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## G = C3×D4.8D10order 480 = 25·3·5

### Direct product of C3 and D4.8D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D4.8D10
 Chief series C1 — C5 — C10 — C20 — C60 — C3×D20 — C3×C4○D20 — C3×D4.8D10
 Lower central C5 — C10 — C20 — C3×D4.8D10
 Upper central C1 — C12 — C2×C12 — C3×C4○D4

Generators and relations for C3×D4.8D10
G = < a,b,c,d,e | a3=b4=c2=1, d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d9 >

Subgroups: 368 in 124 conjugacy classes, 58 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, C6, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8, D5, C10, C10 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C15, C2×C8, D8, SD16 [×2], Q16, C4○D4, C4○D4, Dic5, C20 [×2], C20, D10, C2×C10, C2×C10, C24 [×2], C2×C12, C2×C12 [×2], C3×D4, C3×D4 [×3], C3×Q8, C3×Q8, C3×D5, C30, C30 [×2], C4○D8, C52C8 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×C24, C3×D8, C3×SD16 [×2], C3×Q16, C3×C4○D4, C3×C4○D4, C3×Dic5, C60 [×2], C60, C6×D5, C2×C30, C2×C30, C2×C52C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, C3×C4○D8, C3×C52C8 [×2], C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, D4.8D10, C6×C52C8, C3×D4⋊D5, C3×D4.D5, C3×Q8⋊D5, C3×C5⋊Q16, C3×C4○D20, C15×C4○D4, C3×D4.8D10
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], C2×D4, D10 [×3], C3×D4 [×2], C22×C6, C3×D5, C4○D8, C5⋊D4 [×2], C22×D5, C6×D4, C6×D5 [×3], C2×C5⋊D4, C3×C4○D8, C3×C5⋊D4 [×2], D5×C2×C6, D4.8D10, C6×C5⋊D4, C3×D4.8D10

Smallest permutation representation of C3×D4.8D10
On 240 points
Generators in S240
(1 56 39)(2 57 40)(3 58 21)(4 59 22)(5 60 23)(6 41 24)(7 42 25)(8 43 26)(9 44 27)(10 45 28)(11 46 29)(12 47 30)(13 48 31)(14 49 32)(15 50 33)(16 51 34)(17 52 35)(18 53 36)(19 54 37)(20 55 38)(61 118 83)(62 119 84)(63 120 85)(64 101 86)(65 102 87)(66 103 88)(67 104 89)(68 105 90)(69 106 91)(70 107 92)(71 108 93)(72 109 94)(73 110 95)(74 111 96)(75 112 97)(76 113 98)(77 114 99)(78 115 100)(79 116 81)(80 117 82)(121 176 156)(122 177 157)(123 178 158)(124 179 159)(125 180 160)(126 161 141)(127 162 142)(128 163 143)(129 164 144)(130 165 145)(131 166 146)(132 167 147)(133 168 148)(134 169 149)(135 170 150)(136 171 151)(137 172 152)(138 173 153)(139 174 154)(140 175 155)(181 228 209)(182 229 210)(183 230 211)(184 231 212)(185 232 213)(186 233 214)(187 234 215)(188 235 216)(189 236 217)(190 237 218)(191 238 219)(192 239 220)(193 240 201)(194 221 202)(195 222 203)(196 223 204)(197 224 205)(198 225 206)(199 226 207)(200 227 208)
(1 16 11 6)(2 17 12 7)(3 18 13 8)(4 19 14 9)(5 20 15 10)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)(41 56 51 46)(42 57 52 47)(43 58 53 48)(44 59 54 49)(45 60 55 50)(61 66 71 76)(62 67 72 77)(63 68 73 78)(64 69 74 79)(65 70 75 80)(81 86 91 96)(82 87 92 97)(83 88 93 98)(84 89 94 99)(85 90 95 100)(101 106 111 116)(102 107 112 117)(103 108 113 118)(104 109 114 119)(105 110 115 120)(121 126 131 136)(122 127 132 137)(123 128 133 138)(124 129 134 139)(125 130 135 140)(141 146 151 156)(142 147 152 157)(143 148 153 158)(144 149 154 159)(145 150 155 160)(161 166 171 176)(162 167 172 177)(163 168 173 178)(164 169 174 179)(165 170 175 180)(181 196 191 186)(182 197 192 187)(183 198 193 188)(184 199 194 189)(185 200 195 190)(201 216 211 206)(202 217 212 207)(203 218 213 208)(204 219 214 209)(205 220 215 210)(221 236 231 226)(222 237 232 227)(223 238 233 228)(224 239 234 229)(225 240 235 230)
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 61)(20 62)(21 87)(22 88)(23 89)(24 90)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 97)(32 98)(33 99)(34 100)(35 81)(36 82)(37 83)(38 84)(39 85)(40 86)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(57 101)(58 102)(59 103)(60 104)(121 190)(122 191)(123 192)(124 193)(125 194)(126 195)(127 196)(128 197)(129 198)(130 199)(131 200)(132 181)(133 182)(134 183)(135 184)(136 185)(137 186)(138 187)(139 188)(140 189)(141 203)(142 204)(143 205)(144 206)(145 207)(146 208)(147 209)(148 210)(149 211)(150 212)(151 213)(152 214)(153 215)(154 216)(155 217)(156 218)(157 219)(158 220)(159 201)(160 202)(161 222)(162 223)(163 224)(164 225)(165 226)(166 227)(167 228)(168 229)(169 230)(170 231)(171 232)(172 233)(173 234)(174 235)(175 236)(176 237)(177 238)(178 239)(179 240)(180 221)
(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)
(1 129 11 139)(2 138 12 128)(3 127 13 137)(4 136 14 126)(5 125 15 135)(6 134 16 124)(7 123 17 133)(8 132 18 122)(9 121 19 131)(10 130 20 140)(21 142 31 152)(22 151 32 141)(23 160 33 150)(24 149 34 159)(25 158 35 148)(26 147 36 157)(27 156 37 146)(28 145 38 155)(29 154 39 144)(30 143 40 153)(41 169 51 179)(42 178 52 168)(43 167 53 177)(44 176 54 166)(45 165 55 175)(46 174 56 164)(47 163 57 173)(48 172 58 162)(49 161 59 171)(50 170 60 180)(61 185 71 195)(62 194 72 184)(63 183 73 193)(64 192 74 182)(65 181 75 191)(66 190 76 200)(67 199 77 189)(68 188 78 198)(69 197 79 187)(70 186 80 196)(81 215 91 205)(82 204 92 214)(83 213 93 203)(84 202 94 212)(85 211 95 201)(86 220 96 210)(87 209 97 219)(88 218 98 208)(89 207 99 217)(90 216 100 206)(101 239 111 229)(102 228 112 238)(103 237 113 227)(104 226 114 236)(105 235 115 225)(106 224 116 234)(107 233 117 223)(108 222 118 232)(109 231 119 221)(110 240 120 230)

