direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Q8○D12, C30.96C24, C60.243C23, C15⋊132- (1+4), (S3×Q8)⋊5C10, C4○D12⋊9C10, (C5×D4).39D6, (C5×Q8).60D6, D4⋊2S3⋊5C10, D4.10(S3×C10), (C2×C20).254D6, Q8.16(S3×C10), (C10×Dic6)⋊30C2, (C2×Dic6)⋊14C10, D12.14(C2×C10), C3⋊2(C5×2- (1+4)), C10.81(S3×C23), C6.13(C23×C10), D6.7(C22×C10), (S3×C10).43C23, (S3×C20).41C22, (C2×C30).261C23, C12.27(C22×C10), (C2×C60).377C22, C20.240(C22×S3), Dic6.14(C2×C10), (D4×C15).49C22, (C5×D12).53C22, (Q8×C15).54C22, (C5×Dic6).56C22, (C5×Dic3).45C23, Dic3.9(C22×C10), (C10×Dic3).153C22, (C5×S3×Q8)⋊12C2, C4○D4⋊6(C5×S3), C4.27(S3×C2×C10), C3⋊D4.(C2×C10), (C5×C4○D4)⋊13S3, (C3×C4○D4)⋊5C10, C22.5(S3×C2×C10), (C5×C4○D12)⋊19C2, (C15×C4○D4)⋊15C2, (C4×S3).6(C2×C10), (C2×C4).23(S3×C10), C2.14(S3×C22×C10), (C5×D4⋊2S3)⋊12C2, (C2×C12).51(C2×C10), (C3×D4).10(C2×C10), (C2×C6).5(C22×C10), (C3×Q8).11(C2×C10), (C5×C3⋊D4).4C22, (C2×C10).24(C22×S3), (C2×Dic3).16(C2×C10), SmallGroup(480,1162)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 532 in 292 conjugacy classes, 170 normal (24 characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C5, S3 [×2], C6, C6 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×7], Q8, Q8 [×9], C10, C10 [×5], Dic3 [×6], C12, C12 [×3], D6 [×2], C2×C6 [×3], C15, C2×Q8 [×5], C4○D4, C4○D4 [×9], C20, C20 [×3], C20 [×6], C2×C10 [×3], C2×C10 [×2], Dic6 [×9], C4×S3 [×6], D12, C2×Dic3 [×6], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C5×S3 [×2], C30, C30 [×3], 2- (1+4), C2×C20 [×3], C2×C20 [×12], C5×D4 [×3], C5×D4 [×7], C5×Q8, C5×Q8 [×9], C2×Dic6 [×3], C4○D12 [×3], D4⋊2S3 [×6], S3×Q8 [×2], C3×C4○D4, C5×Dic3 [×6], C60, C60 [×3], S3×C10 [×2], C2×C30 [×3], Q8×C10 [×5], C5×C4○D4, C5×C4○D4 [×9], Q8○D12, C5×Dic6 [×9], S3×C20 [×6], C5×D12, C10×Dic3 [×6], C5×C3⋊D4 [×6], C2×C60 [×3], D4×C15 [×3], Q8×C15, C5×2- (1+4), C10×Dic6 [×3], C5×C4○D12 [×3], C5×D4⋊2S3 [×6], C5×S3×Q8 [×2], C15×C4○D4, C5×Q8○D12
Quotients:
C1, C2 [×15], C22 [×35], C5, S3, C23 [×15], C10 [×15], D6 [×7], C24, C2×C10 [×35], C22×S3 [×7], C5×S3, 2- (1+4), C22×C10 [×15], S3×C23, S3×C10 [×7], C23×C10, Q8○D12, S3×C2×C10 [×7], C5×2- (1+4), S3×C22×C10, C5×Q8○D12
Generators and relations
G = < a,b,c,d,e | a5=b4=e2=1, c2=d6=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d5 >
►On 240 points
Generators in S240
