direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Q8.7D6, C40.63D6, C120.74C22, C60.223C23, (S3×C8)⋊5C10, D4⋊S3⋊4C10, D6.2(C5×D4), (S3×C40)⋊14C2, C24⋊C2⋊6C10, C15⋊33(C4○D8), C8.11(S3×C10), (C5×SD16)⋊7S3, SD16⋊3(C5×S3), C3⋊Q16⋊2C10, (C5×D4).29D6, D4.5(S3×C10), C6.33(D4×C10), Q8.7(S3×C10), (C5×Q8).45D6, D4⋊2S3⋊3C10, C24.11(C2×C10), Q8⋊3S3⋊2C10, (C3×SD16)⋊4C10, D12.3(C2×C10), (S3×C10).26D4, C30.369(C2×D4), C10.187(S3×D4), (C15×SD16)⋊12C2, C12.7(C22×C10), Dic6.3(C2×C10), Dic3.13(C5×D4), (C5×Dic3).50D4, (S3×C20).59C22, C20.196(C22×S3), (C5×D12).32C22, (D4×C15).34C22, (Q8×C15).33C22, (C5×Dic6).34C22, C3⋊3(C5×C4○D8), C4.7(S3×C2×C10), C2.21(C5×S3×D4), C3⋊C8.6(C2×C10), (C5×D4⋊S3)⋊12C2, (C5×C24⋊C2)⋊14C2, (C5×Q8⋊3S3)⋊9C2, (C5×C3⋊Q16)⋊10C2, (C3×D4).5(C2×C10), (C5×C3⋊C8).42C22, (C3×Q8).2(C2×C10), (C5×D4⋊2S3)⋊10C2, (C4×S3).10(C2×C10), SmallGroup(480,795)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Q8.7D6
G = < a,b,c,d,e | a5=b4=d6=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe-1=b-1, dcd-1=b-1c, ece-1=bc, ede-1=b2d-1 >
Subgroups: 308 in 124 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, D8, SD16, SD16, Q16, C4○D4, C20, C20, C2×C10, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C5×S3, C30, C30, C4○D8, C40, C40, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D4⋊2S3, Q8⋊3S3, C5×Dic3, C5×Dic3, C60, C60, S3×C10, S3×C10, C2×C30, C2×C40, C5×D8, C5×SD16, C5×SD16, C5×Q16, C5×C4○D4, Q8.7D6, C5×C3⋊C8, C120, C5×Dic6, S3×C20, S3×C20, C5×D12, C5×D12, C10×Dic3, C5×C3⋊D4, D4×C15, Q8×C15, C5×C4○D8, S3×C40, C5×C24⋊C2, C5×D4⋊S3, C5×C3⋊Q16, C15×SD16, C5×D4⋊2S3, C5×Q8⋊3S3, C5×Q8.7D6
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C22×S3, C5×S3, C4○D8, C5×D4, C22×C10, S3×D4, S3×C10, D4×C10, Q8.7D6, S3×C2×C10, C5×C4○D8, C5×S3×D4, C5×Q8.7D6
►On 240 points
Generators in S240
(1 34 28 22 16)(2 35 29 23 17)(3 36 30 24 18)(4 31 25 19 13)(5 32 26 20 14)(6 33 27 21 15)(7 178 172 166 160)(8 179 173 167 161)(9 180 174 168 162)(10 175 169 163 157)(11 176 170 164 158)(12 177 171 165 159)(37 61 55 49 43)(38 62 56 50 44)(39 63 57 51 45)(40 64 58 52 46)(41 65 59 53 47)(42 66 60 54 48)(67 91 85 79 73)(68 92 86 80 74)(69 93 87 81 75)(70 94 88 82 76)(71 95 89 83 77)(72 96 90 84 78)(97 125 115 109 103)(98 126 116 110 104)(99 121 117 111 105)(100 122 118 112 106)(101 123 119 113 107)(102 124 120 114 108)(127 151 145 139 133)(128 152 146 140 134)(129 153 147 141 135)(130 154 148 142 136)(131 155 149 143 137)(132 156 150 144 138)(181 205 199 193 187)(182 206 200 194 188)(183 207 201 195 189)(184 208 202 196 190)(185 209 203 197 191)(186 210 204 198 192)(211 235 229 223 217)(212 236 230 224 218)(213 237 231 225 219)(214 238 232 226 220)(215 239 233 227 221)(216 240 234 228 222)
