], D5×C2×C6 [×4], S3×C2×C10 [×6], S3×C2×C10 [×4], C22×C30, C22×C5⋊D4, C2×C15⋊D4 [×12], C22×Dic15, D5×C22×C6, S3×C22×C10, C22×C15⋊D4
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], C22×D4, C5⋊D4 [×4], C22×D5 [×7], C2×C3⋊D4 [×6], S3×C23, S3×D5, C2×C5⋊D4 [×6], C23×D5, C22×C3⋊D4, C15⋊D4 [×4], C2×S3×D5 [×3], C22×C5⋊D4, C2×C15⋊D4 [×6], C22×S3×D5, C22×C15⋊D4
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=c-1, ece=c4, ede=d-1 >
►On 240 points
Generators in S240
(1 108)(2 109)(3 110)(4 111)(5 112)(6 113)(7 114)(8 115)(9 116)(10 117)(11 118)(12 119)(13 120)(14 106)(15 107)(16 102)(17 103)(18 104)(19 105)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 100)(30 101)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 62)(47 63)(48 64)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)(57 73)(58 74)(59 75)(60 61)(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)(136 224)(137 225)(138 211)(139 212)(140 213)(141 214)(142 215)(143 216)(144 217)(145 218)(146 219)(147 220)(148 221)(149 222)(150 223)(151 205)(152 206)(153 207)(154 208)(155 209)(156 210)(157 196)(158 197)(159 198)(160 199)(161 200)(162 201)(163 202)(164 203)(165 204)(166 192)(167 193)(168 194)(169 195)(170 181)(171 182)(172 183)(173 184)(174 185)(175 186)(176 187)(177 188)(178 189)(179 190)(180 191)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(61 113)(62 114)(63 115)(64 116)(65 117)(66 118)(67 119)(68 120)(69 106)(70 107)(71 108)(72 109)(73 110)(74 111)(75 112)(76 96)(77 97)(78 98)(79 99)(80 100)(81 101)(82 102)(83 103)(84 104)(85 105)(86 91)(87 92)(88 93)(89 94)(90 95)(121 177)(122 178)(123 179)(124 180)(125 166)(126 167)(127 168)(128 169)(129 170)(130 171)(131 172)(132 173)(133 174)(134 175)(135 176)(136 164)(137 165)(138 151)(139 152)(140 153)(141 154)(142 155)(143 156)(144 157)(145 158)(146 159)(147 160)(148 161)(149 162)(150 163)(181 234)(182 235)(183 236)(184 237)(185 238)(186 239)(187 240)(188 226)(189 227)(190 228)(191 229)(192 230)(193 231)(194 232)(195 233)(196 217)(197 218)(198 219)(199 220)(200 221)(201 222)(202 223)(203 224)(204 225)(205 211)(206 212)(207 213)(208 214)(209 215)(210 216)
(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)
(1 121 22 136)(2 135 23 150)(3 134 24 149)(4 133 25 148)(5 132 26 147)(6 131 27 146)(7 130 28 145)(8 129 29 144)(9 128 30 143)(10 127 16 142)(11 126 17 141)(12 125 18 140)(13 124 19 139)(14 123 20 138)(15 122 21 137)(31 155 49 168)(32 154 50 167)(33 153 51 166)(34 152 52 180)(35 151 53 179)(36 165 54 178)(37 164 55 177)(38 163 56 176)(39 162 57 175)(40 161 58 174)(41 160 59 173)(42 159 60 172)(43 158 46 171)(44 157 47 170)(45 156 48 169)(61 183 78 198)(62 182 79 197)(63 181 80 196)(64 195 81 210)(65 194 82 209)(66 193 83 208)(67 192 84 207)(68 191 85 206)(69 190 86 205)(70 189 87 204)(71 188 88 203)(72 187 89 202)(73 186 90 201)(74 185 76 200)(75 184 77 199)(91 211 106 228)(92 225 107 227)(93 224 108 226)(94 223 109 240)(95 222 110 239)(96 221 111 238)(97 220 112 237)(98 219 113 236)(99 218 114 235)(100 217 115 234)(101 216 116 233)(102 215 117 232)(103 214 118 231)(104 213 119 230)(105 212 120 229)