(1 161 93 153 140)(2 162 94 154 141)(3 163 95 155 142)(4 164 96 156 143)(5 165 85 145 144)(6 166 86 146 133)(7 167 87 147 134)(8 168 88 148 135)(9 157 89 149 136)(10 158 90 150 137)(11 159 91 151 138)(12 160 92 152 139)(13 240 46 225 207)(14 229 47 226 208)(15 230 48 227 209)(16 231 37 228 210)(17 232 38 217 211)(18 233 39 218 212)(19 234 40 219 213)(20 235 41 220 214)(21 236 42 221 215)(22 237 43 222 216)(23 238 44 223 205)(24 239 45 224 206)(25 50 107 123 200)(26 51 108 124 201)(27 52 97 125 202)(28 53 98 126 203)(29 54 99 127 204)(30 55 100 128 193)(31 56 101 129 194)(32 57 102 130 195)(33 58 103 131 196)(34 59 104 132 197)(35 60 105 121 198)(36 49 106 122 199)(61 80 169 114 187)(62 81 170 115 188)(63 82 171 116 189)(64 83 172 117 190)(65 84 173 118 191)(66 73 174 119 192)(67 74 175 120 181)(68 75 176 109 182)(69 76 177 110 183)(70 77 178 111 184)(71 78 179 112 185)(72 79 180 113 186)
(1 103 7 97)(2 104 8 98)(3 105 9 99)(4 106 10 100)(5 107 11 101)(6 108 12 102)(13 78 19 84)(14 79 20 73)(15 80 21 74)(16 81 22 75)(17 82 23 76)(18 83 24 77)(25 151 31 145)(26 152 32 146)(27 153 33 147)(28 154 34 148)(29 155 35 149)(30 156 36 150)(37 115 43 109)(38 116 44 110)(39 117 45 111)(40 118 46 112)(41 119 47 113)(42 120 48 114)(49 137 55 143)(50 138 56 144)(51 139 57 133)(52 140 58 134)(53 141 59 135)(54 142 60 136)(61 215 67 209)(62 216 68 210)(63 205 69 211)(64 206 70 212)(65 207 71 213)(66 208 72 214)(85 200 91 194)(86 201 92 195)(87 202 93 196)(88 203 94 197)(89 204 95 198)(90 193 96 199)(121 157 127 163)(122 158 128 164)(123 159 129 165)(124 160 130 166)(125 161 131 167)(126 162 132 168)(169 236 175 230)(170 237 176 231)(171 238 177 232)(172 239 178 233)(173 240 179 234)(174 229 180 235)(181 227 187 221)(182 228 188 222)(183 217 189 223)(184 218 190 224)(185 219 191 225)(186 220 192 226)
(1 38 7 44)(2 39 8 45)(3 40 9 46)(4 41 10 47)(5 42 11 48)(6 43 12 37)(13 155 19 149)(14 156 20 150)(15 145 21 151)(16 146 22 152)(17 147 23 153)(18 148 24 154)(25 80 31 74)(26 81 32 75)(27 82 33 76)(28 83 34 77)(29 84 35 78)(30 73 36 79)(49 180 55 174)(50 169 56 175)(51 170 57 176)(52 171 58 177)(53 172 59 178)(54 173 60 179)(61 194 67 200)(62 195 68 201)(63 196 69 202)(64 197 70 203)(65 198 71 204)(66 199 72 193)(85 215 91 209)(86 216 92 210)(87 205 93 211)(88 206 94 212)(89 207 95 213)(90 208 96 214)(97 116 103 110)(98 117 104 111)(99 118 105 112)(100 119 106 113)(101 120 107 114)(102 109 108 115)(121 185 127 191)(122 186 128 192)(123 187 129 181)(124 188 130 182)(125 189 131 183)(126 190 132 184)(133 237 139 231)(134 238 140 232)(135 239 141 233)(136 240 142 234)(137 229 143 235)(138 230 144 236)(157 225 163 219)(158 226 164 220)(159 227 165 221)(160 228 166 222)(161 217 167 223)(162 218 168 224)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 23)(14 22)(15 21)(16 20)(17 19)(25 31)(26 30)(27 29)(32 36)(33 35)(37 41)(38 40)(42 48)(43 47)(44 46)(49 57)(50 56)(51 55)(52 54)(58 60)(61 67)(62 66)(63 65)(68 72)(69 71)(73 81)(74 80)(75 79)(76 78)(82 84)(85 91)(86 90)(87 