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), C23.70(S3×D5), C6.46(C23×D5), (C22×S3)⋊13D10, C10.46(S3×C23), (C2×C30).249C23, (C22×C6).101D10, (C22×C10).118D6, (C22×Dic15)⋊20C2, (C2×Dic15)⋊38C22, (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
Generators and relations for C22×C15⋊D4
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 >
Subgroups: 2044 in 472 conjugacy classes, 148 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C10, Dic3, D6, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C24, Dic5, D10, D10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C22×S3, C22×S3, C22×C6, C22×C6, C5×S3, C3×D5, C30, C30, C22×D4, C2×Dic5, C5⋊D4, C22×D5, C22×D5, C22×C10, C22×C10, C22×Dic3, C2×C3⋊D4, S3×C23, C23×C6, Dic15, C6×D5, C6×D5, S3×C10, S3×C10, C2×C30, C22×Dic5, C2×C5⋊D4, C23×D5, C23×C10, C22×C3⋊D4, C15⋊D4, C2×Dic15, D5×C2×C6, D5×C2×C6, S3×C2×C10, S3×C2×C10, C22×C30, C22×C5⋊D4, C2×C15⋊D4, C22×Dic15, D5×C22×C6, S3×C22×C10, C22×C15⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C24, D10, C3⋊D4, C22×S3, C22×D4, C5⋊D4, C22×D5, C2×C3⋊D4, S3×C23, S3×D5, C2×C5⋊D4, C23×D5, C22×C3⋊D4, C15⋊D4, C2×S3×D5, C22×C5⋊D4, C2×C15⋊D4, C22×S3×D5, C22×C15⋊D4
►On 240 points
Generators in S240
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 113)(15 114)(16 105)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 89)(32 90)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 74)(47 75)(48 61)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(121 230)(122 231)(123 232)(124 233)(125 234)(126 235)(127 236)(128 237)(129 238)(130 239)(131 240)(132 226)(133 227)(134 228)(135 229)(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)(151 200)(152 201)(153 202)(154 203)(155 204)(156 205)(157 206)(158 207)(159 208)(160 209)(161 210)(162 196)(163 197)(164 198)(165 199)(166 194)(167 195)(168 181)(169 182)(170 183)(171 184)(172 185)(173 186)(174 187)(175 188)(176 189)(177 190)(178 191)(179 192)(180 193)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(61 107)(62 108)(63 109)(64 110)(65 111)(66 112)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 119)(74 120)(75 106)(76 98)(77 99)(78 100)(79 101)(80 102)(81 103)(82 104)(83 105)(84 91)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(121 170)(122 171)(123 172)(124 173)(125 174)(126 175)(127 176)(128 177)(129 178)(130 179)(131 180)(132 166)(133 167)(134 168)(135 169)(136 151)(137 152)(138 153)(139 154)(140 155)(141 156)(142 157)(143 158)(144 159)(145 160)(146 161)(147 162)(148 163)(149 164)(150 165)(181 228)(182 229)(183 230)(184 231)(185 232)(186 233)(187 234)(188 235)(189 236)(190 237)(191 238)(192 239)(193 240)(194 226)(195 227)(196 222)(197 223)(198 224)(199 225)(200 211)(201 212)(202 213)(203 214)(204 215)(205 216)(206 217)(207 218)(208 219)(209 220)(210 221)
(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)
