direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×Q8⋊2D5, C30.77C24, C60.212C23, Q8⋊6(C6×D5), D20⋊9(C2×C6), (C6×Q8)⋊13D5, (C2×D20)⋊12C6, (C6×D20)⋊28C2, (Q8×C30)⋊13C2, (Q8×C10)⋊10C6, (C3×Q8)⋊24D10, C30⋊18(C4○D4), C10.9(C23×C6), C6.77(C23×D5), (C2×C12).372D10, (D5×C12)⋊24C22, (C3×D20)⋊39C22, C20.23(C22×C6), (Q8×C15)⋊27C22, D10.4(C22×C6), (C6×D5).55C23, (C2×C60).307C22, (C2×C30).385C23, C12.212(C22×D5), (C3×Dic5).76C23, Dic5.16(C22×C6), (C6×Dic5).285C22, (C2×C4×D5)⋊5C6, C5⋊3(C6×C4○D4), C4.23(D5×C2×C6), (D5×C2×C12)⋊15C2, (C4×D5)⋊5(C2×C6), C10⋊3(C3×C4○D4), C15⋊27(C2×C4○D4), (C5×Q8)⋊8(C2×C6), (C2×Q8)⋊8(C3×D5), (C2×C4).62(C6×D5), C2.10(D5×C22×C6), C22.32(D5×C2×C6), (C2×C20).44(C2×C6), (D5×C2×C6).140C22, (C2×C10).67(C22×C6), (C2×Dic5).65(C2×C6), (C22×D5).35(C2×C6), (C2×C6).379(C22×D5), SmallGroup(480,1143)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 976 in 328 conjugacy classes, 178 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×6], C4 [×2], C22, C22 [×12], C5, C6, C6 [×2], C6 [×6], C2×C4 [×3], C2×C4 [×13], D4 [×12], Q8 [×4], C23 [×3], D5 [×6], C10, C10 [×2], C12 [×6], C12 [×2], C2×C6, C2×C6 [×12], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], C20 [×6], D10 [×6], D10 [×6], C2×C10, C2×C12 [×3], C2×C12 [×13], C3×D4 [×12], C3×Q8 [×4], C22×C6 [×3], C3×D5 [×6], C30, C30 [×2], C2×C4○D4, C4×D5 [×12], D20 [×12], C2×Dic5, C2×C20 [×3], C5×Q8 [×4], C22×D5 [×3], C22×C12 [×3], C6×D4 [×3], C6×Q8, C3×C4○D4 [×8], C3×Dic5 [×2], C60 [×6], C6×D5 [×6], C6×D5 [×6], C2×C30, C2×C4×D5 [×3], C2×D20 [×3], Q8⋊2D5 [×8], Q8×C10, C6×C4○D4, D5×C12 [×12], C3×D20 [×12], C6×Dic5, C2×C60 [×3], Q8×C15 [×4], D5×C2×C6 [×3], C2×Q8⋊2D5, D5×C2×C12 [×3], C6×D20 [×3], C3×Q8⋊2D5 [×8], Q8×C30, C6×Q8⋊2D5
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], D5, C2×C6 [×35], C4○D4 [×2], C24, D10 [×7], C22×C6 [×15], C3×D5, C2×C4○D4, C22×D5 [×7], C3×C4○D4 [×2], C23×C6, C6×D5 [×7], Q8⋊2D5 [×2], C23×D5, C6×C4○D4, D5×C2×C6 [×7], C2×Q8⋊2D5, C3×Q8⋊2D5 [×2], D5×C22×C6, C6×Q8⋊2D5
Generators and relations
G = < a,b,c,d,e | a6=b4=d5=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >
►On 240 points
Generators in S240
(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)
(1 107 187 51)(2 108 188 52)(3 103 189 53)(4 104 190 54)(5 105 191 49)(6 106 192 50)(7 78 150 116)(8 73 145 117)(9 74 146 118)(10 75 147 119)(11 76 148 120)(12 77 149 115)(13 225 201 153)(14 226 202 154)(15 227 203 155)(16 228 204 156)(17 223 199 151)(18 224 200 152)(19 170 194 98)(20 171 195 99)(21 172 196 100)(22 173 197 101)(23 174 198 102)(24 169 193 97)(25 113 207 35)(26 114 208 36)(27 109 209 31)(28 110 210 32)(29 111 205 33)(30 112 206 34)(37 61 239 130)(38 62 240 131)(39 63 235 132)(40 64 236 127)(41 65 237 128)(42 66 238 129)(43 211 87 159)(44 212 88 160)(45 213 89 161)(46 214 90 162)(47 215 85 157)(48 216 86 158)(55 185 233 124)(56 186 234 125)(57 181 229 126)(58 182 230 121)(59 183 231 122)(60 184 232 123)(67 163 133 91)(68 164 134 92)(69 165 135 93)(70 166 136 94)(71 167 137 95)(72 168 138 96)(79 221 142 179)(80 222 143 180)(81 217 144 175)(82 218 139 176)(83 219 140 177)(84 220 141 178)