89)(92 96)(93 95)(97 99)(100 108)(101 107)(102 106)(103 105)(109 113)(110 112)(114 120)(115 119)(116 118)(121 131)(122 130)(123 129)(124 128)(125 127)(133 137)(134 136)(138 144)(139 143)(140 142)(145 151)(146 150)(147 149)(152 156)(153 155)(157 167)(158 166)(159 165)(160 164)(161 163)(169 175)(170 174)(171 173)(176 180)(177 179)(181 187)(182 186)(183 185)(188 192)(189 191)(193 201)(194 200)(195 199)(196 198)(202 204)(205 207)(208 216)(209 215)(210 214)(211 213)(217 219)(220 228)(221 227)(222 226)(223 225)(229 237)(230 236)(231 235)(232 234)(238 240)
G:=sub<Sym(240)| (1,161,93,153,140)(2,162,94,154,141)(3,163,95,155,142)(4,164,96,156,143)(5,165,85,145,144)(6,166,86,146,133)(7,167,87,147,134)(8,168,88,148,135)(9,157,89,149,136)(10,158,90,150,137)(11,159,91,151,138)(12,160,92,152,139)(13,240,46,225,207)(14,229,47,226,208)(15,230,48,227,209)(16,231,37,228,210)(17,232,38,217,211)(18,233,39,218,212)(19,234,40,219,213)(20,235,41,220,214)(21,236,42,221,215)(22,237,43,222,216)(23,238,44,223,205)(24,239,45,224,206)(25,50,107,123,200)(26,51,108,124,201)(27,52,97,125,202)(28,53,98,126,203)(29,54,99,127,204)(30,55,100,128,193)(31,56,101,129,194)(32,57,102,130,195)(33,58,103,131,196)(34,59,104,132,197)(35,60,105,121,198)(36,49,106,122,199)(61,80,169,114,187)(62,81,170,115,188)(63,82,171,116,189)(64,83,172,117,190)(65,84,173,118,191)(66,73,174,119,192)(67,74,175,120,181)(68,75,176,109,182)(69,76,177,110,183)(70,77,178,111,184)(71,78,179,112,185)(72,79,180,113,186), (1,103,7,97)(2,104,8,98)(3,105,9,99)(4,106,10,100)(5,107,11,101)(6,108,12,102)(13,78,19,84)(14,79,20,73)(15,80,21,74)(16,81,22,75)(17,82,23,76)(18,83,24,77)(25,151,31,145)(26,152,32,146)(27,153,33,147)(28,154,34,148)(29,155,35,149)(30,156,36,150)(37,115,43,109)(38,116,44,110)(39,117,45,111)(40,118,46,112)(41,119,47,113)(42,120,48,114)(49,137,55,143)(50,138,56,144)(51,139,57,133)(52,140,58,134)(53,141,59,135)(54,142,60,136)(61,215,67,209)(62,216,68,210)(63,205,69,211)(64,206,70,212)(65,207,71,213)(66,208,72,214)(85,200,91,194)(86,201,92,195)(87,202,93,196)(88,203,94,197)(89,204,95,198)(90,193,96,199)(121,157,127,163)(122,158,128,164)(123,159,129,165)(124,160,130,166)(125,161,131,167)(126,162,132,168)(169,236,175,230)(170,237,176,231)(171,238,177,232)(172,239,178,233)(173,240,179,234)(174,229,180,235)(181,227,187,221)(182,228,188,222)(183,217,189,223)(184,218,190,224)(185,219,191,225)(186,220,192,226), (1,38,7,44)(2,39,8,45)(3,40,9,46)(4,41,10,47)(5,42,11,48)(6,43,12,37)(13,155,19,149)(14,156,20,150)(15,145,21,151)(16,146,22,152)(17,147,23,153)(18,148,24,154)(25,80,31,74)(26,81,32,75)(27,82,33,76)(28,83,34,77)(29,84,35,78)(30,73,36,79)(49,180,55,174)(50,169,56,175)