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## G = C3×C8.D10order 480 = 25·3·5

### Direct product of C3 and C8.D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×C8.D10
 Chief series C1 — C5 — C10 — C20 — C60 — C3×D20 — C3×C4○D20 — C3×C8.D10
 Lower central C5 — C10 — C20 — C3×C8.D10
 Upper central C1 — C6 — C2×C12 — C3×M4(2)

Generators and relations for C3×C8.D10
G = < a,b,c,d | a3=b8=1, c10=d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b-1, dcd-1=c9 >

Subgroups: 416 in 120 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], D5, C10, C10, C12 [×2], C12 [×3], C2×C6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5 [×3], C20 [×2], D10, C2×C10, C24 [×2], C2×C12, C2×C12 [×2], C3×D4 [×2], C3×Q8 [×4], C3×D5, C30, C30, C8.C22, C40 [×2], Dic10, Dic10 [×2], Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C3×M4(2), C3×SD16 [×2], C3×Q16 [×2], C6×Q8, C3×C4○D4, C3×Dic5 [×3], C60 [×2], C6×D5, C2×C30, C40⋊C2 [×2], Dic20 [×2], C5×M4(2), C2×Dic10, C4○D20, C3×C8.C22, C120 [×2], C3×Dic10, C3×Dic10 [×2], C3×Dic10, D5×C12, C3×D20, C6×Dic5, C3×C5⋊D4, C2×C60, C8.D10, C3×C40⋊C2 [×2], C3×Dic20 [×2], C15×M4(2), C6×Dic10, C3×C4○D20, C3×C8.D10
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, D20 [×2], C22×D5, C6×D4, C6×D5 [×3], C2×D20, C3×C8.C22, C3×D20 [×2], D5×C2×C6, C8.D10, C6×D20, C3×C8.D10

Smallest permutation representation of C3×C8.D10
On 240 points
Generators in S240
(1 48 36)(2 49 37)(3 50 38)(4 51 39)(5 52 40)(6 53 21)(7 54 22)(8 55 23)(9 56 24)(10 57 25)(11 58 26)(12 59 27)(13 60 28)(14 41 29)(15 42 30)(16 43 31)(17 44 32)(18 45 33)(19 46 34)(20 47 35)(61 105 89)(62 106 90)(63 107 91)(64 108 92)(65 109 93)(66 110 94)(67 111 95)(68 112 96)(69 113 97)(70 114 98)(71 115 99)(72 116 100)(73 117 81)(74 118 82)(75 119 83)(76 120 84)(77 101 85)(78 102 86)(79 103 87)(80 104 88)(121 180 145)(122 161 146)(123 162 147)(124 163 148)(125 164 149)(126 165 150)(127 166 151)(128 167 152)(129 168 153)(130 169 154)(131 170 155)(132 171 156)(133 172 157)(134 173 158)(135 174 159)(136 175 160)(137 176 141)(138 177 142)(139 178 143)(140 179 144)(181 233 213)(182 234 214)(183 235 215)(184 236 216)(185 237 217)(186 238 218)(187 239 219)(188 240 220)(189 221 201)(190 222 202)(191 223 203)(192 224 204)(193 225 205)(194 226 206)(195 227 207)(196 228 208)(197 229 209)(198 230 210)(199 231 211)(200 232 212)
(1 73 16 78 11 63 6 68)(2 64 17 69 12 74 7 79)(3 75 18 80 13 65 8 70)(4 66 19 71 14 76 9 61)(5 77 20 62 15 67 10 72)(21 96 36 81 31 86 26 91)(22 87 37 92 32 97 27 82)(23 98 38 83 33 88 28 93)(24 89 39 94 34 99 29 84)(25 100 40 85 35 90 30 95)(41 120 56 105 51 110 46 115)(42 111 57 116 52 101 47 106)(43 102 58 107 53 112 48 117)(44 113 59 118 54 103 49 108)(45 104 60 109 55 114 50 119)(121 189 126 184 131 199 136 194)(122 200 127 195 132 190 137 185)(123 191 128 186 133 181 138 196)(124 182 129 197 134 192 139 187)(125 193 130 188 135 183 140 198)(141 217 146 212 151 207 156 202)(142 208 147 203 152 218 157 213)(143 219 148 214 153 209 158 204)(144 210 149 205 154 220 159 215)(145 201 150 216 155 211 160 206)(161 232 166 227 171 222 176 237)(162 223 167 238 172 233 177 228)(163 234 168 229 173 224 178 239)(164 225 169 240 174 235 179 230)(165 236 170 231 175 226 180 221)
(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)
(1 131 11 121)(2 140 12 130)(3 129 13 139)(4 138 14 128)(5 127 15 137)(6 136 16 126)(7 125 17 135)(8 134 18 124)(9 123 19 133)(10 132 20 122)(21 160 31 150)(22 149 32 159)(23 158 33 148)(24 147 34 157)(25 156 35 146)(26 145 36 155)(27 154 37 144)(28 143 38 153)(29 152 39 142)(30 141 40 151)(41 167 51 177)(42 176 52 166)(43 165 53 175)(44 174 54 164)(45 163 55 173)(46 172 56 162)(47 161 57 171)(48 170 58 180)(49 179 59 169)(50 168 60 178)(61 196 71 186)(62 185 72 195)(63 194 73 184)(64 183 74 193)(65 192 75 182)(66 181 76 191)(67 190 77 200)(68 199 78 189)(69 188 79 198)(70 197 80 187)(81 216 91 206)(82 205 92 215)(83 214 93 204)(84 203 94 213)(85 212 95 202)(86 201 96 211)(87 210 97 220)(88 219 98 209)(89 208 99 218)(90 217 100 207)(101 232 111 222)(102 221 112 231)(103 230 113 240)(104 239 114 229)(105 228 115 238)(106 237 116 227)(107 226 117 236)(108 235 118 225)(109 224 119 234)(110 233 120 223)

