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## G = C5×Q8.7D6order 480 = 25·3·5

### Direct product of C5 and Q8.7D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×Q8.7D6
 Chief series C1 — C3 — C6 — C12 — C60 — S3×C20 — C5×D4⋊2S3 — C5×Q8.7D6
 Lower central C3 — C6 — C12 — C5×Q8.7D6
 Upper central C1 — C10 — C20 — C5×SD16

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 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×2], C6, C6, C8, C8, C2×C4 [×3], D4, D4 [×3], Q8, Q8, C10, C10 [×3], Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, D8, SD16, SD16, Q16, C4○D4 [×2], C20, C20 [×3], C2×C10 [×3], C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C5×S3 [×2], C30, C30, C4○D8, C40, C40, C2×C20 [×3], C5×D4, C5×D4 [×3], C5×Q8, C5×Q8, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D42S3, Q83S3, 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 [×2], 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×D42S3, C5×Q83S3, C5×Q8.7D6
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], C22×S3, C5×S3, C4○D8, C5×D4 [×2], C22×C10, S3×D4, S3×C10 [×3], D4×C10, Q8.7D6, S3×C2×C10, C5×C4○D8, C5×S3×D4, C5×Q8.7D6

Smallest permutation representation of C5×Q8.7D6
On 240 points
Generators in S240
(1 32 26 20 14)(2 33 27 21 15)(3 34 28 22 16)(4 35 29 23 17)(5 36 30 24 18)(6 31 25 19 13)(7 176 170 164 158)(8 177 171 165 159)(9 178 172 166 160)(10 179 173 167 161)(11 180 174 168 162)(12 175 169 163 157)(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 70 185 129)(2 130 186 71)(3 72 181 131)(4 132 182 67)(5 68 183 127)(6 128 184 69)(7 61 121 238)(8 239 122 62)(9 63 123 240)(10 235 124 64)(11 65 125 236)(12 237 126 66)(13 134 190 75)(14 76 191 135)(15 136 192 77)(16 78 187 137)(17 138 188 73)(18 74 189 133)(19 140 196 81)(20 82 197 141)(21 142 198 83)(22 84 193 143)(23 144 194 79)(24 80 195 139)(25 146 202 87)(26 88 203 147)(27 148 204 89)(28 90 199 149)(29 150 200 85)(30 86 201 145)(31 152 208 93)(32 94 209 153)(33 154 210 95)(34 96 205 155)(35 156 206 91)(36 92 207 151)(37 99 214 158)(38 159 215 100)(39 101 216 160)(40 161 211 102)(41 97 212 162)(42 157 213 98)(43 105 220 164)(44 165 221 106)(45 107 222 166)(46 167 217 108)(47 103 218 168)(48 163 219 104)(49 111 226 170)(50 171 227 112)(51 113 228 172)(52 173 223 114)(53 109 224 174)(54 169 225 110)(55 117 232 176)(56 177 233 118)(57 119 234 178)(58 179 229 120)(59 115 230 180)(60 175 231 116)
(1 215 185 38)(2 101 186 160)(3 211 181 40)(4 97 182 162)(5 213 183 42)(6 99 184 158)(7 31 121 208)(8 153 122 94)(9 33 123 210)(10 155 124 96)(11 35 125 206)(12 151 126 92)(13 105 190 164)(14 221 191 44)(15 107 192 166)(16 217 187 46)(17 103 188 168)(18 219 189 48)(19 111 196 170)(20 227 197 50)(21 113 198 172)(22 223 193 52)(23 109 194 174)(24 225 195 54)(25 117 202 176)(26 233 203 56)(27 119 204 178)(28 229 199 58)(29 115 200 180)(30 231 201 60)(32 239 209 62)(34 235 205 64)(36 237 207 66)(37 69 214 128)(39 71 216 130)(41 67 212 132)(43 75 220 134)(45 77 222 136)(47 73 218 138)(49 81 226 140)(51 83 228 142)(53 79 224 144)(55 87 232 146)(57 89 234 148)(59 85 230 150)(61 93 238 152)(63 95 240 154)(65 91 236 156)(68 157 127 98)(70 159 129 100)(72 161 131 102)(74 163 133 104)(76 165 135 106)(78 167 137 108)(80 169 139 110)(82 171 141 112)(84 173 143 114)(86 175 145 116)(88 177 147 118)(90 179 149 120)
(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)(13 191 190 14)(15 189 192 18)(16 17 187 188)(19 197 196 20)(21 195 198 24)(22 23 193 194)(25 203 202 26)(27 201 204 30)(28 29 199 200)(31 209 208 32)(33 207 210 36)(34 35 205 206)(37 38 214 215)(39 42 216 213)(40 212 211 41)(43 44 220 221)(45 48 222 219)(46 218 217 47)(49 50 226 227)(51 54 228 225)(52 224 223 53)(55 56 232 233)(57 60 234 231)(58 230 229 59)(61 62 238 239)(63 66 240 237)(64 236 235 65)(67 131 132 72)(68 71 127 130)(69 129 128 70)(73 137 138 78)(74 77 133 136)(75 135 134 76)(79 143 144 84)(80 83 139 142)(81 141 140 82)(85 149 150 90)(86 89 145 148)(87 147 146 88)(91 155 156 96)(92 95 151 154)(93 153 152 94)(97 102 162 161)(98 160 157 101)(99 100 158 159)(103 108 168 167)(104 166 163 107)(105 106 164 165)(109 114 174 173)(110 172 169 113)(111 112 170 171)(115 120 180 179)(116 178 175 119)(117 118 176 177)

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

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

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

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

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