(1 217 187 175)(2 218 188 176)(3 219 189 177)(4 220 190 178)(5 221 191 179)(6 222 192 180)(7 216 150 158)(8 211 145 159)(9 212 146 160)(10 213 147 161)(11 214 148 162)(12 215 149 157)(13 231 201 59)(14 232 202 60)(15 233 203 55)(16 234 204 56)(17 229 199 57)(18 230 200 58)(19 92 194 164)(20 93 195 165)(21 94 196 166)(22 95 197 167)(23 96 198 168)(24 91 193 163)(25 237 207 41)(26 238 208 42)(27 239 209 37)(28 240 210 38)(29 235 205 39)(30 236 206 40)(31 130 109 61)(32 131 110 62)(33 132 111 63)(34 127 112 64)(35 128 113 65)(36 129 114 66)(43 117 87 73)(44 118 88 74)(45 119 89 75)(46 120 90 76)(47 115 85 77)(48 116 86 78)(49 142 105 79)(50 143 106 80)(51 144 107 81)(52 139 108 82)(53 140 103 83)(54 141 104 84)(67 169 133 97)(68 170 134 98)(69 171 135 99)(70 172 136 100)(71 173 137 101)(72 174 138 102)(121 224 182 152)(122 225 183 153)(123 226 184 154)(124 227 185 155)(125 228 186 156)(126 223 181 151)
(1 89 17 71 29)(2 90 18 72 30)(3 85 13 67 25)(4 86 14 68 26)(5 87 15 69 27)(6 88 16 70 28)(7 184 19 66 84)(8 185 20 61 79)(9 186 21 62 80)(10 181 22 63 81)(11 182 23 64 82)(12 183 24 65 83)(31 49 211 155 93)(32 50 212 156 94)(33 51 213 151 95)(34 52 214 152 96)(35 53 215 153 91)(36 54 216 154 92)(37 179 117 55 99)(38 180 118 56 100)(39 175 119 57 101)(40 176 120 58 102)(41 177 115 59 97)(42 178 116 60 98)(43 203 135 209 191)(44 204 136 210 192)(45 199 137 205 187)(46 200 138 206 188)(47 201 133 207 189)(48 202 134 208 190)(73 233 171 239 221)(74 234 172 240 222)(75 229 173 235 217)(76 230 174 236 218)(77 231 169 237 219)(78 232 170 238 220)(103 157 225 163 113)(104 158 226 164 114)(105 159 227 165 109)(106 160 228 166 110)(107 161 223 167 111)(108 162 224 168 112)(121 198 127 139 148)(122 193 128 140 149)(123 194 129 141 150)(124 195 130 142 145)(125 196 131 143 146)(126 197 132 144 147)
(1 63)(2 64)(3 65)(4 66)(5 61)(6 62)(7 68)(8 69)(9 70)(10 71)(11 72)(12 67)(13 183)(14 184)(15 185)(16 186)(17 181)(18 182)(19 86)(20 87)(21 88)(22 89)(23 90)(24 85)(25 83)(26 84)(27 79)(28 80)(29 81)(30 82)(31 221)(32 222)(33 217)(34 218)(35 219)(36 220)(37 105)(38 106)(39 107)(40 108)(41 103)(42 104)(43 195)(44 196)(45 197)(46 198)(47 193)(48 194)(49 239)(50 240)(51 235)(52 236)(53 237)(54 238)(55 227)(56 228)(57 223)(58 224)(59 225)(60 226)(73 93)(74 94)(75 95)(76 96)(77 91)(78 92)(97 157)(98 158)(99 159)(100 160)(101 161)(102 162)(109 179)(110 180)(111 175)(112 176)(113 177)(114 178)(115 163)(116 164)(117 165)(118 166)(119 167)(120 168)(121 200)(122 201)(123 202)(124 203)(125 204)(126 199)(127 188)(128 189)(129 190)(130 191)(131 192)(132 187)(133 149)(134 150)(135 145)(136 146)(137 147)(138 148)(139 206)(140 207)(141 208)(142 209)(143 210)(144 205)(151 229)(152 230)(153 231)(154 232)(155 233)(156 234)(169 215)(170 216)(171 211)(172 212)(173 213)(174 214)