(1 130 21 143)(2 129 22 142)(3 128 23 141)(4 127 24 140)(5 126 25 139)(6 125 26 138)(7 124 27 137)(8 123 28 136)(9 122 29 150)(10 121 30 149)(11 135 16 148)(12 134 17 147)(13 133 18 146)(14 132 19 145)(15 131 20 144)(31 157 57 178)(32 156 58 177)(33 155 59 176)(34 154 60 175)(35 153 46 174)(36 152 47 173)(37 151 48 172)(38 165 49 171)(39 164 50 170)(40 163 51 169)(41 162 52 168)(42 161 53 167)(43 160 54 166)(44 159 55 180)(45 158 56 179)(61 185 80 200)(62 184 81 199)(63 183 82 198)(64 182 83 197)(65 181 84 196)(66 195 85 210)(67 194 86 209)(68 193 87 208)(69 192 88 207)(70 191 89 206)(71 190 90 205)(72 189 76 204)(73 188 77 203)(74 187 78 202)(75 186 79 201)(91 222 111 228)(92 221 112 227)(93 220 113 226)(94 219 114 240)(95 218 115 239)(96 217 116 238)(97 216 117 237)(98 215 118 236)(99 214 119 235)(100 213 120 234)(101 212 106 233)(102 211 107 232)(103 225 108 231)(104 224 109 230)(105 223 110 229)
(1 95)(2 99)(3 103)(4 92)(5 96)(6 100)(7 104)(8 93)(9 97)(10 101)(11 105)(12 94)(13 98)(14 102)(15 91)(16 110)(17 114)(18 118)(19 107)(20 111)(21 115)(22 119)(23 108)(24 112)(25 116)(26 120)(27 109)(28 113)(29 117)(30 106)(31 73)(32 62)(33 66)(34 70)(35 74)(36 63)(37 67)(38 71)(39 75)(40 64)(41 68)(42 72)(43 61)(44 65)(45 69)(46 78)(47 82)(48 86)(49 90)(50 79)(51 83)(52 87)(53 76)(54 80)(55 84)(56 88)(57 77)(58 81)(59 85)(60 89)(121 233)(122 237)(123 226)(124 230)(125 234)(126 238)(127 227)(128 231)(129 235)(130 239)(131 228)(132 232)(133 236)(134 240)(135 229)(136 220)(137 224)(138 213)(139 217)(140 221)(141 225)(142 214)(143 218)(144 222)(145 211)(146 215)(147 219)(148 223)(149 212)(150 216)(151 209)(152 198)(153 202)(154 206)(155 210)(156 199)(157 203)(158 207)(159 196)(160 200)(161 204)(162 208)(163 197)(164 201)(165 205)(166 185)(167 189)(168 193)(169 182)(170 186)(171 190)(172 194)(173 183)(174 187)(175 191)(176 195)(177 184)(178 188)(179 192)(180 181)
G:=sub<Sym(240)| (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,113)(15,114)(16,105)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,89)(32,90)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,74)(47,75)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(121,230)(122,231)(123,232)(124,233)(125,234)(126,235)(127,236)(128,237)(129,238)(130,239)(131,240)(132,226)(133,227)(134,228)(135,229)(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)(151,200)(152,201)(153,202)(154,203)(155,204)(156,205)(157,206)(158,207)(159,208)(160,209)(161,210)(162,196)(163,197)(164,198)(165,199)(166,194)(167,195)(168,181)(169,182)(170,183)(171,184)(172,185)(173,186)(174,187)(175,188)(176,189)(177,190)(178,191)(179,192)(180,193), (1,56)(2,57)(3,58)(4,59)(5,60)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(61,107)(62,108)(63,109)(64,110)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,119)(74,120)(75,106)(76,98)(77,99)(78,100)(79,101)(80,102)(81,103)(82,104)(83,105)(84,91)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,176)(128,177)(129,178)(130,179)(131,180)(132,166)(133,167)(134,168)(135,169)(136,151)(137,152)(138,153)(139,154)(140,155)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,162)(148,163)(149,164)(150,165)(181,228)(182,229)(183,230)(184,231)(185,232)(186,233)(187,234)(188,235)(189,236)(190,237)(191,238)(192,239)(193,240)(194,226)(195,227)(196,222)(197,223)(198,224)(199,225)(200,211)(201,212)(202,213)(203,214)(204,215)(205,216)(206,217)(207,218)(208,219)(209,220)(210,221), (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), (1,130,21,143)(2,129,22,142)(3,128,23,141)(4,127,24,140)(5,126,25