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

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

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

102 conjugacy classes

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

102 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 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 D4 D4 D5 D10 D10 D10 C3×D4 C3×D4 C3×D5 C4○D8 C5⋊D4 C5⋊D4 C6×D5 C6×D5 C6×D5 C3×C4○D8 C3×C5⋊D4 C3×C5⋊D4 D4.8D10 C3×D4.8D10 kernel C3×D4.8D10 C6×C5⋊2C8 C3×D4⋊D5 C3×D4.D5 C3×Q8⋊D5 C3×C5⋊Q16 C3×C4○D20 C15×C4○D4 D4.8D10 C2×C5⋊2C8 D4⋊D5 D4.D5 Q8⋊D5 C5⋊Q16 C4○D20 C5×C4○D4 C60 C2×C30 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C20 C2×C10 C4○D4 C15 C12 C2×C6 C2×C4 D4 Q8 C5 C4 C22 C3 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8 8 4 8

Matrix representation of C3×D4.8D10 in GL6(𝔽241)

 225 0 0 0 0 0 0 225 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 1
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 64 0 0 0 0 0 64 177
,
 17 2 0 0 0 0 97 224 0 0 0 0 0 0 214 192 0 0 0 0 187 27 0 0 0 0 0 0 177 128 0 0 0 0 177 64
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 52 190 0 0 0 0 52 0 0 0 0 0 0 0 177 0 0 0 0 0 0 177
,
 206 209 0 0 0 0 219 35 0 0 0 0 0 0 199 9 0 0 0 0 179 42 0 0 0 0 0 0 30 181 0 0 0 0 11 211

G:=sub<GL(6,GF(241))| [225,0,0,0,0,0,0,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,64,64,0,0,0,0,0,177],[17,97,0,0,0,0,2,224,0,0,0,0,0,0,214,187,0,0,0,0,192,27,0,0,0,0,0,0,177,177,0,0,0,0,128,64],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,52,52,0,0,0,0,190,0,0,0,0,0,0,0,177,0,0,0,0,0,0,177],[206,219,0,0,0,0,209,35,0,0,0,0,0,0,199,179,0,0,0,0,9,42,0,0,0,0,0,0,30,11,0,0,0,0,181,211] >;

C3×D4.8D10 in GAP, Magma, Sage, TeX

C_3\times D_4._8D_{10}
% in TeX

G:=Group("C3xD4.8D10");
// GroupNames label

G:=SmallGroup(480,743);
// by ID

G=gap.SmallGroup(480,743);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,2524,648,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=c^2=1,d^10=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^9>;
// generators/relations

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