(1 93)(2 97)(3 101)(4 105)(5 94)(6 98)(7 102)(8 91)(9 95)(10 99)(11 103)(12 92)(13 96)(14 100)(15 104)(16 114)(17 118)(18 107)(19 111)(20 115)(21 119)(22 108)(23 112)(24 116)(25 120)(26 109)(27 113)(28 117)(29 106)(30 110)(31 62)(32 66)(33 70)(34 74)(35 63)(36 67)(37 71)(38 75)(39 64)(40 68)(41 72)(42 61)(43 65)(44 69)(45 73)(46 82)(47 86)(48 90)(49 79)(50 83)(51 87)(52 76)(53 80)(54 84)(55 88)(56 77)(57 81)(58 85)(59 89)(60 78)(121 226)(122 230)(123 234)(124 238)(125 227)(126 231)(127 235)(128 239)(129 228)(130 232)(131 236)(132 240)(133 229)(134 233)(135 237)(136 224)(137 213)(138 217)(139 221)(140 225)(141 214)(142 218)(143 222)(144 211)(145 215)(146 219)(147 223)(148 212)(149 216)(150 220)(151 196)(152 200)(153 204)(154 208)(155 197)(156 201)(157 205)(158 209)(159 198)(160 202)(161 206)(162 210)(163 199)(164 203)(165 207)(166 189)(167 193)(168 182)(169 186)(170 190)(171 194)(172 183)(173 187)(174 191)(175 195)(176 184)(177 188)(178 192)(179 181)(180 185)
G:=sub<Sym(240)| (1,108)(2,109)(3,110)(4,111)(5,112)(6,113)(7,114)(8,115)(9,116)(10,117)(11,118)(12,119)(13,120)(14,106)(15,107)(16,102)(17,103)(18,104)(19,105)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,62)(47,63)(48,64)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,73)(58,74)(59,75)(60,61)(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)(136,224)(137,225)(138,211)(139,212)(140,213)(141,214)(142,215)(143,216)(144,217)(145,218)(146,219)(147,220)(148,221)(149,222)(150,223)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,196)(158,197)(159,198)(160,199)(161,200)(162,201)(163,202)(164,203)(165,204)(166,192)(167,193)(168,194)(169,195)(170,181)(171,182)(172,183)(173,184)(174,185)(175,186)(176,187)(177,188)(178,189)(179,190)(180,191), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,96)(77,97)(78,98)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104)(85,105)(86,91)(87,92)(88,93)(89,94)(90,95)(121,177)(122,178)(123,179)(124,180)(125,166)(126,167)(127,168)(128,169)(129,170)(130,171)(131,172)(132,173)(133,174)(134,175)(135,176)(136,164)(137,165)(138,151)(139,152)(140,153)(141,154)(142,155)(143,156)(144,157)(145,158)(146,159)(147,160)(148,161)(149,162)(150,163)(181,234)(182,235)(183,236)(184,237)(185,238)(186,239)(187,240)(188,226)(189,227)(190,228)(191,229)(192,230)(193,231)(194,232)(195,233)(196,217)(197,218)(198,219)(199,220)(200,221)(201,222)(202,223)(203,224)(204,225)(205,211)(206,212)(207,213)(208,214)(209,215)(210,216), (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), (1,121,22,136)(2,135,23,150)(3,134,24,149)(4,133,25,148)(5,132,26,147)(6,131,27,146)(7,130,28,145)(8,129,29,144)(9,128,30,143)(10,127,16,142)(11,126,17,141)(12,125,18,140)(13,124,19,139)(14,123,20,138)(15,122,21,137)(31,155,49,168)(32,154,50,167)(33,153,51,166)(34,152,52,180)(35,151,53,179)(36,165,54,178)(37,164,55,177)(38,163,56,176)(39,162,57,175)(40,161,58,174)(41,160,59,173)(42,159,60,172)(43,158,46,171)(44,157,47,170)(45,156,48,169)(61,183,78,198)(62,182,79,197)(63,181,80,196)(64,195,81,210)(65,194,82,209)(66,193,83,208)(67,192,84,207)(68,191,85,206)(69,190,86,205)(70,189,87,204)(71,188,88,203)(72,187,89,202)(73,186,90,201)(74,185,76,200)(75,184,77,199)(91,211,106,228)(92,225,107,227)(93,224,108,226)(94,223,109,240)(95,222,110,239)(96,221,111,238)(97,220,112,237)(98,219,113,236)(99,218,114,235)(100,217,115,234)(101,216,116,233)(102,215,117,232)(103,214,118,231)(104,213,119,230)(105,212,120,229), (1,93)(2,97)(3,101)(4,105)(5,94)(6,98)(7,102)(8,91)(9,95)(10,99)(11,103)(12,92)(13,96)(14,100)(15,104)(16,114)(17,118)(18,107)(19,111)(20,115)(21,119