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## G = C3×Q8.10D10order 480 = 25·3·5

### Direct product of C3 and Q8.10D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×Q8.10D10
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C12 — C3×Q8×D5 — C3×Q8.10D10
 Lower central C5 — C10 — C3×Q8.10D10
 Upper central C1 — C6 — C6×Q8

Generators and relations for C3×Q8.10D10
G = < a,b,c,d,e | a3=b4=1, c2=d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=ece-1=b2c, ede-1=d9 >

Subgroups: 784 in 292 conjugacy classes, 170 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, Q8, Q8, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×Q8, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C3×D5, C30, C30, 2- 1+4, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×Q8, C6×Q8, C6×Q8, C3×C4○D4, C3×Dic5, C60, C6×D5, C2×C30, C4○D20, Q8×D5, Q82D5, Q8×C10, C3×2- 1+4, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, Q8×C15, Q8.10D10, C3×C4○D20, C3×Q8×D5, C3×Q82D5, Q8×C30, C3×Q8.10D10
Quotients: C1, C2, C3, C22, C6, C23, D5, C2×C6, C24, D10, C22×C6, C3×D5, 2- 1+4, C22×D5, C23×C6, C6×D5, C23×D5, C3×2- 1+4, D5×C2×C6, Q8.10D10, D5×C22×C6, C3×Q8.10D10

Smallest permutation representation of C3×Q8.10D10
On 240 points
Generators in S240
(1 137 65)(2 138 66)(3 139 67)(4 140 68)(5 121 69)(6 122 70)(7 123 71)(8 124 72)(9 125 73)(10 126 74)(11 127 75)(12 128 76)(13 129 77)(14 130 78)(15 131 79)(16 132 80)(17 133 61)(18 134 62)(19 135 63)(20 136 64)(21 110 209)(22 111 210)(23 112 211)(24 113 212)(25 114 213)(26 115 214)(27 116 215)(28 117 216)(29 118 217)(30 119 218)(31 120 219)(32 101 220)(33 102 201)(34 103 202)(35 104 203)(36 105 204)(37 106 205)(38 107 206)(39 108 207)(40 109 208)(41 182 228)(42 183 229)(43 184 230)(44 185 231)(45 186 232)(46 187 233)(47 188 234)(48 189 235)(49 190 236)(50 191 237)(51 192 238)(52 193 239)(53 194 240)(54 195 221)(55 196 222)(56 197 223)(57 198 224)(58 199 225)(59 200 226)(60 181 227)(81 178 141)(82 179 142)(83 180 143)(84 161 144)(85 162 145)(86 163 146)(87 164 147)(88 165 148)(89 166 149)(90 167 150)(91 168 151)(92 169 152)(93 170 153)(94 171 154)(95 172 155)(96 173 156)(97 174 157)(98 175 158)(99 176 159)(100 177 160)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)(41 46 51 56)(42 47 52 57)(43 48 53 58)(44 49 54 59)(45 50 55 60)(61 66 71 76)(62 67 72 77)(63 68 73 78)(64 69 74 79)(65 70 75 80)(81 96 91 86)(82 97 92 87)(83 98 93 88)(84 99 94 89)(85 100 95 90)(101 116 111 106)(102 117 112 107)(103 118 113 108)(104 119 114 109)(105 120 115 110)(121 126 131 136)(122 127 132 137)(123 128 133 138)(124 129 134 139)(125 130 135 140)(141 156 151 146)(142 157 152 147)(143 158 153 148)(144 159 154 149)(145 160 155 150)(161 176 171 166)(162 177 172 167)(163 178 173 168)(164 179 174 169)(165 180 175 170)(181 186 191 196)(182 187 192 197)(183 188 193 198)(184 189 194 199)(185 190 195 200)(201 216 211 206)(202 217 212 207)(203 218 213 208)(204 219 214 209)(205 220 215 210)(221 226 231 236)(222 227 232 237)(223 228 233 238)(224 229 234 239)(225 230 235 240)
(1 58 11 48)(2 49 12 59)(3 60 13 50)(4 51 14 41)(5 42 15 52)(6 53 16 43)(7 44 17 54)(8 55 18 45)(9 46 19 56)(10 57 20 47)(21 148 31 158)(22 159 32 149)(23 150 33 160)(24 141 34 151)(25 152 35 142)(26 143 36 153)(27 154 37 144)(28 145 38 155)(29 156 39 146)(30 147 40 157)(61 221 71 231)(62 232 72 222)(63 223 73 233)(64 234 74 224)(65 225 75 235)(66 236 76 226)(67 227 77 237)(68 238 78 228)(69 229 79 239)(70 240 80 230)(81 103 91 113)(82 114 92 104)(83 105 93 115)(84 116 94 106)(85 107 95 117)(86 118 96 108)(87 109 97 119)(88 120 98 110)(89 111 99 101)(90 102 100 112)(121 183 131 193)(122 194 132 184)(123 185 133 195)(124 196 134 186)(125 187 135 197)(126 198 136 188)(127 189 137 199)(128 200 138 190)(129 191 139 181)(130 182 140 192)(161 215 171 205)(162 206 172 216)(163 217 173 207)(164 208 174 218)(165 219 175 209)(166 210 176 220)(167 201 177 211)(168 212 178 202)(169 203 179 213)(170 214 180 204)
(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 31 11 21)(2 40 12 30)(3 29 13 39)(4 38 14 28)(5 27 15 37)(6 36 16 26)(7 25 17 35)(8 34 18 24)(9 23 19 33)(10 32 20 22)(41 155 51 145)(42 144 52 154)(43 153 53 143)(44 142 54 152)(45 151 55 141)(46 160 56 150)(47 149 57 159)(48 158 58 148)(49 147 59 157)(50 156 60 146)(61 203 71 213)(62 212 72 202)(63 201 73 211)(64 210 74 220)(65 219 75 209)(66 208 76 218)(67 217 77 207)(68 206 78 216)(69 215 79 205)(70 204 80 214)(81 186 91 196)(82 195 92 185)(83 184 93 194)(84 193 94 183)(85 182 95 192)(86 191 96 181)(87 200 97 190)(88 189 98 199)(89 198 99 188)(90 187 100 197)(101 136 111 126)(102 125 112 135)(103 134 113 124)(104 123 114 133)(105 132 115 122)(106 121 116 131)(107 130 117 140)(108 139 118 129)(109 128 119 138)(110 137 120 127)(161 239 171 229)(162 228 172 238)(163 237 173 227)(164 226 174 236)(165 235 175 225)(166 224 176 234)(167 233 177 223)(168 222 178 232)(169 231 179 221)(170 240 180 230)

