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## G = C10×C8⋊S3order 480 = 25·3·5

### Direct product of C10 and C8⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C10×C8⋊S3
 Chief series C1 — C3 — C6 — C12 — C60 — S3×C20 — S3×C2×C20 — C10×C8⋊S3
 Lower central C3 — C6 — C10×C8⋊S3
 Upper central C1 — C2×C20 — C2×C40

Generators and relations for C10×C8⋊S3
G = < a,b,c,d | a10=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 260 in 136 conjugacy classes, 82 normal (38 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, C10, C10 [×2], C10 [×2], Dic3 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C15, C2×C8, C2×C8, M4(2) [×4], C22×C4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C5×S3 [×2], C30, C30 [×2], C2×M4(2), C40 [×2], C40 [×2], C2×C20, C2×C20 [×5], C22×C10, C8⋊S3 [×4], C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3 [×2], C60 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C40, C2×C40, C5×M4(2) [×4], C22×C20, C2×C8⋊S3, C5×C3⋊C8 [×2], C120 [×2], S3×C20 [×4], C10×Dic3, C2×C60, S3×C2×C10, C10×M4(2), C5×C8⋊S3 [×4], C10×C3⋊C8, C2×C120, S3×C2×C20, C10×C8⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], C23, C10 [×7], D6 [×3], M4(2) [×2], C22×C4, C20 [×4], C2×C10 [×7], C4×S3 [×2], C22×S3, C5×S3, C2×M4(2), C2×C20 [×6], C22×C10, C8⋊S3 [×2], S3×C2×C4, S3×C10 [×3], C5×M4(2) [×2], C22×C20, C2×C8⋊S3, S3×C20 [×2], S3×C2×C10, C10×M4(2), C5×C8⋊S3 [×2], S3×C2×C20, C10×C8⋊S3

Smallest permutation representation of C10×C8⋊S3
On 240 points
Generators in S240
(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)
(1 133 211 45 170 53 91 198)(2 134 212 46 161 54 92 199)(3 135 213 47 162 55 93 200)(4 136 214 48 163 56 94 191)(5 137 215 49 164 57 95 192)(6 138 216 50 165 58 96 193)(7 139 217 41 166 59 97 194)(8 140 218 42 167 60 98 195)(9 131 219 43 168 51 99 196)(10 132 220 44 169 52 100 197)(11 87 31 156 112 189 202 177)(12 88 32 157 113 190 203 178)(13 89 33 158 114 181 204 179)(14 90 34 159 115 182 205 180)(15 81 35 160 116 183 206 171)(16 82 36 151 117 184 207 172)(17 83 37 152 118 185 208 173)(18 84 38 153 119 186 209 174)(19 85 39 154 120 187 210 175)(20 86 40 155 111 188 201 176)(21 128 232 224 63 79 104 146)(22 129 233 225 64 80 105 147)(23 130 234 226 65 71 106 148)(24 121 235 227 66 72 107 149)(25 122 236 228 67 73 108 150)(26 123 237 229 68 74 109 141)(27 124 238 230 69 75 110 142)(28 125 239 221 70 76 101 143)(29 126 240 222 61 77 102 144)(30 127 231 223 62 78 103 145)
(1 101 160)(2 102 151)(3 103 152)(4 104 153)(5 105 154)(6 106 155)(7 107 156)(8 108 157)(9 109 158)(10 110 159)(11 59 227)(12 60 228)(13 51 229)(14 52 230)(15 53 221)(16 54 222)(17 55 223)(18 56 224)(19 57 225)(20 58 226)(21 186 214)(22 187 215)(23 188 216)(24 189 217)(25 190 218)(26 181 219)(27 182 220)(28 183 211)(29 184 212)(30 185 213)(31 194 72)(32 195 73)(33 196 74)(34 197 75)(35 198 76)(36 199 77)(37 200 78)(38 191 79)(39 192 80)(40 193 71)(41 121 202)(42 122 203)(43 123 204)(44 124 205)(45 125 206)(46 126 207)(47 127 208)(48 128 209)(49 129 210)(50 130 201)(61 82 92)(62 83 93)(63 84 94)(64 85 95)(65 86 96)(66 87 97)(67 88 98)(68 89 99)(69 90 100)(70 81 91)(111 138 148)(112 139 149)(113 140 150)(114 131 141)(115 132 142)(116 133 143)(117 134 144)(118 135 145)(119 136 146)(120 137 147)(161 240 172)(162 231 173)(163 232 174)(164 233 175)(165 234 176)(166 235 177)(167 236 178)(168 237 179)(169 238 180)(170 239 171)
(11 149)(12 150)(13 141)(14 142)(15 143)(16 144)(17 145)(18 146)(19 147)(20 148)(21 186)(22 187)(23 188)(24 189)(25 190)(26 181)(27 182)(28 183)(29 184)(30 185)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 127)(38 128)(39 129)(40 130)(41 194)(42 195)(43 196)(44 197)(45 198)(46 199)(47 200)(48 191)(49 192)(50 193)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 81)(71 201)(72 202)(73 203)(74 204)(75 205)(76 206)(77 207)(78 208)(79 209)(80 210)(101 160)(102 151)(103 152)(104 153)(105 154)(106 155)(107 156)(108 157)(109 158)(110 159)(111 226)(112 227)(113 228)(114 229)(115 230)(116 221)(117 222)(118 223)(119 224)(120 225)(171 239)(172 240)(173 231)(174 232)(175 233)(176 234)(177 235)(178 236)(179 237)(180 238)