)(22,108)(23,112)(24,116)(25,120)(26,109)(27,113)(28,117)(29,106)(30,110)(31,62)(32,66)(33,70)(34,74)(35,63)(36,67)(37,71)(38,75)(39,64)(40,68)(41,72)(42,61)(43,65)(44,69)(45,73)(46,82)(47,86)(48,90)(49,79)(50,83)(51,87)(52,76)(53,80)(54,84)(55,88)(56,77)(57,81)(58,85)(59,89)(60,78)(121,226)(122,230)(123,234)(124,238)(125,227)(126,231)(127,235)(128,239)(129,228)(130,232)(131,236)(132,240)(133,229)(134,233)(135,237)(136,224)(137,213)(138,217)(139,221)(140,225)(141,214)(142,218)(143,222)(144,211)(145,215)(146,219)(147,223)(148,212)(149,216)(150,220)(151,196)(152,200)(153,204)(154,208)(155,197)(156,201)(157,205)(158,209)(159,198)(160,202)(161,206)(162,210)(163,199)(164,203)(165,207)(166,189)(167,193)(168,182)(169,186)(170,190)(171,194)(172,183)(173,187)(174,191)(175,195)(176,184)(177,188)(178,192)(179,181)(180,185)>;
G:=Group( (1,108)(2,109)(3,110)(4,111)(5,112)(6,113)(7,114)(8,115)(9,116)(10,117)(11,118)(12,119)(13,120)(14,106)(15,107)(16,102)(17,103)(18,104)(19,105)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,62)(47,63)(48,64)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,73)(58,74)(59,75)(60,61)(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)(136,224)(137,225)(138,211)(139,212)(140,213)(141,214)(142,215)(143,216)(144,217)(145,218)(146,219)(147,220)(148,221)(149,222)(150,223)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,196)(158,197)(159,198)(160,199)(161,200)(162,201)(163,202)(164,203)(165,204)(166,192)(167,193)(168,194)(169,195)(170,181)(171,182)(172,183)(173,184)(174,185)(175,186)(176,187)(177,188)(178,189)(179,190)(180,191), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,96)(77,97)(78,98)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104)(85,105)(86,91)(87,92)(88,93)(89,94)(90,95)(121,177)(122,178)(123,179)(124,180)(125,166)(126,167)(127,168)(128,169)(129,170)(130,171)(131,172)(132,173)(133,174)(134,175)(135,176)(136,164)(137,165)(138,151)(139,152)(140,153)(141,154)(142,155)(143,156)(144,157)(145,158)(146,159)(147,160)(148,161)(149,162)(150,163)(181,234)(182,235)(183,236)(184,237)(185,238)(186,239)(187,240)(188,226)(189,227)(190,228)(191,229)(192,230)(193,231)(194,232)(195,233)(196,217)(197,218)(198,219)(199,220)(200,221)(201,222)(202,223)(203,224)(204,225)(205,211)(206,212)(207,213)(208,214)(209,215)(210,216), (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), (1,121,22,136)(2,135,23,150)(3,134,24,149)(4,133,25,148)(5,132,26,147)(6,131,27,146)(7,130,28,145)(8,129,29,144)(9,128,30,143)(10,127,16,142)(11,126,17,141)(12,125,18,140)(13,124,19,139)(14,123,20,138)(15,122,21,137)(31,155,49,168)(32,154,50,167)(33,153,51,166)(34,152,52,180)(35,151,53,179)(36,165,54,178)(37,164,55,177)(38,163,56,176)(39,162,57,175)(40,161,58,174)(41,160,59,173)(42,159,60,172)(43,158,46,171)(44,157,47,170)(45,156,48,169)(61,183,78,198)(62,182,79,197)(63,181,80,196)(64,195,81,210)(65,194,82,209)(66,193,83,208)(67,192,84,207)(68,191,85,206)(69,190,86,205)(70,189,87,204)(71,188,88,203)(72,187,89,202)(73,186,90,201)(74,185,76,200)(75,184,77,199)(91,211,106,228)(92,225,107,227)(93,224,108,226)(94,223,109,240)(95,222,110,239)(96,221,111,238)(97,220,112,237)(98,219,113,236)(99,218,114,235)(100,217,115,234)(101,216,116,233)(102,215,117,232)(103,214,118,231)(104,213,119,230)(105,212,120,229), (1,93)(2,97)(3,101)(4,105)(5,94)(6,98)(7,102)(8,91)(9,95)(