(1 70 185 131)(2 132 186 71)(3 72 181 127)(4 128 182 67)(5 68 183 129)(6 130 184 69)(7 63 121 238)(8 239 122 64)(9 65 123 240)(10 235 124 66)(11 61 125 236)(12 237 126 62)(13 134 188 73)(14 74 189 135)(15 136 190 75)(16 76 191 137)(17 138 192 77)(18 78 187 133)(19 140 194 79)(20 80 195 141)(21 142 196 81)(22 82 197 143)(23 144 198 83)(24 84 193 139)(25 146 200 85)(26 86 201 147)(27 148 202 87)(28 88 203 149)(29 150 204 89)(30 90 199 145)(31 152 206 91)(32 92 207 153)(33 154 208 93)(34 94 209 155)(35 156 210 95)(36 96 205 151)(37 97 212 158)(38 159 213 98)(39 99 214 160)(40 161 215 100)(41 101 216 162)(42 157 211 102)(43 103 218 164)(44 165 219 104)(45 105 220 166)(46 167 221 106)(47 107 222 168)(48 163 217 108)(49 109 224 170)(50 171 225 110)(51 111 226 172)(52 173 227 112)(53 113 228 174)(54 169 223 114)(55 115 230 176)(56 177 231 116)(57 117 232 178)(58 179 233 118)(59 119 234 180)(60 175 229 120)
(1 215 185 40)(2 101 186 162)(3 211 181 42)(4 97 182 158)(5 213 183 38)(6 99 184 160)(7 33 121 208)(8 155 122 94)(9 35 123 210)(10 151 124 96)(11 31 125 206)(12 153 126 92)(13 103 188 164)(14 219 189 44)(15 105 190 166)(16 221 191 46)(17 107 192 168)(18 217 187 48)(19 109 194 170)(20 225 195 50)(21 111 196 172)(22 227 197 52)(23 113 198 174)(24 223 193 54)(25 115 200 176)(26 231 201 56)(27 117 202 178)(28 233 203 58)(29 119 204 180)(30 229 199 60)(32 237 207 62)(34 239 209 64)(36 235 205 66)(37 67 212 128)(39 69 214 130)(41 71 216 132)(43 73 218 134)(45 75 220 136)(47 77 222 138)(49 79 224 140)(51 81 226 142)(53 83 228 144)(55 85 230 146)(57 87 232 148)(59 89 234 150)(61 91 236 152)(63 93 238 154)(65 95 240 156)(68 159 129 98)(70 161 131 100)(72 157 127 102)(74 165 135 104)(76 167 137 106)(78 163 133 108)(80 171 141 110)(82 173 143 112)(84 169 139 114)(86 177 147 116)(88 179 149 118)(90 175 145 120)
(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 6 185 184)(2 183 186 5)(3 4 181 182)(7 8 121 122)(9 12 123 126)(10 125 124 11)(13 187 188 18)(14 17 189 192)(15 191 190 16)(19 193 194 24)(20 23 195 198)(21 197 196 22)(25 199 200 30)(26 29 201 204)(27 203 202 28)(31 205 206 36)(32 35 207 210)(33 209 208 34)(37 42 212 211)(38 216 213 41)(39 40 214 215)(43 48 218 217)(44 222 219 47)(45 46 220 221)(49 54 224 223)(50 228 225 53)(51 52 226 227)(55 60 230 229)(56 234 231 59)(57 58 232 233)(61 66 236 235)(62 240 237 65)(63 64 238 239)(67 127 128 72)(68 71 129 132)(69 131 130 70)(73 133 134 78)(74 77 135 138)(75 137 136 76)(79 139 140 84)(80 83 141 144)(81 143 142 82)(85 145 146 90)(86 89 147 150)(87 149 148 88)(91 151 152 96)(92 95 153 156)(93 155 154 94)(97 102 158 157)(98 162 159 101)(99 100 160 161)(103 108 164 163)(104 168 165 107)(105 106 166 167)(109 114 170 169)(110 174 171 113)(111 112 172 173)(115 120 176 175)(116 180 177 119)(117 118 178 179)