(51,170,57,176)(52,171,58,177)(53,172,59,178)(54,173,60,179)(61,194,67,200)(62,195,68,201)(63,196,69,202)(64,197,70,203)(65,198,71,204)(66,199,72,193)(85,215,91,209)(86,216,92,210)(87,205,93,211)(88,206,94,212)(89,207,95,213)(90,208,96,214)(97,116,103,110)(98,117,104,111)(99,118,105,112)(100,119,106,113)(101,120,107,114)(102,109,108,115)(121,185,127,191)(122,186,128,192)(123,187,129,181)(124,188,130,182)(125,189,131,183)(126,190,132,184)(133,237,139,231)(134,238,140,232)(135,239,141,233)(136,240,142,234)(137,229,143,235)(138,230,144,236)(157,225,163,219)(158,226,164,220)(159,227,165,221)(160,228,166,222)(161,217,167,223)(162,218,168,224), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,31)(26,30)(27,29)(32,36)(33,35)(37,41)(38,40)(42,48)(43,47)(44,46)(49,57)(50,56)(51,55)(52,54)(58,60)(61,67)(62,66)(63,65)(68,72)(69,71)(73,81)(74,80)(75,79)(76,78)(82,84)(85,91)(86,90)(87,89)(92,96)(93,95)(97,99)(100,108)(101,107)(102,106)(103,105)(109,113)(110,112)(114,120)(115,119)(116,118)(121,131)(122,130)(123,129)(124,128)(125,127)(133,137)(134,136)(138,144)(139,143)(140,142)(145,151)(146,150)(147,149)(152,156)(153,155)(157,167)(158,166)(159,165)(160,164)(161,163)(169,175)(170,174)(171,173)(176,180)(177,179)(181,187)(182,186)(183,185)(188,192)(189,191)(193,201)(194,200)(195,199)(196,198)(202,204)(205,207)(208,216)(209,215)(210,214)(211,213)(217,219)(220,228)(221,227)(222,226)(223,225)(229,237)(230,236)(231,235)(232,234)(238,240)>;
G:=Group( (1,161,93,153,140)(2,162,94,154,141)(3,163,95,155,142)(4,164,96,156,143)(5,165,85,145,144)(6,166,86,146,133)(7,167,87,147,134)(8,168,88,148,135)(9,157,89,149,136)(10,158,90,150,137)(11,159,91,151,138)(12,160,92,152,139)(13,240,46,225,207)(14,229,47,226,208)(15,230,48,227,209)(16,231,37,228,210)(17,232,38,217,211)(18,233,39,218,212)(19,234,40,219,213)(20,235,41,220,214)(21,236,42,221,215)(22,237,43,222,216)(23,238,44,223,205)(24,239,45,224,206)(25,50,107,123,200)(26,51,108,124,201)(27,52,97,125,202)(28,53,98,126,203)(29,54,99,127,204)(30,55,100,128,193)(31,56,101,129,194)(32,57,102,130,195)(33,58,103,131,196)(34,59,104,132,197)(35,60,105,121,198)(36,49,106,122,199)(61,80,169,114,187)(62,81,170,115,188)(63,82,171,116,189)(64,83,172,117,190)(65,84,173,118,191)(66,73,174,119,192)(67,74,175,120,181)(68,75,176,109,182)(69,76,177,110,183)(70,77,178,111,184)(71,78,179,112,185)(72,79,180,113,186), (1,103,7,97)(2,104,8,98)(3,105,9,99)(4,106,10,100)(5,107,11,101)(6,108,12,102)(13,78,19,84)(14,79,20,73)(15,80,21,74)(16,81,22,75)(17,82,23,76)(18,83,24,77)(25,151,31,145)(26,152,32,146)(27,153,33,147)(28,154,34,148)(29,155,35,149)(30,156,36,150)(37,115,43,109)(38,116,44,110)(39,117,45,111)(40