direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×Q8×D5, C30.76C24, C60.211C23, C30⋊9(C2×Q8), C10⋊2(C6×Q8), (Q8×C10)⋊9C6, (Q8×C30)⋊12C2, C15⋊10(C22×Q8), Dic10⋊9(C2×C6), C10.8(C23×C6), C6.76(C23×D5), (C2×Dic10)⋊13C6, (C6×Dic10)⋊29C2, (C2×C12).371D10, C20.22(C22×C6), (Q8×C15)⋊26C22, (C6×D5).75C23, (C2×C30).384C23, (C2×C60).306C22, D10.16(C22×C6), C12.211(C22×D5), Dic5.5(C22×C6), (C3×Dic10)⋊36C22, (D5×C12).110C22, (C3×Dic5).57C23, (C6×Dic5).258C22, C5⋊2(Q8×C2×C6), (C2×C4×D5).5C6, C4.22(D5×C2×C6), (C5×Q8)⋊7(C2×C6), C2.9(D5×C22×C6), (D5×C2×C12).16C2, (C2×C4).61(C6×D5), C22.31(D5×C2×C6), (C2×C20).43(C2×C6), (C4×D5).21(C2×C6), (D5×C2×C6).157C22, (C2×C10).66(C22×C6), (C2×Dic5).46(C2×C6), (C22×D5).46(C2×C6), (C2×C6).378(C22×D5), SmallGroup(480,1142)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 784 in 312 conjugacy classes, 194 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×6], C4 [×6], C22, C22 [×6], C5, C6, C6 [×2], C6 [×4], C2×C4 [×3], C2×C4 [×15], Q8 [×4], Q8 [×12], C23, D5 [×4], C10, C10 [×2], C12 [×6], C12 [×6], C2×C6, C2×C6 [×6], C15, C22×C4 [×3], C2×Q8, C2×Q8 [×11], Dic5 [×6], C20 [×6], D10 [×6], C2×C10, C2×C12 [×3], C2×C12 [×15], C3×Q8 [×4], C3×Q8 [×12], C22×C6, C3×D5 [×4], C30, C30 [×2], C22×Q8, Dic10 [×12], C4×D5 [×12], C2×Dic5 [×3], C2×C20 [×3], C5×Q8 [×4], C22×D5, C22×C12 [×3], C6×Q8, C6×Q8 [×11], C3×Dic5 [×6], C60 [×6], C6×D5 [×6], C2×C30, C2×Dic10 [×3], C2×C4×D5 [×3], Q8×D5 [×8], Q8×C10, Q8×C2×C6, C3×Dic10 [×12], D5×C12 [×12], C6×Dic5 [×3], C2×C60 [×3], Q8×C15 [×4], D5×C2×C6, C2×Q8×D5, C6×Dic10 [×3], D5×C2×C12 [×3], C3×Q8×D5 [×8], Q8×C30, C6×Q8×D5
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], Q8 [×4], C23 [×15], D5, C2×C6 [×35], C2×Q8 [×6], C24, D10 [×7], C3×Q8 [×4], C22×C6 [×15], C3×D5, C22×Q8, C22×D5 [×7], C6×Q8 [×6], C23×C6, C6×D5 [×7], Q8×D5 [×2], C23×D5, Q8×C2×C6, D5×C2×C6 [×7], C2×Q8×D5, C3×Q8×D5 [×2], D5×C22×C6, C6×Q8×D5
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=b-1, bd=db, be=eb, 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)
(1 217 187 173)(2 218 188 174)(3 219 189 169)(4 220 190 170)(5 221 191 171)(6 222 192 172)(7 226 74 44)(8 227 75 45)(9 228 76 46)(10 223 77 47)(11 224 78 48)(12 225 73 43)(13 231 201 49)(14 232 202 50)(15 233 203 51)(16 234 204 52)(17 229 199 53)(18 230 200 54)(19 108 62 32)(20 103 63 33)(21 104 64 34)(22 105 65 35)(23 106 66 36)(24 107 61 31)(25 111 81 41)(26 112 82 42)(27 113 83 37)(28 114 84 38)(29 109 79 39)(30 110 80 40)(55 144 91 85)(56 139 92 86)(57 140 93 87)(58 141 94 88)(59 142 95 89)(60 143 96 90)(67 175 145 97)(68 176 146 98)(69 177 147 99)(70 178 148 100)(71 179 149 101)(72 180 150 102)(115 135 161 132)(116 136 162 127)(117 137 157 128)(118 138 158 129)(119 133 159 130)(120 134 160 131)(121 214 194 164)(122 215 195 165)(123 216 196 166)(124 211 197 167)(125 212 198 168)(126 213 193 163)(151 237 207 181)(152 238 208 182)(153 239 209 183)(154 240 210 184)(155 235 205 185)(156 236 206 186)