,139)(6,125,26,138)(7,124,27,137)(8,123,28,136)(9,122,29,150)(10,121,30,149)(11,135,16,148)(12,134,17,147)(13,133,18,146)(14,132,19,145)(15,131,20,144)(31,157,57,178)(32,156,58,177)(33,155,59,176)(34,154,60,175)(35,153,46,174)(36,152,47,173)(37,151,48,172)(38,165,49,171)(39,164,50,170)(40,163,51,169)(41,162,52,168)(42,161,53,167)(43,160,54,166)(44,159,55,180)(45,158,56,179)(61,185,80,200)(62,184,81,199)(63,183,82,198)(64,182,83,197)(65,181,84,196)(66,195,85,210)(67,194,86,209)(68,193,87,208)(69,192,88,207)(70,191,89,206)(71,190,90,205)(72,189,76,204)(73,188,77,203)(74,187,78,202)(75,186,79,201)(91,222,111,228)(92,221,112,227)(93,220,113,226)(94,219,114,240)(95,218,115,239)(96,217,116,238)(97,216,117,237)(98,215,118,236)(99,214,119,235)(100,213,120,234)(101,212,106,233)(102,211,107,232)(103,225,108,231)(104,224,109,230)(105,223,110,229), (1,95)(2,99)(3,103)(4,92)(5,96)(6,100)(7,104)(8,93)(9,97)(10,101)(11,105)(12,94)(13,98)(14,102)(15,91)(16,110)(17,114)(18,118)(19,107)(20,111)(21,115)(22,119)(23,108)(24,112)(25,116)(26,120)(27,109)(28,113)(29,117)(30,106)(31,73)(32,62)(33,66)(34,70)(35,74)(36,63)(37,67)(38,71)(39,75)(40,64)(41,68)(42,72)(43,61)(44,65)(45,69)(46,78)(47,82)(48,86)(49,90)(50,79)(51,83)(52,87)(53,76)(54,80)(55,84)(56,88)(57,77)(58,81)(59,85)(60,89)(121,233)(122,237)(123,226)(124,230)(125,234)(126,238)(127,227)(128,231)(129,235)(130,239)(131,228)(132,232)(133,236)(134,240)(135,229)(136,220)(137,224)(138,213)(139,217)(140,221)(141,225)(142,214)(143,218)(144,222)(145,211)(146,215)(147,219)(148,223)(149,212)(150,216)(151,209)(152,198)(153,202)(154,206)(155,210)(156,199)(157,203)(158,207)(159,196)(160,200)(161,204)(162,208)(163,197)(164,201)(165,205)(166,185)(167,189)(168,193)(169,182)(170,186)(171,190)(172,194)(173,183)(174,187)(175,191)(176,195)(177,184)(178,188)(179,192)(180,181)>;
G:=Group( (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,113)(15,114)(16,105)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,89)(32,90)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,74)(47,75)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(121,230)(122,231)(123,232)(124,233)(125,234)(126,235)(127,236)(128,237)(129,238)(130,239)(131,240)(132,226)(133,227)(134,228)(135,229)(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)(151,200)(152,201)(153,202)(154,203)(155,204)(156,205)(157,206)(158,207)(159,208)(160,209)(161,210)(162,196)(163,197)(164,198)(165,199)(166,194)(167,195)(168,181)(169,182)(170,183)(171,184)(172,185)(173,186)(174,187)(175,188)(176,189)(177,190)(178,191)(179,192)(180,193), (1,56)(2,57)(3,58)(4,59)(5,60)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(61,107)(62,108)(63,109)(64,110)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,119)(74,120)(75,106)(76,98)(77,99)(78,100)(79,101)(80,102)(81,103)(82,104)(83,105)(84,91)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,176)(128,177)(129,178)(130,179)(131,180)(132,166)(133,167)(134,168)(135,169)(136,151)(137,152)(138,153)(139,154)(140,155)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,162)(148,163)(149,164)(150,165)(181,228)(182,229)(183,230)(184,231)(185,232)(186,233)(187,234)(188,235)(189,236)(190,237)(191,238)(192,239)(193,240