,118,46,112)(41,119,47,113)(42,120,48,114)(49,137,55,143)(50,138,56,144)(51,139,57,133)(52,140,58,134)(53,141,59,135)(54,142,60,136)(61,215,67,209)(62,216,68,210)(63,205,69,211)(64,206,70,212)(65,207,71,213)(66,208,72,214)(85,200,91,194)(86,201,92,195)(87,202,93,196)(88,203,94,197)(89,204,95,198)(90,193,96,199)(121,157,127,163)(122,158,128,164)(123,159,129,165)(124,160,130,166)(125,161,131,167)(126,162,132,168)(169,236,175,230)(170,237,176,231)(171,238,177,232)(172,239,178,233)(173,240,179,234)(174,229,180,235)(181,227,187,221)(182,228,188,222)(183,217,189,223)(184,218,190,224)(185,219,191,225)(186,220,192,226), (1,38,7,44)(2,39,8,45)(3,40,9,46)(4,41,10,47)(5,42,11,48)(6,43,12,37)(13,155,19,149)(14,156,20,150)(15,145,21,151)(16,146,22,152)(17,147,23,153)(18,148,24,154)(25,80,31,74)(26,81,32,75)(27,82,33,76)(28,83,34,77)(29,84,35,78)(30,73,36,79)(49,180,55,174)(50,169,56,175)(51,170,57,176)(52,171,58,177)(53,172,59,178)(54,173,60,179)(61,194,67,200)(62,195,68,201)(63,196,69,202)(64,197,70,203)(65,198,71,204)(66,199,72,193)(85,215,91,209)(86,216,92,210)(87,205,93,211)(88,206,94,212)(89,207,95,213)(90,208,96,214)(97,116,103,110)(98,117,104,111)(99,118,105,112)(100,119,106,113)(101,120,107,114)(102,109,108,115)(121,185,127,191)(122,186,128,192)(123,187,129,181)(124,188,130,182)(125,189,131,183)(126,190,132,184)(133,237,139,231)(134,238,140,232)(135,239,141,233)(136,240,142,234)(137,229,143,235)(138,230,144,236)(157,225,163,219)(158,226,164,220)(159,227,165,221)(160,228,166,222)(161,217,167,223)(162,218,168,224), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,31)(26,30)(27,29)(32,36)(33,35)(37,41)(38,40)(42,48)(43,47)(44,46)(49,57)(50,56)(51,55)(52,54)(58,60)(61,67)(62,66)(63,65)(68,72)(69,71)(73,81)(74,80)(75,79)(76,78)(82,84)(85,91)(86,90)(87,89)(92,96)(93,95)(97,99)(100,108)(101,107)(102,106)(103,105)(109,113)(110,112)(114,120)(115,119)(116,118)(121,131)(122,130)(123,129)(124,128)(125,127)(133,137)(134,136)(138,144)(139,143)(140,142)(145,151)(146,150)(147,149)(152,156)(153,155)(157,167)(158,166)(159,165)(160,164)(161,163)(169,175)(170,174)(171,173)(176,180)(177,179)(181,187)(182,186)(183,185)(188,192)(189,191)(193,201)(194,200)(195,199)(196,198)(202,204)(205,207)(208,216)(209,215)(210,214)(211,213)(217,219)(220,228)(221,227)(222,226)(223,225)(229,237)(230,236)(231,235)(232,234)(238,240) );
