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## G = C5×Q8.14D6order 480 = 25·3·5

### Direct product of C5 and Q8.14D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×Q8.14D6
 Chief series C1 — C3 — C6 — C12 — C60 — C5×Dic6 — C10×Dic6 — C5×Q8.14D6
 Lower central C3 — C6 — C12 — C5×Q8.14D6
 Upper central C1 — C10 — C2×C20 — C5×C4○D4

Generators and relations for C5×Q8.14D6
G = < a,b,c,d,e | a5=b4=d6=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=bc, ede-1=d-1 >

Subgroups: 244 in 120 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4, D4, Q8, Q8 [×3], C10, C10 [×2], Dic3 [×2], C12 [×2], C12, C2×C6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C20 [×2], C20 [×3], C2×C10, C2×C10, C3⋊C8 [×2], Dic6 [×2], Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C30, C30 [×2], C8.C22, C40 [×2], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C5×Q8 [×3], C4.Dic3, D4.S3 [×2], C3⋊Q16 [×2], C2×Dic6, C3×C4○D4, C5×Dic3 [×2], C60 [×2], C60, C2×C30, C2×C30, C5×M4(2), C5×SD16 [×2], C5×Q16 [×2], Q8×C10, C5×C4○D4, Q8.14D6, C5×C3⋊C8 [×2], C5×Dic6 [×2], C5×Dic6, C10×Dic3, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, C5×C8.C22, C5×C4.Dic3, C5×D4.S3 [×2], C5×C3⋊Q16 [×2], C10×Dic6, C15×C4○D4, C5×Q8.14D6
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], C3⋊D4 [×2], C22×S3, C5×S3, C8.C22, C5×D4 [×2], C22×C10, C2×C3⋊D4, S3×C10 [×3], D4×C10, Q8.14D6, C5×C3⋊D4 [×2], S3×C2×C10, C5×C8.C22, C10×C3⋊D4, C5×Q8.14D6

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

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

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

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

105 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 5A 5B 5C 5D 6A 6B 6C 6D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 12A 12B 12C 12D 12E 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 20M ··· 20T 30A 30B 30C 30D 30E ··· 30P 40A ··· 40H 60A ··· 60H 60I ··· 60T order 1 2 2 2 3 4 4 4 4 4 5 5 5 5 6 6 6 6 8 8 10 10 10 10 10 10 10 10 10 10 10 10 12 12 12 12 12 15 15 15 15 20 ··· 20 20 20 20 20 20 ··· 20 30 30 30 30 30 ··· 30 40 ··· 40 60 ··· 60 60 ··· 60 size 1 1 2 4 2 2 2 4 12 12 1 1 1 1 2 4 4 4 12 12 1 1 1 1 2 2 2 2 4 4 4 4 2 2 4 4 4 2 2 2 2 2 ··· 2 4 4 4 4 12 ··· 12 2 2 2 2 4 ··· 4 12 ··· 12 2 ··· 2 4 ··· 4

105 irreducible representations

 dim 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 2 2 4 4 4 4 type + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 S3 D4 D4 D6 D6 D6 C3⋊D4 C3⋊D4 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 S3×C10 C5×C3⋊D4 C5×C3⋊D4 C8.C22 Q8.14D6 C5×C8.C22 C5×Q8.14D6 kernel C5×Q8.14D6 C5×C4.Dic3 C5×D4.S3 C5×C3⋊Q16 C10×Dic6 C15×C4○D4 Q8.14D6 C4.Dic3 D4.S3 C3⋊Q16 C2×Dic6 C3×C4○D4 C5×C4○D4 C60 C2×C30 C2×C20 C5×D4 C5×Q8 C20 C2×C10 C4○D4 C12 C2×C6 C2×C4 D4 Q8 C4 C22 C15 C5 C3 C1 # reps 1 1 2 2 1 1 4 4 8 8 4 4 1 1 1 1 1 1 2 2 4 4 4 4 4 4 8 8 1 2 4 8

Matrix representation of C5×Q8.14D6 in GL4(𝔽241) generated by

 91 0 0 0 0 91 0 0 0 0 91 0 0 0 0 91
,
 99 43 0 0 198 142 0 0 0 43 99 198 198 0 43 142
,
 0 0 1 240 240 240 2 1 160 160 1 0 161 160 1 0
,
 240 1 0 0 240 0 0 0 80 81 0 240 79 80 1 1
,
 5 19 0 0 24 236 0 0 124 119 220 160 103 98 181 21
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,91,0,0,0,0,91],[99,198,0,198,43,142,43,0,0,0,99,43,0,0,198,142],[0,240,160,161,0,240,160,160,1,2,1,1,240,1,0,0],[240,240,80,79,1,0,81,80,0,0,0,1,0,0,240,1],[5,24,124,103,19,236,119,98,0,0,220,181,0,0,160,21] >;

C5×Q8.14D6 in GAP, Magma, Sage, TeX

C_5\times Q_8._{14}D_6
% in TeX

G:=Group("C5xQ8.14D6");
// GroupNames label

G:=SmallGroup(480,830);
// by ID

G=gap.SmallGroup(480,830);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,926,891,4204,1068,102,15686]);
// 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=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations

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