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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 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4, D4, Q8, Q8 [×3], C10, C10 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5 [×2], C20 [×2], C20, C2×C10, C2×C10, C24 [×2], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C3×Q8 [×3], C30, C30 [×2], C8.C22, C52C8 [×2], Dic10 [×2], Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C3×M4(2), C3×SD16 [×2], C3×Q16 [×2], C6×Q8, C3×C4○D4, C3×Dic5 [×2], C60 [×2], C60, C2×C30, C2×C30, C4.Dic5, D4.D5 [×2], C5⋊Q16 [×2], C2×Dic10, C5×C4○D4, C3×C8.C22, C3×C52C8 [×2], C3×Dic10 [×2], C3×Dic10, C6×Dic5, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, D4.9D10, C3×C4.Dic5, C3×D4.D5 [×2], C3×C5⋊Q16 [×2], C6×Dic10, C15×C4○D4, C3×D4.9D10
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, C8.C22, C5⋊D4 [×2], C22×D5, C6×D4, C6×D5 [×3], C2×C5⋊D4, C3×C8.C22, C3×C5⋊D4 [×2], 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 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 76 115)(62 77 111)(63 78 112)(64 79 113)(65 80 114)(66 71 120)(67 72 116)(68 73 117)(69 74 118)(70 75 119)(81 95 102)(82 91 103)(83 92 104)(84 93 105)(85 94 101)(86 97 107)(87 98 108)(88 99 109)(89 100 110)(90 96 106)(121 163 235)(122 164 236)(123 165 237)(124 166 238)(125 167 239)(126 168 240)(127 169 231)(128 170 232)(129 161 233)(130 162 234)(131 149 159)(132 150 160)(133 141 151)(134 142 152)(135 143 153)(136 144 154)(137 145 155)(138 146 156)(139 147 157)(140 148 158)(171 215 195)(172 216 196)(173 217 197)(174 218 198)(175 219 199)(176 220 200)(177 211 191)(178 212 192)(179 213 193)(180 214 194)(181 221 201)(182 222 202)(183 223 203)(184 224 204)(185 225 205)(186 226 206)(187 227 207)(188 228 208)(189 229 209)(190 230 210)
(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 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 101 68 108)(62 102 69 109)(63 103 70 110)(64 104 66 106)(65 105 67 107)(71 90 79 83)(72 86 80 84)(73 87 76 85)(74 88 77 81)(75 89 78 82)(91 119 100 112)(92 120 96 113)(93 116 97 114)(94 117 98 115)(95 118 99 111)(121 141 126 146)(122 142 127 147)(123 143 128 148)(124 144 129 149)(125 145 130 150)(131 238 136 233)(132 239 137 234)(133 240 138 235)(134 231 139 236)(135 232 140 237)(151 168 156 163)(152 169 157 164)(153 170 158 165)(154 161 159 166)(155 162 160 167)(171 190 176 185)(172 181 177 186)(173 182 178 187)(174 183 179 188)(175 184 180 189)(191 206 196 201)(192 207 197 202)(193 208 198 203)(194 209 199 204)(195 210 200 205)(211 226 216 221)(212 227 217 222)(213 228 218 223)(214 229 219 224)(215 230 220 225)
(1 130)(2 126)(3 122)(4 128)(5 124)(6 129)(7 125)(8 121)(9 127)(10 123)(11 144)(12 150)(13 146)(14 142)(15 148)(16 149)(17 145)(18 141)(19 147)(20 143)(21 238)(22 234)(23 240)(24 236)(25 232)(26 233)(27 239)(28 235)(29 231)(30 237)(31 136)(32 132)(33 138)(34 134)(35 140)(36 131)(37 137)(38 133)(39 139)(40 135)(41 166)(42 162)(43 168)(44 164)(45 170)(46 161)(47 167)(48 163)(49 169)(50 165)(51 154)(52 160)(53 156)(54 152)(55 158)(56 159)(57 155)(58 151)(59 157)(60 153)(61 192)(62 198)(63 194)(64 200)(65 196)(66 195)(67 191)(68 197)(69 193)(70 199)(71 171)(72 177)(73 173)(74 179)(75 175)(76 178)(77 174)(78 180)(79 176)(80 172)(81 188)(82 184)(83 190)(84 186)(85 182)(86 181)(87 187)(88 183)(89 189)(90 185)(91 224)(92 230)(93 226)(94 222)(95 228)(96 225)(97 221)(98 227)(99 223)(100 229)(101 202)(102 208)(103 204)(104 210)(105 206)(106 205)(107 201)(108 207)(109 203)(110 209)(111 218)(112 214)(113 220)(114 216)(115 212)(116 211)(117 217)(118 213)(119 219)(120 215)
(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 73 7 76)(2 72 8 80)(3 71 9 79)(4 75 10 78)(5 74 6 77)(11 88 16 81)(12 87 17 85)(13 86 18 84)(14 90 19 83)(15 89 20 82)(21 69 26 62)(22 68 27 61)(23 67 28 65)(24 66 29 64)(25 70 30 63)(31 109 36 102)(32 108 37 101)(33 107 38 105)(34 106 39 104)(35 110 40 103)(41 118 46 111)(42 117 47 115)(43 116 48 114)(44 120 49 113)(45 119 50 112)(51 99 56 95)(52 98 57 94)(53 97 58 93)(54 96 59 92)(55 100 60 91)(121 181 126 186)(122 190 127 185)(123 189 128 184)(124 188 129 183)(125 187 130 182)(131 198 136 193)(132 197 137 192)(133 196 138 191)(134 195 139 200)(135 194 140 199)(141 172 146 177)(142 171 147 176)(143 180 148 175)(144 179 149 174)(145 178 150 173)(151 216 156 211)(152 215 157 220)(153 214 158 219)(154 213 159 218)(155 212 160 217)(161 223 166 228)(162 222 167 227)(163 221 168 226)(164 230 169 225)(165 229 170 224)(201 240 206 235)(202 239 207 234)(203 238 208 233)(204 237 209 232)(205 236 210 231)

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

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

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

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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