(1 214 187 164)(2 215 188 165)(3 216 189 166)(4 211 190 167)(5 212 191 168)(6 213 192 163)(7 53 74 229)(8 54 75 230)(9 49 76 231)(10 50 77 232)(11 51 78 233)(12 52 73 234)(13 228 201 46)(14 223 202 47)(15 224 203 48)(16 225 204 43)(17 226 199 44)(18 227 200 45)(19 41 62 111)(20 42 63 112)(21 37 64 113)(22 38 65 114)(23 39 66 109)(24 40 61 110)(25 108 81 32)(26 103 82 33)(27 104 83 34)(28 105 84 35)(29 106 79 36)(30 107 80 31)(55 148 91 70)(56 149 92 71)(57 150 93 72)(58 145 94 67)(59 146 95 68)(60 147 96 69)(85 100 144 178)(86 101 139 179)(87 102 140 180)(88 97 141 175)(89 98 142 176)(90 99 143 177)(115 208 161 152)(116 209 162 153)(117 210 157 154)(118 205 158 155)(119 206 159 156)(120 207 160 151)(121 173 194 217)(122 174 195 218)(123 169 196 219)(124 170 197 220)(125 171 198 221)(126 172 193 222)(127 183 136 239)(128 184 137 240)(129 185 138 235)(130 186 133 236)(131 181 134 237)(132 182 135 238)
(1 155 71 29 199)(2 156 72 30 200)(3 151 67 25 201)(4 152 68 26 202)(5 153 69 27 203)(6 154 70 28 204)(7 194 138 139 66)(8 195 133 140 61)(9 196 134 141 62)(10 197 135 142 63)(11 198 136 143 64)(12 193 137 144 65)(13 189 207 145 81)(14 190 208 146 82)(15 191 209 147 83)(16 192 210 148 84)(17 187 205 149 79)(18 188 206 150 80)(19 76 123 131 88)(20 77 124 132 89)(21 78 125 127 90)(22 73 126 128 85)(23 74 121 129 86)(24 75 122 130 87)(31 227 165 159 93)(32 228 166 160 94)(33 223 167 161 95)(34 224 168 162 96)(35 225 163 157 91)(36 226 164 158 92)(37 233 171 183 99)(38 234 172 184 100)(39 229 173 185 101)(40 230 174 186 102)(41 231 169 181 97)(42 232 170 182 98)(43 213 117 55 105)(44 214 118 56 106)(45 215 119 57 107)(46 216 120 58 108)(47 211 115 59 103)(48 212 116 60 104)(49 219 237 175 111)(50 220 238 176 112)(51 221 239 177 113)(52 222 240 178 114)(53 217 235 179 109)(54 218 236 180 110)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 13)(7 124)(8 125)(9 126)(10 121)(11 122)(12 123)(19 137)(20 138)(21 133)(22 134)(23 135)(24 136)(25 210)(26 205)(27 206)(28 207)(29 208)(30 209)(31 116)(32 117)(33 118)(34 119)(35 120)(36 115)(37 236)(38 237)(39 238)(40 239)(41 240)(42 235)(43 166)(44 167)(45 168)(46 163)(47 164)(48 165)(49 172)(50 173)(51 174)(52 169)(53 170)(54 171)(55 94)(56 95)(57 96)(58 91)(59 92)(60 93)(61 127)(62 128)(63 129)(64 130)(65 131)(66 132)(67 148)(68 149)(69 150)(70 145)(71 146)(72 147)(73 196)(74 197)(75 198)(76 193)(77 194)(78 195)(79 152)(80 153)(81 154)(82 155)(83 156)(84 151)(85 141)(86 142)(87 143)(88 144)(89 139)(90 140)(97 178)(98 179)(99 180)(100 175)(101 176)(102 177)(103 158)(104 159)(105 160)(106 161)(107 162)(108 157)(109 182)(110 183)(111 184)(112 185)(113 186)(114 181)(187 202)(188 203)(189 204)(190 199)(191 200)(192 201)(211 226)(212 227)(213 228)(214 223)(215 224)(216 225)(217 232)(218 233)(219 234)(220 229)(221 230)(222 231)