)(194,226)(195,227)(196,222)(197,223)(198,224)(199,225)(200,211)(201,212)(202,213)(203,214)(204,215)(205,216)(206,217)(207,218)(208,219)(209,220)(210,221), (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), (1,130,21,143)(2,129,22,142)(3,128,23,141)(4,127,24,140)(5,126,25,139)(6,125,26,138)(7,124,27,137)(8,123,28,136)(9,122,29,150)(10,121,30,149)(11,135,16,148)(12,134,17,147)(13,133,18,146)(14,132,19,145)(15,131,20,144)(31,157,57,178)(32,156,58,177)(33,155,59,176)(34,154,60,175)(35,153,46,174)(36,152,47,173)(37,151,48,172)(38,165,49,171)(39,164,50,170)(40,163,51,169)(41,162,52,168)(42,161,53,167)(43,160,54,166)(44,159,55,180)(45,158,56,179)(61,185,80,200)(62,184,81,199)(63,183,82,198)(64,182,83,197)(65,181,84,196)(66,195,85,210)(67,194,86,209)(68,193,87,208)(69,192,88,207)(70,191,89,206)(71,190,90,205)(72,189,76,204)(73,188,77,203)(74,187,78,202)(75,186,79,201)(91,222,111,228)(92,221,112,227)(93,220,113,226)(94,219,114,240)(95,218,115,239)(96,217,116,238)(97,216,117,237)(98,215,118,236)(99,214,119,235)(100,213,120,234)(101,212,106,233)(102,211,107,232)(103,225,108,231)(104,224,109,230)(105,223,110,229), (1,95)(2,99)(3,103)(4,92)(5,96)(6,100)(7,104)(8,93)(9,97)(10,101)(11,105)(12,94)(13,98)(14,102)(15,91)(16,110)(17,114)(18,118)(19,107)(20,111)(21,115)(22,119)(23,108)(24,112)(25,116)(26,120)(27,109)(28,113)(29,117)(30,106)(31,73)(32,62)(33,66)(34,70)(35,74)(36,63)(37,67)(38,71)(39,75)(40,64)(41,68)(42,72)(43,61)(44,65)(45,69)(46,78)(47,82)(48,86)(49,90)(50,79)(51,83)(52,87)(53,76)(54,80)(55,84)(56,88)(57,77)(58,81)(59,85)(60,89)(121,233)(122,237)(123,226)(124,230)(125,234)(126,238)(127,227)(128,231)(129,235)(130,239)(131,228)(132,232)(133,236)(134,240)(135,229)(136,220)(137,224)(138,213)(139,217)(140,221)(141,225)(142,214)(143,218)(144,222)(145,211)(146,215)(147,219)(148,223)(149,212)(150,216)(151,209)(152,198)(153,202)(154,206)(155,210)(156,199)(157,203)(158,207)(159,196)(160,200)(161,204)(162,208)(163,197)(164,201)(165,205)(166,185)(167,189)(168,193)(169,182)(170,186)(171,190)(172,194)(173,183)(174,187)(175,191)(176,195)(177,184)(178,188)(179,192)(180,181) );
G=PermutationGroup([[(1,115),(2,116),(3,117),(4,118),(5,119),(6,120),(7,106),(8,107),(9,108),(10,109),(11,110),(12,111),(13,112),(14,113),(15,114),(16,105),(17,91),(18,92),(19,93),(20,94),(21,95),(22,96),(23,97),(24,98),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,89),(32,90),(33,76),(34,77),(35,78),(36,79),(37,80),(38,81),(39,82),(40,83),(41,84),(42,85),(43,86),(44,87),(45,88),(46,74),(47,75),(48,61),(49,62),(50,63),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,73),(121,230),(122,231),(123,232),(124,233),(125,234),(126,235),(127,236),(128,237),(129,238),(130,239),(131,240),(132,226),(133,227),(134,228),(135,229),(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),(151,200),(152,201),(153,202),(154,203),(155,204),(156,205),(157,206),(158,207),(159,208),(160,209),(161,210),(162,196),(163,197),(164,198),(165,199),(166,194),(167,195),(168,181),(169,182),(170,183),(171,184),(172,185),(173,186),(174,187),(175,188),(176,189),(177,190),(178,191),(179,192),(180,193)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39),(61,107),(62,108),(63,109),(64,