G:=sub<Sym(240)| (1,34,28,22,16)(2,35,29,23,17)(3,36,30,24,18)(4,31,25,19,13)(5,32,26,20,14)(6,33,27,21,15)(7,178,172,166,160)(8,179,173,167,161)(9,180,174,168,162)(10,175,169,163,157)(11,176,170,164,158)(12,177,171,165,159)(37,61,55,49,43)(38,62,56,50,44)(39,63,57,51,45)(40,64,58,52,46)(41,65,59,53,47)(42,66,60,54,48)(67,91,85,79,73)(68,92,86,80,74)(69,93,87,81,75)(70,94,88,82,76)(71,95,89,83,77)(72,96,90,84,78)(97,125,115,109,103)(98,126,116,110,104)(99,121,117,111,105)(100,122,118,112,106)(101,123,119,113,107)(102,124,120,114,108)(127,151,145,139,133)(128,152,146,140,134)(129,153,147,141,135)(130,154,148,142,136)(131,155,149,143,137)(132,156,150,144,138)(181,205,199,193,187)(182,206,200,194,188)(183,207,201,195,189)(184,208,202,196,190)(185,209,203,197,191)(186,210,204,198,192)(211,235,229,223,217)(212,236,230,224,218)(213,237,231,225,219)(214,238,232,226,220)(215,239,233,227,221)(216,240,234,228,222), (1,70,185,131)(2,132,186,71)(3,72,181,127)(4,128,182,67)(5,68,183,129)(6,130,184,69)(7,63,121,238)(8,239,122,64)(9,65,123,240)(10,235,124,66)(11,61,125,236)(12,237,126,62)(13,134,188,73)(14,74,189,135)(15,136,190,75)(16,76,191,137)(17,138,192,77)(18,78,187,133)(19,140,194,79)(20,80,195,141)(21,142,196,81)(22,82,197,143)(23,144,198,83)(24,84,193,139)(25,146,200,85)(26,86,201,147)(27,148,202,87)(28,88,203,149)(29,150,204,89)(30,90,199,145)(31,152,206,91)(32,92,207,153)(33,154,208,93)(34,94,209,155)(35,156,210,95)(36,96,205,151)(37,97,212,158)(38,159,213,98)(39,99,214,160)(40,161,215,100)(41,101,216,162)(42,157,211,102)(43,103,218,164)(44,165,219,104)(45,105,220,166)(46,167,221,106)(47,107,222,168)(48,163,217,108)(49,109,224,170)(50,171,225,110)(51,111,226,172)(52,173,227,112)(53,113,228,174)(54,169,223,114)(55,115,230,176)(56,177,231,116)(57,117,232,178)(58,179,233,118)(59,119,234,180)(60,175,229,120), (1,215,185,40)(2,101,186,162)(3,211,181,42)(4,97,182,158)(5,213,183,38)(6,99,184,160)(7,33,121,208)(8,155,122,94)(9,35,123,210)(10,151,124,96)(11,31,125,206)(12,153,126,92)(13,103,188,164)(14,219,189,44)(15,105,190,166)(16,221,191,46)(17,107,192,168)(18,217,187,48)(19,109,194,170)(20,225,195,50)(21,111,196,172)(22,227,197,52)(23,113,198,174)(24,223,193,54)(25,115,200,176)(26,231,201,56)(27,117,202,178)(28,233,203,58)(29,119,204,180)(30,229,199,60)(32,237,207,62)(34,239,209,64)(36,235,205,66)(37,67,212,128)(39,69,214,130)(41,71,216,132)(43,73,218,134)(45,75,220,136)(47,77,222,138)(49,79,224,140)(51,81,226,142)(53,83,228,144)(55,85,230,146)(57,87,232,148)(59,89,234,150)(61,91,236,152)(63,93,238,154)(65,95,240,156)(68,159,129,98)(70,161,131,100)(72,157,127,102)(74,165,135,104)(76,167,137,106)(78,163,133,108)(