G=PermutationGroup([(1,161,93,153,140),(2,162,94,154,141),(3,163,95,155,142),(4,164,96,156,143),(5,165,85,145,144),(6,166,86,146,133),(7,167,87,147,134),(8,168,88,148,135),(9,157,89,149,136),(10,158,90,150,137),(11,159,91,151,138),(12,160,92,152,139),(13,240,46,225,207),(14,229,47,226,208),(15,230,48,227,209),(16,231,37,228,210),(17,232,38,217,211),(18,233,39,218,212),(19,234,40,219,213),(20,235,41,220,214),(21,236,42,221,215),(22,237,43,222,216),(23,238,44,223,205),(24,239,45,224,206),(25,50,107,123,200),(26,51,108,124,201),(27,52,97,125,202),(28,53,98,126,203),(29,54,99,127,204),(30,55,100,128,193),(31,56,101,129,194),(32,57,102,130,195),(33,58,103,131,196),(34,59,104,132,197),(35,60,105,121,198),(36,49,106,122,199),(61,80,169,114,187),(62,81,170,115,188),(63,82,171,116,189),(64,83,172,117,190),(65,84,173,118,191),(66,73,174,119,192),(67,74,175,120,181),(68,75,176,109,182),(69,76,177,110,183),(70,77,178,111,184),(71,78,179,112,185),(72,79,180,113,186)], [(1,103,7,97),(2,104,8,98),(3,105,9,99),(4,106,10,100),(5,107,11,101),(6,108,12,102),(13,78,19,84),(14,79,20,73),(15,80,21,74),(16,81,22,75),(17,82,23,76),(18,83,24,77),(25,151,31,145),(26,152,32,146),(27,153,33,147),(28,154,34,148),(29,155,35,149),(30,156,36,150),(37,115,43,109),(38,116,44,110),(39,117,45,111),(40,118,46,112),(41,119,47,113),(42,120,48,114),(49,137,55,143),(50,138,56,144),(51,139,57,133),(52,140,58,134),(53,141,59,135),(54,142,60,136),(61,215,67,209),(62,216,68,210),(63,205,69,211),(64,206,70,212),(65,207,71,213),(66,208,72,214),(85,200,91,194),(86,201,92,195),(87,202,93,196),(88,203,94,197),(89,204,95,198),(90,193,96,199),(121,157,127,163),(122,158,128,164),(123,159,129,165),(124,160,130,166),(125,161,131,167),(126,162,132,168),(169,236,175,230),(170,237,176,231),(171,238,177,232),(172,239,178,233),(173,240,179,234),(174,229,180,235),(181,227,187,221),(182,228,188,222),(183,217,189,223),(184,218,190,224),(185,219,191,225),(186,220,192,226)], [(1,38,7,44),(2,39,8,45),(3,40,9,46),(4,41,10,47),(5,42,11,48),(6,43,12,37),(13,155,19,149),(14,156,20,150),(15,145,21,151),(16,146,22,152),(17,147,23,153),(18,148,24,154),(25,80,31,74),(26,81,32,75),(27,82,33,76),(28,83,34,77),(29,84,35,78),(30,73,36,79),(49,180,55,174),(50,169,56,175),(51,170,57,176),(52,171,58,177),(53,172,59,178),(54,173,60,179),(61,194,67,200),(62,195,68,201),(63,196,69,202),(64,197,70,203),(65,198,71,204),(66,199,72,193),(85,215,91,209),(86,216,92,210),(87,205,93,211),(88,206,94,212),(89,207,95,213),(90,208,96,214),(97,116,103,110),(98,117,104,111),(99,118,105,112),(100,119,106,113),(101,120,107,114),(102,109,108,115),(121,185,127,191),(122,186,128,192),(123,187,129,181),(124,188,130,182),(125,189,131,183),(126,190,132,184),(133,237,139,231),(134,238,140,232),(135,239,141,233),(136,240,142,234),(137,229,143,235),(138,230,144,236),(157,225,163,219),(158,226,164,220),(159,227,165,221),(160,228,166,222),(161,217,167,223),(162,218,168,224)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