G:=sub<Sym(240)| (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), (1,107,187,51)(2,108,188,52)(3,103,189,53)(4,104,190,54)(5,105,191,49)(6,106,192,50)(7,78,150,116)(8,73,145,117)(9,74,146,118)(10,75,147,119)(11,76,148,120)(12,77,149,115)(13,225,201,153)(14,226,202,154)(15,227,203,155)(16,228,204,156)(17,223,199,151)(18,224,200,152)(19,170,194,98)(20,171,195,99)(21,172,196,100)(22,173,197,101)(23,174,198,102)(24,169,193,97)(25,113,207,35)(26,114,208,36)(27,109,209,31)(28,110,210,32)(29,111,205,33)(30,112,206,34)(37,61,239,130)(38,62,240,131)(39,63,235,132)(40,64,236,127)(41,65,237,128)(42,66,238,129)(43,211,87,159)(44,212,88,160)(45,213,89,161)(46,214,90,162)(47,215,85,157)(48,216,86,158)(55,185,233,124)(56,186,234,125)(57,181,229,126)(58,182,230,121)(59,183,231,122)(60,184,232,123)(67,163,133,91)(68,164,134,92)(69,165,135,93)(70,166,136,94)(71,167,137,95)(72,168,138,96)(79,221,142,179)(80,222,143,180)(81,217,144,175)(82,218,139,176)(83,219,140,177)(84,220,141,178), (1,217,187,175)(2,218,188,176)(3,219,189,177)(4,220,190,178)(5,221,191,179)(6,222,192,180)(7,216,150,158)(8,211,145,159)(9,212,146,160)(10,213,147,161)(11,214,148,162)(12,215,149,157)(13,231,201,59)(14,232,202,60)(15,233,203,55)(16,234,204,56)(17,229,199,57)(18,230,200,58)(19,92,194,164)(20,93,195,165)(21,94,196,166)(22,95,197,167)(23,96,198,168)(24,91,193,163)(25,237,207,41)(26,238,208,42)(27,239,209,37)(28,240,210,38)(29,235,205,39)(30,236,206,40)(31,130,109,61)(32,131,110,62)(33,132,111,63)(34,127,112,64)(35,128,113,65)(36,129,114,66)(43,117,87,73)(44,118,88,74)(45,119,89,75)(46,120,90,76)(47,115,85,77)(48,116,86,78)(49,142,105,79)(50,143,106,80)(51,144,107,81)(52,139,108,82)(53,140,103,83)(54,141,104,84)(67,169,133,97)(68,170,134,98)(69,171,135,99)(70,172,136,100)(71,173,137,101)(72,174,138,102)(121,224,182,152)(122,225,183,153)(123,226,184,154)(124,227,185,155)(125,228,186,156)(126,223,181,151), (1,89,17,71,29)(2,90,18,72,30)(3,85,13,67,25)(4,86,14,68,26)(5,87,15,69,27)(6,88,16,70,28)(7,184,19,66,84)(8,185,20,61,79)(9,186,21,62,80)(10,181,22,63,81)(11,182,23,64,82)(12,183,24,65,83)(31,49,211,155,93)(32,50,212,156,94)(33,51,213,151,95)(34,52,214,152,96)(35,53,215,153,91)(36,54,216,154,92)(37,179,117,55,99)(38,180,118,56,100)(39,175,119,57,101)(40,176,120,58,102)(41,177,115,59,97)(42,178,116,60,98)(43,203,135,209,191)(44