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

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

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

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 10D 12A 12B 12C 12D 12E ··· 12J 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 24A 24B 24C 24D 30A 30B 30C 30D 30E 30F 30G 30H 40A ··· 40H 60A ··· 60H 60I 60J 60K 60L 120A ··· 120P 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 15 15 15 15 20 20 20 20 20 20 24 24 24 24 30 30 30 30 30 30 30 30 40 ··· 40 60 ··· 60 60 60 60 60 120 ··· 120 size 1 1 2 20 1 1 2 2 20 20 20 2 2 1 1 2 2 20 20 4 4 2 2 4 4 2 2 2 2 20 ··· 20 2 2 2 2 2 2 2 2 4 4 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4 2 ··· 2 4 4 4 4 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 4 4 4 4 type + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D5 D10 D10 C3×D4 C3×D4 C3×D5 D20 D20 C6×D5 C6×D5 C3×D20 C3×D20 C8.C22 C3×C8.C22 C8.D10 C3×C8.D10 kernel C3×C8.D10 C3×C40⋊C2 C3×Dic20 C15×M4(2) C6×Dic10 C3×C4○D20 C8.D10 C40⋊C2 Dic20 C5×M4(2) C2×Dic10 C4○D20 C60 C2×C30 C3×M4(2) C24 C2×C12 C20 C2×C10 M4(2) C12 C2×C6 C8 C2×C4 C4 C22 C15 C5 C3 C1 # reps 1 2 2 1 1 1 2 4 4 2 2 2 1 1 2 4 2 2 2 4 4 4 8 4 8 8 1 2 4 8

Matrix representation of C3×C8.D10 in GL6(𝔽241)

 15 0 0 0 0 0 0 15 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
,
 115 187 0 0 0 0 236 126 0 0 0 0 0 0 170 217 138 89 0 0 156 41 103 138 0 0 114 157 200 24 0 0 237 114 85 71
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 163 163 0 0 0 0 78 44 0 0 0 0 100 220 197 78 0 0 103 100 163 78
,
 126 199 0 0 0 0 5 115 0 0 0 0 0 0 49 178 130 68 0 0 38 37 68 17 0 0 149 45 83 236 0 0 34 22 80 72

G:=sub<GL(6,GF(241))| [15,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[115,236,0,0,0,0,187,126,0,0,0,0,0,0,170,156,114,237,0,0,217,41,157,114,0,0,138,103,200,85,0,0,89,138,24,71],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,163,78,100,103,0,0,163,44,220,100,0,0,0,0,197,163,0,0,0,0,78,78],[126,5,0,0,0,0,199,115,0,0,0,0,0,0,49,38,149,34,0,0,178,37,45,22,0,0,130,68,83,80,0,0,68,17,236,72] >;

C3×C8.D10 in GAP, Magma, Sage, TeX

C_3\times C_8.D_{10}
% in TeX

G:=Group("C3xC8.D10");
// GroupNames label

G:=SmallGroup(480,702);
// by ID

G=gap.SmallGroup(480,702);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,555,142,2524,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=1,c^10=d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations

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