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), (1,217,187,173)(2,218,188,174)(3,219,189,169)(4,220,190,170)(5,221,191,171)(6,222,192,172)(7,226,74,44)(8,227,75,45)(9,228,76,46)(10,223,77,47)(11,224,78,48)(12,225,73,43)(13,231,201,49)(14,232,202,50)(15,233,203,51)(16,234,204,52)(17,229,199,53)(18,230,200,54)(19,108,62,32)(20,103,63,33)(21,104,64,34)(22,105,65,35)(23,106,66,36)(24,107,61,31)(25,111,81,41)(26,112,82,42)(27,113,83,37)(28,114,84,38)(29,109,79,39)(30,110,80,40)(55,144,91,85)(56,139,92,86)(57,140,93,87)(58,141,94,88)(59,142,95,89)(60,143,96,90)(67,175,145,97)(68,176,146,98)(69,177,147,99)(70,178,148,100)(71,179,149,101)(72,180,150,102)(115,135,161,132)(116,136,162,127)(117,137,157,128)(118,138,158,129)(119,133,159,130)(120,134,160,131)(121,214,194,164)(122,215,195,165)(123,216,196,166)(124,211,197,167)(125,212,198,168)(126,213,193,163)(151,237,207,181)(152,238,208,182)(153,239,209,183)(154,240,210,184)(155,235,205,185)(156,236,206,186), (1,214,187,164)(2,215,188,165)(3,216,189,166)(4,211,190,167)(5,212,191,168)(6,213,192,163)(7,53,74,229)(8,54,75,230)(9,49,76,231)(10,50,77,232)(11,51,78,233)(12,52,73,234)(13,228,201,46)(14,223,202,47)(15,224,203,48)(16,225,204,43)(17,226,199,44)(18,227,200,45)(19,41,62,111)(20,42,63,112)(21,37,64,113)(22,38,65,114)(23,39,66,109)(24,40,61,110)(25,108,81,32)(26,103,82,33)(27,104,83,34)(28,105,84,35)(29,106,79,36)(30,107,80,31)(55,148,91,70)(56,149,92,71)(57,150,93,72)(58,145,94,67)(59,146,95,68)(60,147,96,69)(85,100,144,178)(86,101,139,179)(87,102,140,180)(88,97,141,175)(89,98,142,176)(90,99,143,177)(115,208,161,152)(116,209,162,153)(117,210,157,154)(118,205,158,155)(119,206,159,156)(120,207,160,151)(121,173,194,217)(122,174,195,218)(123,169,196,219)(124,170,197,220)(125,171,198,221)(126,172,193,222)(127,183,136,239)(128,184,137,240)(129,185,138,235)(130,186,133,236)(131,181,134,237)(132,182,135,238), (1,155,71,29,199)(2,156,72,30,200)(3,151,67,25,201)(4,152,68,26,202)(5,153,69,27,203)(6,154,70,28,204)(7,194,138,139,66)(8,195,133,140,61)(9,196,134,141,62)(10,197,135,142,63)(11,198,136,143,64)(12,193,137,144,65)(13,189,207,145,81)(14,190,208,146,82)(15,191,209,147,83)(16,192,210,148,84)(17,187,205,149,79)(18,188,206,150,80)(19,76,123,131,88)(20,77,124,132,89)(21,78,125,127,90)(22,73,126,128,85)(23,74,121,129,86)(24,75,122,130,87)(31,227,165,159,93)(32,228,166,160,94)(33,223,167,161,95)(34,224,168,162,96)(35,225,163,157,91)(36,226,164,158,92)(37,233,171