,204,136,210,192)(45,199,137,205,187)(46,200,138,206,188)(47,201,133,207,189)(48,202,134,208,190)(73,233,171,239,221)(74,234,172,240,222)(75,229,173,235,217)(76,230,174,236,218)(77,231,169,237,219)(78,232,170,238,220)(103,157,225,163,113)(104,158,226,164,114)(105,159,227,165,109)(106,160,228,166,110)(107,161,223,167,111)(108,162,224,168,112)(121,198,127,139,148)(122,193,128,140,149)(123,194,129,141,150)(124,195,130,142,145)(125,196,131,143,146)(126,197,132,144,147), (1,63)(2,64)(3,65)(4,66)(5,61)(6,62)(7,68)(8,69)(9,70)(10,71)(11,72)(12,67)(13,183)(14,184)(15,185)(16,186)(17,181)(18,182)(19,86)(20,87)(21,88)(22,89)(23,90)(24,85)(25,83)(26,84)(27,79)(28,80)(29,81)(30,82)(31,221)(32,222)(33,217)(34,218)(35,219)(36,220)(37,105)(38,106)(39,107)(40,108)(41,103)(42,104)(43,195)(44,196)(45,197)(46,198)(47,193)(48,194)(49,239)(50,240)(51,235)(52,236)(53,237)(54,238)(55,227)(56,228)(57,223)(58,224)(59,225)(60,226)(73,93)(74,94)(75,95)(76,96)(77,91)(78,92)(97,157)(98,158)(99,159)(100,160)(101,161)(102,162)(109,179)(110,180)(111,175)(112,176)(113,177)(114,178)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,200)(122,201)(123,202)(124,203)(125,204)(126,199)(127,188)(128,189)(129,190)(130,191)(131,192)(132,187)(133,149)(134,150)(135,145)(136,146)(137,147)(138,148)(139,206)(140,207)(141,208)(142,209)(143,210)(144,205)(151,229)(152,230)(153,231)(154,232)(155,233)(156,234)(169,215)(170,216)(171,211)(172,212)(173,213)(174,214)>;
G:=Group( (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), (1,107,187,51)(2,108,188,52)(3,103,189,53)(4,104,190,54)(5,105,191,49)(6,106,192,50)(7,78,150,116)(8,73,145,117)(9,74,146,118)(10,75,147,119)(11,76,148,120)(12,77,149,115)(13,225,201,153)(14,226,202,154)(15,227,203,155)(16,228,204,156)(17,223,199,151)(18,224,200,152)(19,170,194,98)(20,171,195,99)(21,172,196,100)(22,173,197,101)(23,174,198,102)(24,169,193,97)(25,113,207,35)(26,114,208,36)(27,109,209,31)(28,110,210,32)(29,111,205,33)(30,112,206,34)(37,61,239,130)(38,62,240,131)(39,63,235,132)(40,64,236,127)(41,65,237,128)(42,66,238,129)(43,211,87,159)(44,212,88,160)(45,213,89,161)(46,214,90,162)(47,215,85,157)(48,216,86,158)(55,185,233,124)(56,186,234,125)(57,181,229,126)(58,182,230,121)(59,183,231,122)(60,184,232,123)(67,163,133,91)(68,164,134,92)(69,165,135,93)(70,166,136,94)(71,167,137,95)(72,168,138,96)(79,221,142,179)(80,222,143,180)(81,217,144,175)(82,218,139,176)(83,219,140,177)(84,220,141,178), (1,217,187,175)(2,218,188,176)(3,219,189,177)(4,220,190,178)(5,221,191,179)(6,222,192,180)(7,216,150,158)(8,211,145,159)(9,212,146,160)(10,213,147,161)(11,214,148,162)(12,215,149,157)(13,231,201,59)(14,232,202,60)(15,233,203,55)(16,234,204,56)(17,229,199,57)(18,230,200,58)(19,92,194,164)(20,93,195,165)(21,94,196,166