Copied to
clipboard

## 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, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C30, C8.C22, C40, Dic10, Dic10, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C3×M4(2), C3×SD16, C3×Q16, C6×Q8, C3×C4○D4, C3×Dic5, C60, C6×D5, C2×C30, C40⋊C2, Dic20, C5×M4(2), C2×Dic10, C4○D20, C3×C8.C22, C120, C3×Dic10, C3×Dic10, C3×Dic10, D5×C12, C3×D20, C6×Dic5, C3×C5⋊D4, C2×C60, C8.D10, C3×C40⋊C2, C3×Dic20, C15×M4(2), C6×Dic10, C3×C4○D20, C3×C8.D10
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C8.C22, D20, C22×D5, C6×D4, C6×D5, C2×D20, C3×C8.C22, C3×D20, D5×C2×C6, C8.D10, C6×D20, C3×C8.D10

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

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

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

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

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

׿
×
𝔽