,183,99)(38,234,172,184,100)(39,229,173,185,101)(40,230,174,186,102)(41,231,169,181,97)(42,232,170,182,98)(43,213,117,55,105)(44,214,118,56,106)(45,215,119,57,107)(46,216,120,58,108)(47,211,115,59,103)(48,212,116,60,104)(49,219,237,175,111)(50,220,238,176,112)(51,221,239,177,113)(52,222,240,178,114)(53,217,235,179,109)(54,218,236,180,110), (1,14)(2,15)(3,16)(4,17)(5,18)(6,13)(7,124)(8,125)(9,126)(10,121)(11,122)(12,123)(19,137)(20,138)(21,133)(22,134)(23,135)(24,136)(25,210)(26,205)(27,206)(28,207)(29,208)(30,209)(31,116)(32,117)(33,118)(34,119)(35,120)(36,115)(37,236)(38,237)(39,238)(40,239)(41,240)(42,235)(43,166)(44,167)(45,168)(46,163)(47,164)(48,165)(49,172)(50,173)(51,174)(52,169)(53,170)(54,171)(55,94)(56,95)(57,96)(58,91)(59,92)(60,93)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,148)(68,149)(69,150)(70,145)(71,146)(72,147)(73,196)(74,197)(75,198)(76,193)(77,194)(78,195)(79,152)(80,153)(81,154)(82,155)(83,156)(84,151)(85,141)(86,142)(87,143)(88,144)(89,139)(90,140)(97,178)(98,179)(99,180)(100,175)(101,176)(102,177)(103,158)(104,159)(105,160)(106,161)(107,162)(108,157)(109,182)(110,183)(111,184)(112,185)(113,186)(114,181)(187,202)(188,203)(189,204)(190,199)(191,200)(192,201)(211,226)(212,227)(213,228)(214,223)(215,224)(216,225)(217,232)(218,233)(219,234)(220,229)(221,230)(222,231)>;
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), (1,217,187,173)(2,218,188,174)(3,219,189,169)(4,220,190,170)(5,221,191,171)(6,222,192,172)(7,226,74,44)(8,227,75,45)(9,228,76,46)(10,223,77,47)(11,224,78,48)(12,225,73,43)(13,231,201,49)(14,232,202,50)(15,233,203,51)(16,234,204,52)(17,229,199,53)(18,230,200,54)(19,108,62,32)(20,103,63,33)(21,104,64,34)(22,105,65,35)(23,106,66,36)(24,107,61,31)(25,111,81,41)(26,112,82,42)(27,113,83,37)(28,114,84,38)(29,109,79,39)(30,110,80,40)(55,144,91,85)(56,139,92,86)(57,140,93,87)(58,141,94,88)(59,142,95,89)(60,143,96,90)(67,175,145,97)(68,176,146,98)(69,177,147,99)(70,178,148,100)(71,179,149,101)(72,180,150,102)(115,135,161,132)(116,136,162,127)(117,137,157,128)(118,138,158,129)(119,133,159,130)(120,134,160,131)(121,214,194,164)(122,215,195,165)(123,216,196,166)(124,211,197,167)(125,212,198,168)(126,213,193,163)(151,237,207,181)(152,238,208,182)(153,239,209,183)(154,240,210,184)(155,235,205,185)(156,236,206,186), (1,214,187,164)(2,215,188,165)(3,216,189,166)(4,211,190,167)(5,212,191,168)(6,213,192,163)(7,53,74,229)(8,54,75,230)(9,49,76,231)(10,50,77,232)(11,51,78,233)(12,52,73,234)(13,228,201,46)(14,223,202,47)(15,224,203,48)(16,225,204,43)(17,226,199,44)(18,227,200,45)(19,41,62,111)(20,42,63,112)(21,37,64,113)(22,38,65,114)(23,39,66,109)(24,40,61,110)(25,108,81,32)(26,103,82,33)(27,104,83,34)(28,105,84,35)(29,106,79,36)(30,107,80,31)(55,148,91,