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

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

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

111 conjugacy classes

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

111 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D5 D10 D10 C3×D5 C6×D5 C6×D5 2- 1+4 C3×2- 1+4 Q8.10D10 C3×Q8.10D10 kernel C3×Q8.10D10 C3×C4○D20 C3×Q8×D5 C3×Q8⋊2D5 Q8×C30 Q8.10D10 C4○D20 Q8×D5 Q8⋊2D5 Q8×C10 C6×Q8 C2×C12 C3×Q8 C2×Q8 C2×C4 Q8 C15 C5 C3 C1 # reps 1 6 4 4 1 2 12 8 8 2 2 6 8 4 12 16 1 2 4 8

Matrix representation of C3×Q8.10D10 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 13 0 0 0 0 13
,
 60 15 0 0 8 1 0 0 37 37 0 60 15 43 1 0
,
 8 11 0 0 44 53 0 0 18 36 25 17 19 60 17 36
,
 41 56 0 0 38 20 0 0 49 49 0 58 2 22 3 0
,
 36 36 0 6 4 20 24 37 25 20 19 5 14 30 39 47
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[60,8,37,15,15,1,37,43,0,0,0,1,0,0,60,0],[8,44,18,19,11,53,36,60,0,0,25,17,0,0,17,36],[41,38,49,2,56,20,49,22,0,0,0,3,0,0,58,0],[36,4,25,14,36,20,20,30,0,24,19,39,6,37,5,47] >;

C3×Q8.10D10 in GAP, Magma, Sage, TeX

C_3\times Q_8._{10}D_{10}
% in TeX

G:=Group("C3xQ8.10D10");
// GroupNames label

G:=SmallGroup(480,1144);
// by ID

G=gap.SmallGroup(480,1144);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,344,555,268,1571,18822]);
// Polycyclic

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

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