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## G = C3×D4.9D10order 480 = 25·3·5

### Direct product of C3 and D4.9D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D4.9D10
 Chief series C1 — C5 — C10 — C20 — C60 — C3×Dic10 — C6×Dic10 — C3×D4.9D10
 Lower central C5 — C10 — C20 — C3×D4.9D10
 Upper central C1 — C6 — C2×C12 — C3×C4○D4

Generators and relations for C3×D4.9D10
G = < a,b,c,d,e | a3=b4=c2=d10=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >

Subgroups: 320 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, C12, C12, C2×C6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C30, C30, C8.C22, C52C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C3×M4(2), C3×SD16, C3×Q16, C6×Q8, C3×C4○D4, C3×Dic5, C60, C60, C2×C30, C2×C30, C4.Dic5, D4.D5, C5⋊Q16, C2×Dic10, C5×C4○D4, C3×C8.C22, C3×C52C8, C3×Dic10, C3×Dic10, C6×Dic5, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, D4.9D10, C3×C4.Dic5, C3×D4.D5, C3×C5⋊Q16, C6×Dic10, C15×C4○D4, C3×D4.9D10
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C8.C22, C5⋊D4, C22×D5, C6×D4, C6×D5, C2×C5⋊D4, C3×C8.C22, C3×C5⋊D4, D5×C2×C6, D4.9D10, C6×C5⋊D4, C3×D4.9D10

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

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

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

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

93 conjugacy classes

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

93 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 C3 C6 C6 C6 C6 C6 D4 D4 D5 D10 D10 D10 C3×D4 C3×D4 C3×D5 C5⋊D4 C5⋊D4 C6×D5 C6×D5 C6×D5 C3×C5⋊D4 C3×C5⋊D4 C8.C22 C3×C8.C22 D4.9D10 C3×D4.9D10 kernel C3×D4.9D10 C3×C4.Dic5 C3×D4.D5 C3×C5⋊Q16 C6×Dic10 C15×C4○D4 D4.9D10 C4.Dic5 D4.D5 C5⋊Q16 C2×Dic10 C5×C4○D4 C60 C2×C30 C3×C4○D4 C2×C12 C3×D4 C3×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 2 2 4 4 2 2 1 1 2 2 2 2 2 2 4 4 4 4 4 4 8 8 1 2 4 8

Matrix representation of C3×D4.9D10 in GL4(𝔽241) generated by

 15 0 0 0 0 15 0 0 0 0 15 0 0 0 0 15
,
 41 85 0 0 156 200 0 0 0 0 200 156 0 0 85 41
,
 0 0 200 156 0 0 85 41 41 85 0 0 156 200 0 0
,
 240 189 0 0 52 52 0 0 0 0 1 52 0 0 189 189
,
 85 79 0 0 238 156 0 0 0 0 97 111 0 0 128 144
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[41,156,0,0,85,200,0,0,0,0,200,85,0,0,156,41],[0,0,41,156,0,0,85,200,200,85,0,0,156,41,0,0],[240,52,0,0,189,52,0,0,0,0,1,189,0,0,52,189],[85,238,0,0,79,156,0,0,0,0,97,128,0,0,111,144] >;

C3×D4.9D10 in GAP, Magma, Sage, TeX

C_3\times D_4._9D_{10}
% in TeX

G:=Group("C3xD4.9D10");
// GroupNames label

G:=SmallGroup(480,744);
// by ID

G=gap.SmallGroup(480,744);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,590,555,2524,648,102,18822]);
// Polycyclic

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

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