80,171,141,110)(82,173,143,112)(84,169,139,114)(86,177,147,116)(88,179,149,118)(90,175,145,120), (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), (1,6,185,184)(2,183,186,5)(3,4,181,182)(7,8,121,122)(9,12,123,126)(10,125,124,11)(13,187,188,18)(14,17,189,192)(15,191,190,16)(19,193,194,24)(20,23,195,198)(21,197,196,22)(25,199,200,30)(26,29,201,204)(27,203,202,28)(31,205,206,36)(32,35,207,210)(33,209,208,34)(37,42,212,211)(38,216,213,41)(39,40,214,215)(43,48,218,217)(44,222,219,47)(45,46,220,221)(49,54,224,223)(50,228,225,53)(51,52,226,227)(55,60,230,229)(56,234,231,59)(57,58,232,233)(61,66,236,235)(62,240,237,65)(63,64,238,239)(67,127,128,72)(68,71,129,132)(69,131,130,70)(73,133,134,78)(74,77,135,138)(75,137,136,76)(79,139,140,84)(80,83,141,144)(81,143,142,82)(85,145,146,90)(86,89,147,150)(87,149,148,88)(91,151,152,96)(92,95,153,156)(93,155,154,94)(97,102,158,157)(98,162,159,101)(99,100,160,161)(103,108,164,163)(104,168,165,107)(105,106,166,167)(109,114,170,169)(110,174,171,113)(111,112,172,173)(115,120,176,175)(116,180,177,119)(117,118,178,179)>;
G:=Group( (1,34,28,22,16)(2,35,29,23,17)(3,36,30,24,18)(4,31,25,19,13)(5,32,26,20,14)(6,33,27,21,15)(7,178,172,166,160)(8,179,173,167,161)(9,180,174,168,162)(10,175,169,163,157)(11,176,170,164,158)(12,177,171,165,159)(37,61,55,49,43)(38,62,56,50,44)(39,63,57,51,45)(40,64,58,52,46)(41,65,59,53,47)(42,66,60,54,48)(67,91,85,79,73)(68,92,86,80,74)(69,93,87,81,75)(70,94,88,82,76)(71,95,89,83,77)(72,96,90,84,78)(97,125,115,109,103)(98,126,116,110,104)(99,121,117,111,105)(100,122,118,112,106)(101,123,119,113,107)(102,124,120,114,108)(127,151,145,139,133)(128,152,146,140,134)(129,153,147,141,135)(130,154,148,142,136)(131,155,149,143,137)(132,156,150,144,138)(181,205,199,193,187)(182,206,200,194,188)(183,207,201,195,189)(184,208,202,196,190)(185,209,203,197,191)(186,210,204,198,192)(211,235,229,223,217)(212,236,230,224,218)(213,237,231,225,219)(214,238,232,226,220)(215,239,233,227,221)(216,240,234,228,222), (1,70,185,131)(2,132,186,71)(3,72,181,127)(4,128,182,67)(5,68,183,129)(6,130,184,69)(7,63,121,238)(8,239,122,64)(9,65,123,240)(10,235,124,66)(11,61,125,236)(12,237,126,62)(13,134,188,73)(14,74,189,135)(15,136,190,75)(16,76,191,137)(17,138,192,77)(18,78,187,133)(19,140,194,79)(20,80,195,141)(21,142,196,81)(22,82,197,143)(23,144,198,83)(24,84,193,139)(25,146,200,85)(26,86,201,147)(27,148,202,87)(28,88,203,149)(29,150,204,89)(30,90,199,145)(31,152,206,91)(32,92,207,153)(33,154,208,93)(34,94,209,155)(35,156,210,95)(36,96,205,151)(37,97,212,158)(38,159,213,98)(39,99,214,160)(40,161,215,100)(41,101,216,162)(42,157,211,102)(43,103,218,164)(44,165,219,104)(45,105,220,166)(46,167,221,106)(47,107,222,168)(48,163,217,108)(49,109,224,170)(50