)(22,95,197,167)(23,96,198,168)(24,91,193,163)(25,237,207,41)(26,238,208,42)(27,239,209,37)(28,240,210,38)(29,235,205,39)(30,236,206,40)(31,130,109,61)(32,131,110,62)(33,132,111,63)(34,127,112,64)(35,128,113,65)(36,129,114,66)(43,117,87,73)(44,118,88,74)(45,119,89,75)(46,120,90,76)(47,115,85,77)(48,116,86,78)(49,142,105,79)(50,143,106,80)(51,144,107,81)(52,139,108,82)(53,140,103,83)(54,141,104,84)(67,169,133,97)(68,170,134,98)(69,171,135,99)(70,172,136,100)(71,173,137,101)(72,174,138,102)(121,224,182,152)(122,225,183,153)(123,226,184,154)(124,227,185,155)(125,228,186,156)(126,223,181,151), (1,89,17,71,29)(2,90,18,72,30)(3,85,13,67,25)(4,86,14,68,26)(5,87,15,69,27)(6,88,16,70,28)(7,184,19,66,84)(8,185,20,61,79)(9,186,21,62,80)(10,181,22,63,81)(11,182,23,64,82)(12,183,24,65,83)(31,49,211,155,93)(32,50,212,156,94)(33,51,213,151,95)(34,52,214,152,96)(35,53,215,153,91)(36,54,216,154,92)(37,179,117,55,99)(38,180,118,56,100)(39,175,119,57,101)(40,176,120,58,102)(41,177,115,59,97)(42,178,116,60,98)(43,203,135,209,191)(44,204,136,210,192)(45,199,137,205,187)(46,200,138,206,188)(47,201,133,207,189)(48,202,134,208,190)(73,233,171,239,221)(74,234,172,240,222)(75,229,173,235,217)(76,230,174,236,218)(77,231,169,237,219)(78,232,170,238,220)(103,157,225,163,113)(104,158,226,164,114)(105,159,227,165,109)(106,160,228,166,110)(107,161,223,167,111)(108,162,224,168,112)(121,198,127,139,148)(122,193,128,140,149)(123,194,129,141,150)(124,195,130,142,145)(125,196,131,143,146)(126,197,132,144,147), (1,63)(2,64)(3,65)(4,66)(5,61)(6,62)(7,68)(8,69)(9,70)(10,71)(11,72)(12,67)(13,183)(14,184)(15,185)(16,186)(17,181)(18,182)(19,86)(20,87)(21,88)(22,89)(23,90)(24,85)(25,83)(26,84)(27,79)(28,80)(29,81)(30,82)(31,221)(32,222)(33,217)(34,218)(35,219)(36,220)(37,105)(38,106)(39,107)(40,108)(41,103)(42,104)(43,195)(44,196)(45,197)(46,198)(47,193)(48,194)(49,239)(50,240)(51,235)(52,236)(53,237)(54,238)(55,227)(56,228)(57,223)(58,224)(59,225)(60,226)(73,93)(74,94)(75,95)(76,96)(77,91)(78,92)(97,157)(98,158)(99,159)(100,160)(101,161)(102,162)(109,179)(110,180)(111,175)(112,176)(113,177)(114,178)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,200)(122,201)(123,202)(124,203)(125,204)(126,199)(127,188)(128,189)(129,190)(130,191)(131,192)(132,187)(133,149)(134,150)(135,145)(136,146)(137,147)(138,148)(139,206)(140,207)(141,208)(142,209)(143,210)(144,205)(151,229)(152,230)(153,231)(154,232)(155,233)(156,234)(169,215)(170,216)(171,211)(172,212)(173,213)(174,214) );
