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## G = C6×Q8×D5order 480 = 25·3·5

### Direct product of C6, Q8 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C6×Q8×D5
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C2×C6 — D5×C2×C12 — C6×Q8×D5
 Lower central C5 — C10 — C6×Q8×D5
 Upper central C1 — C2×C6 — C6×Q8

Generators and relations for C6×Q8×D5
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 >

Subgroups: 784 in 312 conjugacy classes, 194 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, Q8, Q8, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C22×C4, C2×Q8, C2×Q8, Dic5, C20, D10, C2×C10, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×C6, C3×D5, C30, C30, C22×Q8, Dic10, C4×D5, C2×Dic5, C2×C20, C5×Q8, C22×D5, C22×C12, C6×Q8, C6×Q8, C3×Dic5, C60, C6×D5, C2×C30, C2×Dic10, C2×C4×D5, Q8×D5, Q8×C10, Q8×C2×C6, C3×Dic10, D5×C12, C6×Dic5, C2×C60, Q8×C15, D5×C2×C6, C2×Q8×D5, C6×Dic10, D5×C2×C12, C3×Q8×D5, Q8×C30, C6×Q8×D5
Quotients: C1, C2, C3, C22, C6, Q8, C23, D5, C2×C6, C2×Q8, C24, D10, C3×Q8, C22×C6, C3×D5, C22×Q8, C22×D5, C6×Q8, C23×C6, C6×D5, Q8×D5, C23×D5, Q8×C2×C6, D5×C2×C6, C2×Q8×D5, C3×Q8×D5, D5×C22×C6, C6×Q8×D5

Smallest permutation representation of C6×Q8×D5
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)
(1 217 187 175)(2 218 188 176)(3 219 189 177)(4 220 190 178)(5 221 191 179)(6 222 192 180)(7 216 150 158)(8 211 145 159)(9 212 146 160)(10 213 147 161)(11 214 148 162)(12 215 149 157)(13 231 201 59)(14 232 202 60)(15 233 203 55)(16 234 204 56)(17 229 199 57)(18 230 200 58)(19 92 194 164)(20 93 195 165)(21 94 196 166)(22 95 197 167)(23 96 198 168)(24 91 193 163)(25 237 207 41)(26 238 208 42)(27 239 209 37)(28 240 210 38)(29 235 205 39)(30 236 206 40)(31 130 109 61)(32 131 110 62)(33 132 111 63)(34 127 112 64)(35 128 113 65)(36 129 114 66)(43 117 87 73)(44 118 88 74)(45 119 89 75)(46 120 90 76)(47 115 85 77)(48 116 86 78)(49 142 105 79)(50 143 106 80)(51 144 107 81)(52 139 108 82)(53 140 103 83)(54 141 104 84)(67 169 133 97)(68 170 134 98)(69 171 135 99)(70 172 136 100)(71 173 137 101)(72 174 138 102)(121 224 182 152)(122 225 183 153)(123 226 184 154)(124 227 185 155)(125 228 186 156)(126 223 181 151)
(1 104 187 54)(2 105 188 49)(3 106 189 50)(4 107 190 51)(5 108 191 52)(6 103 192 53)(7 75 150 119)(8 76 145 120)(9 77 146 115)(10 78 147 116)(11 73 148 117)(12 74 149 118)(13 228 201 156)(14 223 202 151)(15 224 203 152)(16 225 204 153)(17 226 199 154)(18 227 200 155)(19 173 194 101)(20 174 195 102)(21 169 196 97)(22 170 197 98)(23 171 198 99)(24 172 193 100)(25 110 207 32)(26 111 208 33)(27 112 209 34)(28 113 210 35)(29 114 205 36)(30 109 206 31)(37 64 239 127)(38 65 240 128)(39 66 235 129)(40 61 236 130)(41 62 237 131)(42 63 238 132)(43 214 87 162)(44 215 88 157)(45 216 89 158)(46 211 90 159)(47 212 85 160)(48 213 86 161)(55 182 233 121)(56 183 234 122)(57 184 229 123)(58 185 230 124)(59 186 231 125)(60 181 232 126)(67 166 133 94)(68 167 134 95)(69 168 135 96)(70 163 136 91)(71 164 137 92)(72 165 138 93)(79 218 142 176)(80 219 143 177)(81 220 144 178)(82 221 139 179)(83 222 140 180)(84 217 141 175)
(1 29 71 17 89)(2 30 72 18 90)(3 25 67 13 85)(4 26 68 14 86)(5 27 69 15 87)(6 28 70 16 88)(7 84 66 19 184)(8 79 61 20 185)(9 80 62 21 186)(10 81 63 22 181)(11 82 64 23 182)(12 83 65 24 183)(31 93 155 211 49)(32 94 156 212 50)(33 95 151 213 51)(34 96 152 214 52)(35 91 153 215 53)(36 92 154 216 54)(37 99 55 117 179)(38 100 56 118 180)(39 101 57 119 175)(40 102 58 120 176)(41 97 59 115 177)(42 98 60 116 178)(43 191 209 135 203)(44 192 210 136 204)(45 187 205 137 199)(46 188 206 138 200)(47 189 207 133 201)(48 190 208 134 202)(73 221 239 171 233)(74 222 240 172 234)(75 217 235 173 229)(76 218 236 174 230)(77 219 237 169 231)(78 220 238 170 232)(103 113 163 225 157)(104 114 164 226 158)(105 109 165 227 159)(106 110 166 228 160)(107 111 167 223 161)(108 112 168 224 162)(121 148 139 127 198)(122 149 140 128 193)(123 150 141 129 194)(124 145 142 130 195)(125 146 143 131 196)(126 147 144 132 197)
(1 48)(2 43)(3 44)(4 45)(5 46)(6 47)(7 144)(8 139)(9 140)(10 141)(11 142)(12 143)(13 210)(14 205)(15 206)(16 207)(17 208)(18 209)(19 197)(20 198)(21 193)(22 194)(23 195)(24 196)(25 204)(26 199)(27 200)(28 201)(29 202)(30 203)(31 224)(32 225)(33 226)(34 227)(35 228)(36 223)(37 230)(38 231)(39 232)(40 233)(41 234)(42 229)(49 162)(50 157)(51 158)(52 159)(53 160)(54 161)(55 236)(56 237)(57 238)(58 239)(59 240)(60 235)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 136)(68 137)(69 138)(70 133)(71 134)(72 135)(73 176)(74 177)(75 178)(76 179)(77 180)(78 175)(79 148)(80 149)(81 150)(82 145)(83 146)(84 147)(85 192)(86 187)(87 188)(88 189)(89 190)(90 191)(91 166)(92 167)(93 168)(94 163)(95 164)(96 165)(97 172)(98 173)(99 174)(100 169)(101 170)(102 171)(103 212)(104 213)(105 214)(106 215)(107 216)(108 211)(109 152)(110 153)(111 154)(112 155)(113 156)(114 151)(115 222)(116 217)(117 218)(118 219)(119 220)(120 221)(127 185)(128 186)(129 181)(130 182)(131 183)(132 184)

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

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

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

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

Matrix representation of C6×Q8×D5 in 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] >;

C6×Q8×D5 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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