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## G = C6×D4.D5order 480 = 25·3·5

### Direct product of C6 and D4.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C6×D4.D5
 Chief series C1 — C5 — C10 — C20 — C60 — C3×Dic10 — C6×Dic10 — C6×D4.D5
 Lower central C5 — C10 — C20 — C6×D4.D5
 Upper central C1 — C2×C6 — C2×C12 — C6×D4

Generators and relations for C6×D4.D5
G = < a,b,c,d,e | a6=b4=c2=d5=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 368 in 136 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C30, C30, C30, C2×SD16, C52C8, Dic10, Dic10, C2×Dic5, C2×C20, C5×D4, C5×D4, C22×C10, C2×C24, C3×SD16, C6×D4, C6×Q8, C3×Dic5, C60, C2×C30, C2×C30, C2×C52C8, D4.D5, C2×Dic10, D4×C10, C6×SD16, C3×C52C8, C3×Dic10, C3×Dic10, C6×Dic5, C2×C60, D4×C15, D4×C15, C22×C30, C2×D4.D5, C6×C52C8, C3×D4.D5, C6×Dic10, D4×C30, C6×D4.D5
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, SD16, C2×D4, D10, C3×D4, C22×C6, C3×D5, C2×SD16, C5⋊D4, C22×D5, C3×SD16, C6×D4, C6×D5, D4.D5, C2×C5⋊D4, C6×SD16, C3×C5⋊D4, D5×C2×C6, C2×D4.D5, C3×D4.D5, C6×C5⋊D4, C6×D4.D5

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

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 5A 5B 6A ··· 6F 6G 6H 6I 6J 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 15C 15D 20A 20B 20C 20D 24A ··· 24H 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 2 3 3 4 4 4 4 5 5 6 ··· 6 6 6 6 6 8 8 8 8 10 ··· 10 10 ··· 10 12 12 12 12 12 12 12 12 15 15 15 15 20 20 20 20 24 ··· 24 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 4 4 1 1 2 2 20 20 2 2 1 ··· 1 4 4 4 4 10 10 10 10 2 ··· 2 4 ··· 4 2 2 2 2 20 20 20 20 2 2 2 2 4 4 4 4 10 ··· 10 2 ··· 2 4 ··· 4 4 ··· 4

102 irreducible representations

 dim 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 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D4 D5 SD16 D10 D10 C3×D4 C3×D4 C3×D5 C5⋊D4 C5⋊D4 C3×SD16 C6×D5 C6×D5 C3×C5⋊D4 C3×C5⋊D4 D4.D5 C3×D4.D5 kernel C6×D4.D5 C6×C5⋊2C8 C3×D4.D5 C6×Dic10 D4×C30 C2×D4.D5 C2×C5⋊2C8 D4.D5 C2×Dic10 D4×C10 C60 C2×C30 C6×D4 C30 C2×C12 C3×D4 C20 C2×C10 C2×D4 C12 C2×C6 C10 C2×C4 D4 C4 C22 C6 C2 # reps 1 1 4 1 1 2 2 8 2 2 1 1 2 4 2 4 2 2 4 4 4 8 4 8 8 8 4 8

Matrix representation of C6×D4.D5 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 225 0 0 0 0 0 0 225
,
 0 1 0 0 0 0 240 0 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 190 240 0 0 0 0 191 240
,
 19 222 0 0 0 0 222 222 0 0 0 0 0 0 0 240 0 0 0 0 240 0 0 0 0 0 0 0 125 17 0 0 0 0 130 116

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,225,0,0,0,0,0,0,225],[0,240,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,190,191,0,0,0,0,240,240],[19,222,0,0,0,0,222,222,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,0,0,0,125,130,0,0,0,0,17,116] >;

C6×D4.D5 in GAP, Magma, Sage, TeX

C_6\times D_4.D_5
% in TeX

G:=Group("C6xD4.D5");
// GroupNames label

G:=SmallGroup(480,726);
// by ID

G=gap.SmallGroup(480,726);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,590,2524,648,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^4=c^2=d^5=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations

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