,171,225,110)(51,111,226,172)(52,173,227,112)(53,113,228,174)(54,169,223,114)(55,115,230,176)(56,177,231,116)(57,117,232,178)(58,179,233,118)(59,119,234,180)(60,175,229,120), (1,215,185,40)(2,101,186,162)(3,211,181,42)(4,97,182,158)(5,213,183,38)(6,99,184,160)(7,33,121,208)(8,155,122,94)(9,35,123,210)(10,151,124,96)(11,31,125,206)(12,153,126,92)(13,103,188,164)(14,219,189,44)(15,105,190,166)(16,221,191,46)(17,107,192,168)(18,217,187,48)(19,109,194,170)(20,225,195,50)(21,111,196,172)(22,227,197,52)(23,113,198,174)(24,223,193,54)(25,115,200,176)(26,231,201,56)(27,117,202,178)(28,233,203,58)(29,119,204,180)(30,229,199,60)(32,237,207,62)(34,239,209,64)(36,235,205,66)(37,67,212,128)(39,69,214,130)(41,71,216,132)(43,73,218,134)(45,75,220,136)(47,77,222,138)(49,79,224,140)(51,81,226,142)(53,83,228,144)(55,85,230,146)(57,87,232,148)(59,89,234,150)(61,91,236,152)(63,93,238,154)(65,95,240,156)(68,159,129,98)(70,161,131,100)(72,157,127,102)(74,165,135,104)(76,167,137,106)(78,163,133,108)(80,171,141,110)(82,173,143,112)(84,169,139,114)(86,177,147,116)(88,179,149,118)(90,175,145,120), (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), (1,6,185,184)(2,183,186,5)(3,4,181,182)(7,8,121,122)(9,12,123,126)(10,125,124,11)(13,187,188,18)(14,17,189,192)(15,191,190,16)(19,193,194,24)(20,23,195,198)(21,197,196,22)(25,199,200,30)(26,29,201,204)(27,203,202,28)(31,205,206,36)(32,35,207,210)(33,209,208,34)(37,42,212,211)(38,216,213,41)(39,40,214,215)(43,48,218,217)(44,222,219,47)(45,46,220,221)(49,54,224,223)(50,228,225,53)(51,52,226,227)(55,60,230,229)(56,234,231,59)(57,58,232,233)(61,66,236,235)(62,240,237,65)(63,64,238,239)(67,127,128,72)(68,71,129,132)(69,131,130,70)(73,133,134,78)(74,77,135,138)(75,137,136,76)(79,139,140,84)(80,83,141,144)(81,143,142,82)(85,145,146,90)(86,89,147,150)(87,149,148,88)(91,151,152,96)(92,95,153,156)(93,155,154,94)(97,102,158,157)(98,162,159,101)(99,100,160,161)(103,108,164,163)(104,168,165,107)(105,106,166,167)(109,114,170,169)(110,174,171,113)(111,112,172,173)(115,120,176,175)(116,180,177,119)(117,118,178,179) );
G=PermutationGroup([[(1,34,28,22,16),(2,35,29,23,17),(3,36,30,24,18),(4,31,25,19,13),(5,32,26,20,14),(6,33,27,21,15),(7,178,172,166,160),(8,179,173,167,161),(9,180,174,168,162),(10,175,169,163,157),(11,176,170,164,158),(12,177,171,165,159),(37,61,55,49,43),(38,62,56,50,44),(39,63,57,51,45),(40,64,58,52,46),(41,65,59,53,47),(42,66,60,54,48),(67,91,85,79,73),(68,92,86,80,74),(69,93,87,81,75),(70,94,88,82,76),(71,95,89,83,77),(72,96,90,84,78),(97,125,115,109,103),(98,126,116,110,104),(99,121,117,111,105),(100,122,118,112,106),(101,123,119,113,107),(102,124,120,114,108),(127,151,145,139,133),(128,152,146,140,134),(129,153,147,141,135),(130,154,148,142,