10,99)(11,103)(12,92)(13,96)(14,100)(15,104)(16,114)(17,118)(18,107)(19,111)(20,115)(21,119)(22,108)(23,112)(24,116)(25,120)(26,109)(27,113)(28,117)(29,106)(30,110)(31,62)(32,66)(33,70)(34,74)(35,63)(36,67)(37,71)(38,75)(39,64)(40,68)(41,72)(42,61)(43,65)(44,69)(45,73)(46,82)(47,86)(48,90)(49,79)(50,83)(51,87)(52,76)(53,80)(54,84)(55,88)(56,77)(57,81)(58,85)(59,89)(60,78)(121,226)(122,230)(123,234)(124,238)(125,227)(126,231)(127,235)(128,239)(129,228)(130,232)(131,236)(132,240)(133,229)(134,233)(135,237)(136,224)(137,213)(138,217)(139,221)(140,225)(141,214)(142,218)(143,222)(144,211)(145,215)(146,219)(147,223)(148,212)(149,216)(150,220)(151,196)(152,200)(153,204)(154,208)(155,197)(156,201)(157,205)(158,209)(159,198)(160,202)(161,206)(162,210)(163,199)(164,203)(165,207)(166,189)(167,193)(168,182)(169,186)(170,190)(171,194)(172,183)(173,187)(174,191)(175,195)(176,184)(177,188)(178,192)(179,181)(180,185) );
G=PermutationGroup([(1,108),(2,109),(3,110),(4,111),(5,112),(6,113),(7,114),(8,115),(9,116),(10,117),(11,118),(12,119),(13,120),(14,106),(15,107),(16,102),(17,103),(18,104),(19,105),(20,91),(21,92),(22,93),(23,94),(24,95),(25,96),(26,97),(27,98),(28,99),(29,100),(30,101),(31,82),(32,83),(33,84),(34,85),(35,86),(36,87),(37,88),(38,89),(39,90),(40,76),(41,77),(42,78),(43,79),(44,80),(45,81),(46,62),(47,63),(48,64),(49,65),(50,66),(51,67),(52,68),(53,69),(54,70),(55,71),(56,72),(57,73),(58,74),(59,75),(60,61),(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),(136,224),(137,225),(138,211),(139,212),(140,213),(141,214),(142,215),(143,216),(144,217),(145,218),(146,219),(147,220),(148,221),(149,222),(150,223),(151,205),(152,206),(153,207),(154,208),(155,209),(156,210),(157,196),(158,197),(159,198),(160,199),(161,200),(162,201),(163,202),(164,203),(165,204),(166,192),(167,193),(168,194),(169,195),(170,181),(171,182),(172,183),(173,184),(174,185),(175,186),(176,187),(177,188),(178,189),(179,190),(180,191)], [(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,46),(8,47),(9,48),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,31),(17,32),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42),(28,43),(29,44),(30,45),(61,113),(62,114),(63,115),(64,116),(65,117),(66,118),(67,119),(68,120),(69,106),(70,107),(71,108),(72,109),(73,110),(74,111),(75,112),(76,96),(77,97),(78,98),(79,99),(80,100),(81,101),(82,102),(83,103),(84,104),(85,105),(86,91),(87,92),(88,93),(89,94),(90,95),(121,177),(122,178),(123,179),(124,180),(125,166),(126,167),(127,168),(128,169),(129,170),(130,171),(131,172),(132,173),(133,174),(134,175),(135,176),(136,164),(137,165),(138,151),(139,152),(140,153),(141,154),(142,155),(143,156),(144,157),(145,158),(146,159),(147,160),(148,161),(149,162),(150,163),(181,234),(182,235),(183,236),(184,237),(185,238),(186,239),(187,240),(188,226),(189,227),(190,228),(191,229),(192,230),(193,231),(194,232),(195,233),(196,217),(197,218),(198,219),(199,220),(200,221),(201,222),(202,223),(203,224),(204,225),(205,211),(206,212),(207,213),(208,214),(209,215),(210,216)], [(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)], [(1,121,22,136),(2,135,23,150),(3,134,24,149),(4,133,25,148),(5,132,26,147),(6,131,27,146),(7,130,28,145),(8,129,29,144),(9,128,30,143),(10,127,16,142),(11,126,17,141),(12,125,18,140),(13,124,19,139),(14,123,20,138),(15,122,21,137),(31,155,49,168),