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10L 10M ··· 10T 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20X 24A ··· 24H 30A ··· 30L 40A ··· 40P 40Q ··· 40AF 60A ··· 60P 120A ··· 120AF order 1 2 2 2 2 2 3 4 4 4 4 4 4 5 5 5 5 6 6 6 8 8 8 8 8 8 8 8 10 ··· 10 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 1 1 6 6 2 1 1 1 1 6 6 1 1 1 1 2 2 2 2 2 2 2 6 6 6 6 1 ··· 1 6 ··· 6 2 2 2 2 2 2 2 2 1 ··· 1 6 ··· 6 2 ··· 2 2 ··· 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 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 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C5 C10 C10 C10 C10 C20 C20 C20 S3 D6 D6 M4(2) C4×S3 C4×S3 C5×S3 C8⋊S3 S3×C10 S3×C10 C5×M4(2) S3×C20 S3×C20 C5×C8⋊S3 kernel C10×C8⋊S3 C5×C8⋊S3 C10×C3⋊C8 C2×C120 S3×C2×C20 S3×C20 C10×Dic3 S3×C2×C10 C2×C8⋊S3 C8⋊S3 C2×C3⋊C8 C2×C24 S3×C2×C4 C4×S3 C2×Dic3 C22×S3 C2×C40 C40 C2×C20 C30 C20 C2×C10 C2×C8 C10 C8 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 4 16 4 4 4 16 8 8 1 2 1 4 2 2 4 8 8 4 16 8 8 32

Matrix representation of C10×C8⋊S3 in GL5(𝔽241)

 240 0 0 0 0 0 87 0 0 0 0 0 87 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 64 0 0 0 0 0 64 0 0 0 0 0 69 138 0 0 0 103 172
,
 1 0 0 0 0 0 0 1 0 0 0 240 240 0 0 0 0 0 0 1 0 0 0 240 240
,
 240 0 0 0 0 0 1 0 0 0 0 240 240 0 0 0 0 0 1 0 0 0 0 240 240

G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,87,0,0,0,0,0,87,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,69,103,0,0,0,138,172],[1,0,0,0,0,0,0,240,0,0,0,1,240,0,0,0,0,0,0,240,0,0,0,1,240],[240,0,0,0,0,0,1,240,0,0,0,0,240,0,0,0,0,0,1,240,0,0,0,0,240] >;

C10×C8⋊S3 in GAP, Magma, Sage, TeX

C_{10}\times C_8\rtimes S_3
% in TeX

G:=Group("C10xC8:S3");
// GroupNames label

G:=SmallGroup(480,779);
// by ID

G=gap.SmallGroup(480,779);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1766,226,102,15686]);
// Polycyclic

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

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