136),(131,155,149,143,137),(132,156,150,144,138),(181,205,199,193,187),(182,206,200,194,188),(183,207,201,195,189),(184,208,202,196,190),(185,209,203,197,191),(186,210,204,198,192),(211,235,229,223,217),(212,236,230,224,218),(213,237,231,225,219),(214,238,232,226,220),(215,239,233,227,221),(216,240,234,228,222)], [(1,70,185,131),(2,132,186,71),(3,72,181,127),(4,128,182,67),(5,68,183,129),(6,130,184,69),(7,63,121,238),(8,239,122,64),(9,65,123,240),(10,235,124,66),(11,61,125,236),(12,237,126,62),(13,134,188,73),(14,74,189,135),(15,136,190,75),(16,76,191,137),(17,138,192,77),(18,78,187,133),(19,140,194,79),(20,80,195,141),(21,142,196,81),(22,82,197,143),(23,144,198,83),(24,84,193,139),(25,146,200,85),(26,86,201,147),(27,148,202,87),(28,88,203,149),(29,150,204,89),(30,90,199,145),(31,152,206,91),(32,92,207,153),(33,154,208,93),(34,94,209,155),(35,156,210,95),(36,96,205,151),(37,97,212,158),(38,159,213,98),(39,99,214,160),(40,161,215,100),(41,101,216,162),(42,157,211,102),(43,103,218,164),(44,165,219,104),(45,105,220,166),(46,167,221,106),(47,107,222,168),(48,163,217,108),(49,109,224,170),(50,171,225,110),(51,111,226,172),(52,173,227,112),(53,113,228,174),(54,169,223,114),(55,115,230,176),(56,177,231,116),(57,117,232,178),(58,179,233,118),(59,119,234,180),(60,175,229,120)], [(1,215,185,40),(2,101,186,162),(3,211,181,42),(4,97,182,158),(5,213,183,38),(6,99,184,160),(7,33,121,208),(8,155,122,94),(9,35,123,210),(10,151,124,96),(11,31,125,206),(12,153,126,92),(13,103,188,164),(14,219,189,44),(15,105,190,166),(16,221,191,46),(17,107,192,168),(18,217,187,48),(19,109,194,170),(20,225,195,50),(21,111,196,172),(22,227,197,52),(23,113,198,174),(24,223,193,54),(25,115,200,176),(26,231,201,56),(27,117,202,178),(28,233,203,58),(29,119,204,180),(30,229,199,60),(32,237,207,62),(34,239,209,64),(36,235,205,66),(37,67,212,128),(39,69,214,130),(41,71,216,132),(43,73,218,134),(45,75,220,136),(47,77,222,138),(49,79,224,140),(51,81,226,142),(53,83,228,144),(55,85,230,146),(57,87,232,148),(59,89,234,150),(61,91,236,152),(63,93,238,154),(65,95,240,156),(68,159,129,98),(70,161,131,100),(72,157,127,102),(74,165,135,104),(76,167,137,106),(78,163,133,108),(80,171,141,110),(82,173,143,112),(84,169,139,114),(86,177,147,116),(88,179,149,118),(90,175,145,120)], [(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)], [(1,6,185,184),(2,183,186,5),(3,4,181,182),(7,8,121,122),(9,12,123,126),(10,125,124,11),(13,187,188,18),(14,17,189,192),(15,191,190,16),(19,193,194,24),(20,23,195,198),(21,197,196,22),(25,199,200,30),(26,29,201,204),(27,203,202,28),(31,205,206,36),(32,35,207,210),(33,209,208,34),(37,42,212,211),(38,216,213,41),(39,40,214,215),(43,48,218,217),(44,222,219,47),(45,46,220,221),(49,54,224,223),(50