(32,154,50,167),(33,153,51,166),(34,152,52,180),(35,151,53,179),(36,165,54,178),(37,164,55,177),(38,163,56,176),(39,162,57,175),(40,161,58,174),(41,160,59,173),(42,159,60,172),(43,158,46,171),(44,157,47,170),(45,156,48,169),(61,183,78,198),(62,182,79,197),(63,181,80,196),(64,195,81,210),(65,194,82,209),(66,193,83,208),(67,192,84,207),(68,191,85,206),(69,190,86,205),(70,189,87,204),(71,188,88,203),(72,187,89,202),(73,186,90,201),(74,185,76,200),(75,184,77,199),(91,211,106,228),(92,225,107,227),(93,224,108,226),(94,223,109,240),(95,222,110,239),(96,221,111,238),(97,220,112,237),(98,219,113,236),(99,218,114,235),(100,217,115,234),(101,216,116,233),(102,215,117,232),(103,214,118,231),(104,213,119,230),(105,212,120,229)], [(1,93),(2,97),(3,101),(4,105),(5,94),(6,98),(7,102),(8,91),(9,95),(10,99),(11,103),(12,92),(13,96),(14,100),(15,104),(16,114),(17,118),(18,107),(19,111),(20,115),(21,119),(22,108),(23,112),(24,116),(25,120),(26,109),(27,113),(28,117),(29,106),(30,110),(31,62),(32,66),(33,70),(34,74),(35,63),(36,67),(37,71),(38,75),(39,64),(40,68),(41,72),(42,61),(43,65),(44,69),(45,73),(46,82),(47,86),(48,90),(49,79),(50,83),(51,87),(52,76),(53,80),(54,84),(55,88),(56,77),(57,81),(58,85),(59,89),(60,78),(121,226),(122,230),(123,234),(124,238),(125,227),(126,231),(127,235),(128,239),(129,228),(130,232),(131,236),(132,240),(133,229),(134,233),(135,237),(136,224),(137,213),(138,217),(139,221),(140,225),(141,214),(142,218),(143,222),(144,211),(145,215),(146,219),(147,223),(148,212),(149,216),(150,220),(151,196),(152,200),(153,204),(154,208),(155,197),(156,201),(157,205),(158,209),(159,198),(160,202),(161,206),(162,210),(163,199),(164,203),(165,207),(166,189),(167,193),(168,182),(169,186),(170,190),(171,194),(172,183),(173,187),(174,191),(175,195),(176,184),(177,188),(178,192),(179,181),(180,185)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 18 | 0 | 0 |
| 0 | 0 | 43 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 2 | 15 | 0 | 0 | 0 | 0 |
| 20 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 43 | 0 | 0 |
| 0 | 0 | 1 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 45 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 43 | 0 | 0 |
| 0 | 0 | 1 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,43,0,0,0,0,18,18,0,0,0,0,0,0,47,0,0,0,0,0,0,13],[2,20,0,0,0,0,15,59,0,0,0,0,0,0,18,1,0,0,0,0,43,43,0,0,0,0,0,0,0,1,0,0,0,0,60,0],[60,45,0,0,0,0,0,1,0,0,0,0,0,0,18,1,0,0,0,0,43,43,0,0,0,0,0,0,1,0,0,0,0,0,0,60] >;
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | ··· | 6G | 6H | ··· | 6O | 10A | ··· | 10N | 10O | ··· | 10AD | 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 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 6 | ··· | 6 | 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 | C3⋊D4 | C5⋊D4 | S3×D5 | C15⋊D4 | C2×S3×D5 |
| kernel | C22×C15⋊D4 | C2×C15⋊D4 | C22×Dic15 | D5×C22×C6 | S3×C22×C10 | C23×D5 | C2×C30 | S3×C23 | C22×D5 | 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_{15}\rtimes D_4 % in TeX
G:=Group("C2^2xC15:D4"); // GroupNames label
G:=SmallGroup(480,1118);
// by ID
G=gap.SmallGroup(480,1118);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,1356,18822]);
// 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=c^-1,e*c*e=c^4,e*d*e=d^-1>;
// generators/relations
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