70)(56,149,92,71)(57,150,93,72)(58,145,94,67)(59,146,95,68)(60,147,96,69)(85,100,144,178)(86,101,139,179)(87,102,140,180)(88,97,141,175)(89,98,142,176)(90,99,143,177)(115,208,161,152)(116,209,162,153)(117,210,157,154)(118,205,158,155)(119,206,159,156)(120,207,160,151)(121,173,194,217)(122,174,195,218)(123,169,196,219)(124,170,197,220)(125,171,198,221)(126,172,193,222)(127,183,136,239)(128,184,137,240)(129,185,138,235)(130,186,133,236)(131,181,134,237)(132,182,135,238), (1,155,71,29,199)(2,156,72,30,200)(3,151,67,25,201)(4,152,68,26,202)(5,153,69,27,203)(6,154,70,28,204)(7,194,138,139,66)(8,195,133,140,61)(9,196,134,141,62)(10,197,135,142,63)(11,198,136,143,64)(12,193,137,144,65)(13,189,207,145,81)(14,190,208,146,82)(15,191,209,147,83)(16,192,210,148,84)(17,187,205,149,79)(18,188,206,150,80)(19,76,123,131,88)(20,77,124,132,89)(21,78,125,127,90)(22,73,126,128,85)(23,74,121,129,86)(24,75,122,130,87)(31,227,165,159,93)(32,228,166,160,94)(33,223,167,161,95)(34,224,168,162,96)(35,225,163,157,91)(36,226,164,158,92)(37,233,171,183,99)(38,234,172,184,100)(39,229,173,185,101)(40,230,174,186,102)(41,231,169,181,97)(42,232,170,182,98)(43,213,117,55,105)(44,214,118,56,106)(45,215,119,57,107)(46,216,120,58,108)(47,211,115,59,103)(48,212,116,60,104)(49,219,237,175,111)(50,220,238,176,112)(51,221,239,177,113)(52,222,240,178,114)(53,217,235,179,109)(54,218,236,180,110), (1,14)(2,15)(3,16)(4,17)(5,18)(6,13)(7,124)(8,125)(9,126)(10,121)(11,122)(12,123)(19,137)(20,138)(21,133)(22,134)(23,135)(24,136)(25,210)(26,205)(27,206)(28,207)(29,208)(30,209)(31,116)(32,117)(33,118)(34,119)(35,120)(36,115)(37,236)(38,237)(39,238)(40,239)(41,240)(42,235)(43,166)(44,167)(45,168)(46,163)(47,164)(48,165)(49,172)(50,173)(51,174)(52,169)(53,170)(54,171)(55,94)(56,95)(57,96)(58,91)(59,92)(60,93)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,148)(68,149)(69,150)(70,145)(71,146)(72,147)(73,196)(74,197)(75,198)(76,193)(77,194)(78,195)(79,152)(80,153)(81,154)(82,155)(83,156)(84,151)(85,141)(86,142)(87,143)(88,144)(89,139)(90,140)(97,178)(98,179)(99,180)(100,175)(101,176)(102,177)(103,158)(104,159)(105,160)(106,161)(107,162)(108,157)(109,182)(110,183)(111,184)(112,185)(113,186)(114,181)(187,202)(188,203)(189,204)(190,199)(191,200)(192,201)(211,226)(212,227)(213,228)(214,223)(215,224)(216,225)(217,232)(218,233)(219,234)(220,229)(221,230)(222,231) );