,23),(14,22),(15,21),(16,20),(17,19),(25,31),(26,30),(27,29),(32,36),(33,35),(37,41),(38,40),(42,48),(43,47),(44,46),(49,57),(50,56),(51,55),(52,54),(58,60),(61,67),(62,66),(63,65),(68,72),(69,71),(73,81),(74,80),(75,79),(76,78),(82,84),(85,91),(86,90),(87,89),(92,96),(93,95),(97,99),(100,108),(101,107),(102,106),(103,105),(109,113),(110,112),(114,120),(115,119),(116,118),(121,131),(122,130),(123,129),(124,128),(125,127),(133,137),(134,136),(138,144),(139,143),(140,142),(145,151),(146,150),(147,149),(152,156),(153,155),(157,167),(158,166),(159,165),(160,164),(161,163),(169,175),(170,174),(171,173),(176,180),(177,179),(181,187),(182,186),(183,185),(188,192),(189,191),(193,201),(194,200),(195,199),(196,198),(202,204),(205,207),(208,216),(209,215),(210,214),(211,213),(217,219),(220,228),(221,227),(222,226),(223,225),(229,237),(230,236),(231,235),(232,234),(238,240)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 58 | 0 | 0 | 0 |
| 0 | 58 | 0 | 0 |
| 0 | 0 | 58 | 0 |
| 0 | 0 | 0 | 58 |
| 34 | 0 | 35 | 0 |
| 0 | 34 | 0 | 35 |
| 14 | 0 | 27 | 0 |
| 0 | 14 | 0 | 27 |
| 1 | 0 | 28 | 0 |
| 0 | 1 | 0 | 28 |
| 13 | 0 | 60 | 0 |
| 0 | 13 | 0 | 60 |
| 46 | 23 | 0 | 0 |
| 38 | 23 | 0 | 0 |
| 0 | 0 | 46 | 23 |
| 0 | 0 | 38 | 23 |
| 0 | 60 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 60 | 0 |
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,58,0,0,0,0,58],[34,0,14,0,0,34,0,14,35,0,27,0,0,35,0,27],[1,0,13,0,0,1,0,13,28,0,60,0,0,28,0,60],[46,38,0,0,23,23,0,0,0,0,46,38,0,0,23,23],[0,60,0,0,60,0,0,0,0,0,0,60,0,0,60,0] >;
135 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 10Q | ··· | 10X | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20AN | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D6 | D6 | D6 | C5×S3 | S3×C10 | S3×C10 | S3×C10 | 2- (1+4) | Q8○D12 | C5×2- (1+4) | C5×Q8○D12 |
| kernel | C5×Q8○D12 | C10×Dic6 | C5×C4○D12 | C5×D4⋊2S3 | C5×S3×Q8 | C15×C4○D4 | Q8○D12 | C2×Dic6 | C4○D12 | D4⋊2S3 | S3×Q8 | C3×C4○D4 | C5×C4○D4 | C2×C20 | C5×D4 | C5×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C15 | C5 | C3 | C1 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 4 | 12 | 12 | 24 | 8 | 4 | 1 | 3 | 3 | 1 | 4 | 12 | 12 | 4 | 1 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_5\times Q_8\circ D_{12} % in TeX
G:=Group("C5xQ8oD12"); // GroupNames label
G:=SmallGroup(480,1162);
// by ID
G=gap.SmallGroup(480,1162);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,568,891,436,2467,304,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=e^2=1,c^2=d^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^5>;
// generators/relations
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