G=PermutationGroup([(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)], [(1,107,187,51),(2,108,188,52),(3,103,189,53),(4,104,190,54),(5,105,191,49),(6,106,192,50),(7,78,150,116),(8,73,145,117),(9,74,146,118),(10,75,147,119),(11,76,148,120),(12,77,149,115),(13,225,201,153),(14,226,202,154),(15,227,203,155),(16,228,204,156),(17,223,199,151),(18,224,200,152),(19,170,194,98),(20,171,195,99),(21,172,196,100),(22,173,197,101),(23,174,198,102),(24,169,193,97),(25,113,207,35),(26,114,208,36),(27,109,209,31),(28,110,210,32),(29,111,205,33),(30,112,206,34),(37,61,239,130),(38,62,240,131),(39,63,235,132),(40,64,236,127),(41,65,237,128),(42,66,238,129),(43,211,87,159),(44,212,88,160),(45,213,89,161),(46,214,90,162),(47,215,85,157),(48,216,86,158),(55,185,233,124),(56,186,234,125),(57,181,229,126),(58,182,230,121),(59,183,231,122),(60,184,232,123),(67,163,133,91),(68,164,134,92),(69,165,135,93),(70,166,136,94),(71,167,137,95),(72,168,138,96),(79,221,142,179),(80,222,143,180),(81,217,144,175),(82,218,139,176),(83,219,140,177),(84,220,141,178)], [(1,217,187,175),(2,218,188,176),(3,219,189,177),(4,220,190,178),(5,221,191,179),(6,222,192,180),(7,216,150,158),(8,211,145,159),(9,212,146,160),(10,213,147,161),(11,214,148,162),(12,215,149,157),(13,231,201,59),(14,232,202,60),(15,233,203,55),(16,234,204,56),(17,229,199,57),(18,230,200,58),(19,92,194,164),(20,93,195,165),(21,94,196,166),(22,95,197,167),(23,96,198,168),(24,91,193,163),(25,237,207,41),(26,238,208,42),(27,239,209,37),(28,240,210,38),(29,235,205,39),(30,236,206,40),(31,130,109,61),(32,131,110,62),(33,132,111,63),(34,127,112,64),(35,128,113,65),(36,129,114,66),(43,117,87,73),(44,118,88,74),(45,119,89,75),(46,120,90,76),(47,115,85,77),(48,116,86,78),(49,142,105,79),(50,143,106,80),(51,144,107,81),(52,139,108,82),(53,140,103,83),(54,141,104,84),(67,169,133,97),(68,170,134,98),(69,171,135,99),(70,172,136,100),(71,173,137,101),(72,174,138,102),(121,224,182,152),(122,225,183,153),(123,226,184,154),(124,227,185,155),(125,228,186,156),(126,223,181,151)], [(1,89,17,71,29),(2,90,18,72,30),(3,85,13,67,25),(4,86,14,68,26),(5,87,15,69,27),(6,88,16,70,28),(7,184,19,66,84),(8,185,20,61,79),(9,186,21,62,80),(10,181,22,63,81),(11,182,23,64,82),(12,183,24,65,83),(31,49,211,155,93),(32,50,212,156,94),(33,51,213,151,95),(34,52,214,152,96),(35,53,215,153,91),(36,54,216,154,92),(37,179,117,55,99),(38,180,118,56,100),(39,175,119,57,101),(40,176,120,58,102),(41,177,115,59,97),(42,178,116,60,98),(43,203,135,209,191),(44,204,136,210,192),(45,199,137,205,187),(46,200,138,206,188),(47,201,133,207,189),(48,202,134,208,190),(73,233,171,239,221),(74,234,172,240,222),(75,229,173,235,217),(76,230,174,236,218),(77,231,169,237,219),(78,232,170,238,220),(103,157,225,163,113),(104,158,226,164,114),(105,159,227,165,109),(106,160,228,166,110),(107,161,223,167,111),(108,162,224,168,112),(121,198,127,139,148),(122,193,128,140,149),(123,194,129,141,150),(124,195,130,142,145),(125,196,131,143,146),(126,197,132,144,147)], [(1,63),(2,64),(3,65),(4,66