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)], [(1,217,187,173),(2,218,188,174),(3,219,189,169),(4,220,190,170),(5,221,191,171),(6,222,192,172),(7,226,74,44),(8,227,75,45),(9,228,76,46),(10,223,77,47),(11,224,78,48),(12,225,73,43),(13,231,201,49),(14,232,202,50),(15,233,203,51),(16,234,204,52),(17,229,199,53),(18,230,200,54),(19,108,62,32),(20,103,63,33),(21,104,64,34),(22,105,65,35),(23,106,66,36),(24,107,61,31),(25,111,81,41),(26,112,82,42),(27,113,83,37),(28,114,84,38),(29,109,79,39),(30,110,80,40),(55,144,91,85),(56,139,92,86),(57,140,93,87),(58,141,94,88),(59,142,95,89),(60,143,96,90),(67,175,145,97),(68,176,146,98),(69,177,147,99),(70,178,148,100),(71,179,149,101),(72,180,150,102),(115,135,161,132),(116,136,162,127),(117,137,157,128),(118,138,158,129),(119,133,159,130),(120,134,160,131),(121,214,194,164),(122,215,195,165),(123,216,196,166),(124,211,197,167),(125,212,198,168),(126,213,193,163),(151,237,207,181),(152,238,208,182),(153,239,209,183),(154,240,210,184),(155,235,205,185),(156,236,206,186)], [(1,214,187,164),(2,215,188,165),(3,216,189,166),(4,211,190,167),(5,212,191,168),(6,213,192,163),(7,53,74,229),(8,54,75,230),(9,49,76,231),(10,50,77,232),(11,51,78,233),(12,52,73,234),(13,228,201,46),(14,223,202,47),(15,224,203,48),(16,225,204,43),(17,226,199,44),(18,227,200,45),(19,41,62,111),(20,42,63,112),(21,37,64,113),(22,38,65,114),(23,39,66,109),(24,40,61,110),(25,108,81,32),(26,103,82,33),(27,104,83,34),(28,105,84,35),(29,106,79,36),(30,107,80,31),(55,148,91,70),(56,149,92,71),(57,150,93,72),(58,145,94,67),(59,146,95,68),(60,147,96,69),(85,100,144,178),(86,101,139,179),(87,102,140,180),(88,97,141,175),(89,98,142,176),(90,99,143,177),(115,208,161,152),(116,209,162,153),(117,210,157,154),(118,205,158,155),(119,206,159,156),(120,207,160,151),(121,173,194,217),(122,174,195,218),(123,169,196,219),(124,170,197,220),(125,171,198,221),(126,172,193,222),(127,183,136,239),(128,184,137,240),(129,185,138,235),(130,186,133,236),(131,181,134,237),(132,182,135,238)], [(1,155,71,29,199),(2,156,72,30,200),(3,151,67,25,201),(4,152,68,26,202),(5,153,69,27,203),(6,154,70,28,204),(7,194,138,139,66),(8,195,133,140,61),(9,196,134,141,62),(10,197,135,142,63),(11,198,136,143,64),(12,193,137,144,65),(13,189,207,145,81),(14,190,208,146,82),(15,191,209,147,83),(16,192,210,148,84),(17,187,205,149,79),(18,188,206,150,80),(19,76,123,131,88),(20,77,124,132,89),(21,78,125,127,90),(22,73,126,128,85),(23,74,121,129,86),(24,75,122,130,87),(31,227,165,159,93),(32,228,166,160,94),(33,223,167,161,95),(34,224,168,162,96),(35,225,163,157,91),(36,226,164,158,92),(37,233,171,183,99),(38,234,172,184,100),(39,229,173,185,101),(40,230,174,186,102),(41,231,169,181,97),(42,232,170,182,98),(43,213,117,55,105),(44,214,118,56,106),(45,215,119,57,107),(46,216,120,58,108),(47,211,115,59,103),(48,212,116,60,104),(49,219,237,175,111),(50,220,238,176,112),(51,221,239,177,113),(52,222,240,178,114),(53,217,235,179,109),(54,218,236,180,110)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,13),(7,124),(8,125),(9,126),(10,121),(11,122),(12,123),(19,137),(20,138),(21,133),(22,134),(23,135