110),(65,111),(66,112),(67,113),(68,114),(69,115),(70,116),(71,117),(72,118),(73,119),(74,120),(75,106),(76,98),(77,99),(78,100),(79,101),(80,102),(81,103),(82,104),(83,105),(84,91),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(121,170),(122,171),(123,172),(124,173),(125,174),(126,175),(127,176),(128,177),(129,178),(130,179),(131,180),(132,166),(133,167),(134,168),(135,169),(136,151),(137,152),(138,153),(139,154),(140,155),(141,156),(142,157),(143,158),(144,159),(145,160),(146,161),(147,162),(148,163),(149,164),(150,165),(181,228),(182,229),(183,230),(184,231),(185,232),(186,233),(187,234),(188,235),(189,236),(190,237),(191,238),(192,239),(193,240),(194,226),(195,227),(196,222),(197,223),(198,224),(199,225),(200,211),(201,212),(202,213),(203,214),(204,215),(205,216),(206,217),(207,218),(208,219),(209,220),(210,221)], [(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)], [(1,130,21,143),(2,129,22,142),(3,128,23,141),(4,127,24,140),(5,126,25,139),(6,125,26,138),(7,124,27,137),(8,123,28,136),(9,122,29,150),(10,121,30,149),(11,135,16,148),(12,134,17,147),(13,133,18,146),(14,132,19,145),(15,131,20,144),(31,157,57,178),(32,156,58,177),(33,155,59,176),(34,154,60,175),(35,153,46,174),(36,152,47,173),(37,151,48,172),(38,165,49,171),(39,164,50,170),(40,163,51,169),(41,162,52,168),(42,161,53,167),(43,160,54,166),(44,159,55,180),(45,158,56,179),(61,185,80,200),(62,184,81,199),(63,183,82,198),(64,182,83,197),(65,181,84,196),(66,195,85,210),(67,194,86,209),(68,193,87,208),(69,192,88,207),(70,191,89,206),(71,190,90,205),(72,189,76,204),(73,188,77,203),(74,187,78,202),(75,186,79,201),(91,222,111,228),(92,221,112,227),(93,220,113,226),(94,219,114,240),(95,218,115,239),(96,217,116,238),(97,216,117,237),(98,215,118,236),(99,214,119,235),(100,213,120,234),(101,212,106,233),(102,211,107,232),(103,225,108,231),(104,224,109,230),(105,223,110,229)], [(1,95),(2,99),(3,103),(4,92),(5,96),(6,100),(7,104),(8,93),(9,97),(10,101),(11,105),(12,94),(13,98),(14,102),(15,91),(16,110),(17,114),(18,118),(19,107),(20,111),(21,115),(22,119),(23,108),(24,112),(25,116),(26,120),(27,109),(28,113),(29,117),(30,106),(31,73),(32,62),(33,66),(34,70),(35,74),(36,63),(37,67),(38,71),(39,75),(40,64),(41,68),(42,72),(43,61),(44,65),(45,69),(46,78),(47,82),(48,86),(49,90),(50,79),(51,83),(52,87),(53,76),(54,80),(55,84),(56,88),(57,77),(58,81),(59,85),(60,89),(121,233),(122,237),(123,226),(124,230),(125,234),(126,238),(127,227),(128,231),(129,235),(130,239),(131,228),(132,232),(133,236),(134,240),(135,229),(136,220),(137,224),(138,213),(139,217),(140,221),(141,225),(142,214),(143,218),(144,222),(145,211),(146,215),(147,219),(148,223),(149,212),(150,216),(151,209),(152,198),(153,202),(154,206),(155,210),(156,199),(157,203),(158,207),(159,196),(160,200),(161,204),(162,208),(163,197),(164,201),(165,205),(166,185),(167,189),(168,193),(169,182),(170,186),(171,190),(172,194),(173,183),(174,187),(175,191),(176,195),(177,184),(178,188),(179,192),(180,181)]])
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 |
Matrix representation of C22×C15⋊D4 ►in 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] >;
C22×C15⋊D4 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