,228,225,53),(51,52,226,227),(55,60,230,229),(56,234,231,59),(57,58,232,233),(61,66,236,235),(62,240,237,65),(63,64,238,239),(67,127,128,72),(68,71,129,132),(69,131,130,70),(73,133,134,78),(74,77,135,138),(75,137,136,76),(79,139,140,84),(80,83,141,144),(81,143,142,82),(85,145,146,90),(86,89,147,150),(87,149,148,88),(91,151,152,96),(92,95,153,156),(93,155,154,94),(97,102,158,157),(98,162,159,101),(99,100,160,161),(103,108,164,163),(104,168,165,107),(105,106,166,167),(109,114,170,169),(110,174,171,113),(111,112,172,173),(115,120,176,175),(116,180,177,119),(117,118,178,179)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 10O | 10P | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 20M | 20N | 20O | 20P | 20Q | 20R | 20S | 20T | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 40I | ··· | 40P | 60A | 60B | 60C | 60D | 60E | 60F | 60G | 60H | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 4 | 6 | 12 | 2 | 2 | 3 | 3 | 4 | 12 | 1 | 1 | 1 | 1 | 2 | 8 | 2 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 4 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | D6 | C5×S3 | C4○D8 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | S3×C10 | C5×C4○D8 | S3×D4 | Q8.7D6 | C5×S3×D4 | C5×Q8.7D6 |
| kernel | C5×Q8.7D6 | S3×C40 | C5×C24⋊C2 | C5×D4⋊S3 | C5×C3⋊Q16 | C15×SD16 | C5×D4⋊2S3 | C5×Q8⋊3S3 | Q8.7D6 | S3×C8 | C24⋊C2 | D4⋊S3 | C3⋊Q16 | C3×SD16 | D4⋊2S3 | Q8⋊3S3 | C5×SD16 | C5×Dic3 | S3×C10 | C40 | C5×D4 | C5×Q8 | SD16 | C15 | Dic3 | D6 | C8 | D4 | Q8 | C3 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 16 | 1 | 2 | 4 | 8 |
Matrix representation of C5×Q8.7D6 ►in GL4(𝔽241) generated by
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 91 | 0 |
| 0 | 0 | 0 | 91 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 177 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 |
| 0 | 1 | 0 | 0 |
| 240 | 240 | 0 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 0 | 233 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 211 |
| 0 | 0 | 233 | 0 |
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,91,0,0,0,0,91],[1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,177],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,0],[0,240,0,0,1,240,0,0,0,0,0,233,0,0,30,0],[0,1,0,0,1,0,0,0,0,0,0,233,0,0,211,0] >;
C5×Q8.7D6 in GAP, Magma, Sage, TeX
C_5\times Q_8._7D_6
% in TeX
G:=Group("C5xQ8.7D6"); // GroupNames label
G:=SmallGroup(480,795);
// by ID
G=gap.SmallGroup(480,795);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,1766,471,436,2111,1068,102,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=d^6=1,c^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b^-1*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d^-1>;
// generators/relations