),(24,136),(25,210),(26,205),(27,206),(28,207),(29,208),(30,209),(31,116),(32,117),(33,118),(34,119),(35,120),(36,115),(37,236),(38,237),(39,238),(40,239),(41,240),(42,235),(43,166),(44,167),(45,168),(46,163),(47,164),(48,165),(49,172),(50,173),(51,174),(52,169),(53,170),(54,171),(55,94),(56,95),(57,96),(58,91),(59,92),(60,93),(61,127),(62,128),(63,129),(64,130),(65,131),(66,132),(67,148),(68,149),(69,150),(70,145),(71,146),(72,147),(73,196),(74,197),(75,198),(76,193),(77,194),(78,195),(79,152),(80,153),(81,154),(82,155),(83,156),(84,151),(85,141),(86,142),(87,143),(88,144),(89,139),(90,140),(97,178),(98,179),(99,180),(100,175),(101,176),(102,177),(103,158),(104,159),(105,160),(106,161),(107,162),(108,157),(109,182),(110,183),(111,184),(112,185),(113,186),(114,181),(187,202),(188,203),(189,204),(190,199),(191,200),(192,201),(211,226),(212,227),(213,228),(214,223),(215,224),(216,225),(217,232),(218,233),(219,234),(220,229),(221,230),(222,231)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 48 | 0 | 0 | 0 |
| 0 | 48 | 0 | 0 |
| 0 | 0 | 14 | 0 |
| 0 | 0 | 0 | 14 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 22 | 8 |
| 0 | 0 | 8 | 39 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 60 | 43 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 60 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(61))| [48,0,0,0,0,48,0,0,0,0,14,0,0,0,0,14],[60,0,0,0,0,60,0,0,0,0,22,8,0,0,8,39],[60,0,0,0,0,60,0,0,0,0,0,1,0,0,60,0],[0,60,0,0,1,43,0,0,0,0,1,0,0,0,0,1],[0,60,0,0,60,0,0,0,0,0,1,0,0,0,0,1] >;
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | ··· | 4F | 4G | ··· | 4L | 5A | 5B | 6A | ··· | 6F | 6G | ··· | 6N | 10A | ··· | 10F | 12A | ··· | 12L | 12M | ··· | 12X | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 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 | 5 | 5 | 5 | 5 | 1 | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 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 | Q8 | D5 | D10 | D10 | C3×Q8 | C3×D5 | C6×D5 | C6×D5 | Q8×D5 | C3×Q8×D5 |
| kernel | C6×Q8×D5 | C6×Dic10 | D5×C2×C12 | C3×Q8×D5 | Q8×C30 | C2×Q8×D5 | C2×Dic10 | C2×C4×D5 | Q8×D5 | Q8×C10 | C6×D5 | C6×Q8 | C2×C12 | C3×Q8 | D10 | C2×Q8 | C2×C4 | Q8 | C6 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 2 | 6 | 6 | 16 | 2 | 4 | 2 | 6 | 8 | 8 | 4 | 12 | 16 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_6\times Q_8\times D_5
% in TeX
G:=Group("C6xQ8xD5"); // GroupNames label
G:=SmallGroup(480,1142);
// by ID
G=gap.SmallGroup(480,1142);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,268,409,192,18822]);
// 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=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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