),(5,61),(6,62),(7,68),(8,69),(9,70),(10,71),(11,72),(12,67),(13,183),(14,184),(15,185),(16,186),(17,181),(18,182),(19,86),(20,87),(21,88),(22,89),(23,90),(24,85),(25,83),(26,84),(27,79),(28,80),(29,81),(30,82),(31,221),(32,222),(33,217),(34,218),(35,219),(36,220),(37,105),(38,106),(39,107),(40,108),(41,103),(42,104),(43,195),(44,196),(45,197),(46,198),(47,193),(48,194),(49,239),(50,240),(51,235),(52,236),(53,237),(54,238),(55,227),(56,228),(57,223),(58,224),(59,225),(60,226),(73,93),(74,94),(75,95),(76,96),(77,91),(78,92),(97,157),(98,158),(99,159),(100,160),(101,161),(102,162),(109,179),(110,180),(111,175),(112,176),(113,177),(114,178),(115,163),(116,164),(117,165),(118,166),(119,167),(120,168),(121,200),(122,201),(123,202),(124,203),(125,204),(126,199),(127,188),(128,189),(129,190),(130,191),(131,192),(132,187),(133,149),(134,150),(135,145),(136,146),(137,147),(138,148),(139,206),(140,207),(141,208),(142,209),(143,210),(144,205),(151,229),(152,230),(153,231),(154,232),(155,233),(156,234),(169,215),(170,216),(171,211),(172,212),(173,213),(174,214)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 48 | 0 | 0 | 0 |
| 0 | 48 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 50 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 17 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 17 | 60 | 0 | 0 |
| 44 | 44 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 50 | 0 |
G:=sub<GL(4,GF(61))| [48,0,0,0,0,48,0,0,0,0,13,0,0,0,0,13],[60,0,0,0,0,60,0,0,0,0,11,0,0,0,0,50],[60,0,0,0,0,60,0,0,0,0,0,60,0,0,1,0],[17,1,0,0,60,0,0,0,0,0,1,0,0,0,0,1],[17,44,0,0,60,44,0,0,0,0,0,50,0,0,11,0] >;
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | ··· | 6F | 6G | ··· | 6R | 10A | ··· | 10F | 12A | ··· | 12L | 12M | ··· | 12T | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | ··· | 10 | 1 | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 2 | 2 | 1 | ··· | 1 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D5 | C4○D4 | D10 | D10 | C3×D5 | C3×C4○D4 | C6×D5 | C6×D5 | Q8⋊2D5 | C3×Q8⋊2D5 |
| kernel | C6×Q8⋊2D5 | D5×C2×C12 | C6×D20 | C3×Q8⋊2D5 | Q8×C30 | C2×Q8⋊2D5 | C2×C4×D5 | C2×D20 | Q8⋊2D5 | Q8×C10 | C6×Q8 | C30 | C2×C12 | C3×Q8 | C2×Q8 | C10 | C2×C4 | Q8 | C6 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 2 | 6 | 6 | 16 | 2 | 2 | 4 | 6 | 8 | 4 | 8 | 12 | 16 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_6\times Q_8\rtimes_2D_5
% in TeX
G:=Group("C6xQ8:2D5"); // GroupNames label
G:=SmallGroup(480,1143);
// by ID
G=gap.SmallGroup(480,1143);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,268,1571,409,192,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=d^5=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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