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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, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C30, C30, C8.C22, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C4.Dic3, D4.S3, C3⋊Q16, C2×Dic6, C3×C4○D4, C5×Dic3, C60, C60, C2×C30, C2×C30, C5×M4(2), C5×SD16, C5×Q16, Q8×C10, C5×C4○D4, Q8.14D6, C5×C3⋊C8, C5×Dic6, C5×Dic6, C10×Dic3, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, C5×C8.C22, C5×C4.Dic3, C5×D4.S3, C5×C3⋊Q16, C10×Dic6, C15×C4○D4, C5×Q8.14D6
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C8.C22, C5×D4, C22×C10, C2×C3⋊D4, S3×C10, D4×C10, Q8.14D6, C5×C3⋊D4, 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 51 39 66 12)(2 49 37 64 10)(3 50 38 65 11)(4 55 43 31 16)(5 56 44 32 17)(6 57 45 33 18)(7 58 46 34 19)(8 59 47 35 20)(9 60 48 36 21)(13 89 52 40 28)(14 90 53 41 29)(15 88 54 42 30)(22 110 100 62 76)(23 111 101 63 77)(24 109 102 61 78)(25 96 84 72 115)(26 94 82 70 116)(27 95 83 71 117)(67 112 103 91 79)(68 113 104 92 80)(69 114 105 93 81)(73 118 106 97 85)(74 119 107 98 86)(75 120 108 99 87)(121 148 173 151 133)(122 149 174 152 134)(123 150 169 153 135)(124 145 170 154 136)(125 146 171 155 137)(126 147 172 156 138)(127 230 179 157 139)(128 231 180 158 140)(129 232 175 159 141)(130 233 176 160 142)(131 234 177 161 143)(132 229 178 162 144)(163 225 213 201 189)(164 226 214 202 190)(165 227 215 203 191)(166 228 216 204 192)(167 223 211 199 187)(168 224 212 200 188)(181 235 217 205 193)(182 236 218 206 194)(183 237 219 207 195)(184 238 220 208 196)(185 239 221 209 197)(186 240 222 210 198)
(1 57 60 54)(2 55 58 52)(3 56 59 53)(4 7 89 10)(5 8 90 11)(6 9 88 12)(13 64 16 19)(14 65 17 20)(15 66 18 21)(22 70 69 75)(23 71 67 73)(24 72 68 74)(25 104 107 102)(26 105 108 100)(27 103 106 101)(28 37 31 34)(29 38 32 35)(30 39 33 36)(40 49 43 46)(41 50 44 47)(42 51 45 48)(61 96 92 98)(62 94 93 99)(63 95 91 97)(76 82 81 87)(77 83 79 85)(78 84 80 86)(109 115 113 119)(110 116 114 120)(111 117 112 118)(121 233 124 230)(122 234 125 231)(123 229 126 232)(127 133 130 136)(128 134 131 137)(129 135 132 138)(139 151 142 154)(140 152 143 155)(141 153 144 156)(145 179 148 176)(146 180 149 177)(147 175 150 178)(157 173 160 170)(158 174 161 171)(159 169 162 172)(163 186 166 183)(164 181 167 184)(165 182 168 185)(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)(223 238 226 235)(224 239 227 236)(225 240 228 237)
(1 147 60 150)(2 145 58 148)(3 149 59 146)(4 233 89 230)(5 231 90 234)(6 229 88 232)(7 121 10 124)(8 125 11 122)(9 123 12 126)(13 127 16 130)(14 131 17 128)(15 129 18 132)(19 133 64 136)(20 137 65 134)(21 135 66 138)(22 181 69 184)(23 185 67 182)(24 183 68 186)(25 216 107 213)(26 214 108 211)(27 212 106 215)(28 139 31 142)(29 143 32 140)(30 141 33 144)(34 151 37 154)(35 155 38 152)(36 153 39 156)(40 157 43 160)(41 161 44 158)(42 159 45 162)(46 173 49 170)(47 171 50 174)(48 169 51 172)(52 179 55 176)(53 177 56 180)(54 175 57 178)(61 207 92 210)(62 205 93 208)(63 209 91 206)(70 164 75 167)(71 168 73 165)(72 166 74 163)(76 193 81 196)(77 197 79 194)(78 195 80 198)(82 190 87 187)(83 188 85 191)(84 192 86 189)(94 202 99 199)(95 200 97 203)(96 204 98 201)(100 217 105 220)(101 221 103 218)(102 219 104 222)(109 237 113 240)(110 235 114 238)(111 239 112 236)(115 228 119 225)(116 226 120 223)(117 224 118 227)
(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)
(1 117 60 118)(2 116 58 120)(3 115 59 119)(4 22 89 69)(5 24 90 68)(6 23 88 67)(7 75 10 70)(8 74 11 72)(9 73 12 71)(13 81 16 76)(14 80 17 78)(15 79 18 77)(19 87 64 82)(20 86 65 84)(21 85 66 83)(25 47 107 50)(26 46 108 49)(27 48 106 51)(28 93 31 62)(29 92 32 61)(30 91 33 63)(34 99 37 94)(35 98 38 96)(36 97 39 95)(40 105 43 100)(41 104 44 102)(42 103 45 101)(52 114 55 110)(53 113 56 109)(54 112 57 111)(121 181 124 184)(122 186 125 183)(123 185 126 182)(127 187 130 190)(128 192 131 189)(129 191 132 188)(133 193 136 196)(134 198 137 195)(135 197 138 194)(139 199 142 202)(140 204 143 201)(141 203 144 200)(145 238 148 235)(146 237 149 240)(147 236 150 239)(151 205 154 208)(152 210 155 207)(153 209 156 206)(157 211 160 214)(158 216 161 213)(159 215 162 212)(163 231 166 234)(164 230 167 233)(165 229 168 232)(169 221 172 218)(170 220 173 217)(171 219 174 222)(175 227 178 224)(176 226 179 223)(177 225 180 228)

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

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

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

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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