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G = C2×C4×C9⋊S3order 432 = 24·33

Direct product of C2×C4 and C9⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C2×C4×C9⋊S3
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C2×C9⋊S3 — C22×C9⋊S3 — C2×C4×C9⋊S3
 Lower central C3×C9 — C2×C4×C9⋊S3
 Upper central C1 — C2×C4

Generators and relations for C2×C4×C9⋊S3
G = < a,b,c,d,e | a2=b4=c9=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1524 in 270 conjugacy classes, 99 normal (19 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, D9, C18, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C3×C9, Dic9, C36, D18, C2×C18, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, S3×C2×C4, C9⋊S3, C3×C18, C3×C18, C4×D9, C2×Dic9, C2×C36, C22×D9, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C9⋊Dic3, C3×C36, C2×C9⋊S3, C6×C18, C2×C4×D9, C2×C4×C3⋊S3, C4×C9⋊S3, C2×C9⋊Dic3, C6×C36, C22×C9⋊S3, C2×C4×C9⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, D9, C3⋊S3, C4×S3, C22×S3, D18, C2×C3⋊S3, S3×C2×C4, C9⋊S3, C4×D9, C22×D9, C4×C3⋊S3, C22×C3⋊S3, C2×C9⋊S3, C2×C4×D9, C2×C4×C3⋊S3, C4×C9⋊S3, C22×C9⋊S3, C2×C4×C9⋊S3

Smallest permutation representation of C2×C4×C9⋊S3
On 216 points
Generators in S216
(1 132)(2 133)(3 134)(4 135)(5 127)(6 128)(7 129)(8 130)(9 131)(10 113)(11 114)(12 115)(13 116)(14 117)(15 109)(16 110)(17 111)(18 112)(19 125)(20 126)(21 118)(22 119)(23 120)(24 121)(25 122)(26 123)(27 124)(28 136)(29 137)(30 138)(31 139)(32 140)(33 141)(34 142)(35 143)(36 144)(37 145)(38 146)(39 147)(40 148)(41 149)(42 150)(43 151)(44 152)(45 153)(46 154)(47 155)(48 156)(49 157)(50 158)(51 159)(52 160)(53 161)(54 162)(55 163)(56 164)(57 165)(58 166)(59 167)(60 168)(61 169)(62 170)(63 171)(64 172)(65 173)(66 174)(67 175)(68 176)(69 177)(70 178)(71 179)(72 180)(73 181)(74 182)(75 183)(76 184)(77 185)(78 186)(79 187)(80 188)(81 189)(82 190)(83 191)(84 192)(85 193)(86 194)(87 195)(88 196)(89 197)(90 198)(91 199)(92 200)(93 201)(94 202)(95 203)(96 204)(97 205)(98 206)(99 207)(100 208)(101 209)(102 210)(103 211)(104 212)(105 213)(106 214)(107 215)(108 216)
(1 105 51 78)(2 106 52 79)(3 107 53 80)(4 108 54 81)(5 100 46 73)(6 101 47 74)(7 102 48 75)(8 103 49 76)(9 104 50 77)(10 167 194 140)(11 168 195 141)(12 169 196 142)(13 170 197 143)(14 171 198 144)(15 163 190 136)(16 164 191 137)(17 165 192 138)(18 166 193 139)(19 179 206 152)(20 180 207 153)(21 172 199 145)(22 173 200 146)(23 174 201 147)(24 175 202 148)(25 176 203 149)(26 177 204 150)(27 178 205 151)(28 109 55 82)(29 110 56 83)(30 111 57 84)(31 112 58 85)(32 113 59 86)(33 114 60 87)(34 115 61 88)(35 116 62 89)(36 117 63 90)(37 118 64 91)(38 119 65 92)(39 120 66 93)(40 121 67 94)(41 122 68 95)(42 123 69 96)(43 124 70 97)(44 125 71 98)(45 126 72 99)(127 208 154 181)(128 209 155 182)(129 210 156 183)(130 211 157 184)(131 212 158 185)(132 213 159 186)(133 214 160 187)(134 215 161 188)(135 216 162 189)
(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)
(1 32 37)(2 33 38)(3 34 39)(4 35 40)(5 36 41)(6 28 42)(7 29 43)(8 30 44)(9 31 45)(10 21 213)(11 22 214)(12 23 215)(13 24 216)(14 25 208)(15 26 209)(16 27 210)(17 19 211)(18 20 212)(46 63 68)(47 55 69)(48 56 70)(49 57 71)(50 58 72)(51 59 64)(52 60 65)(53 61 66)(54 62 67)(73 90 95)(74 82 96)(75 83 97)(76 84 98)(77 85 99)(78 86 91)(79 87 92)(80 88 93)(81 89 94)(100 117 122)(101 109 123)(102 110 124)(103 111 125)(104 112 126)(105 113 118)(106 114 119)(107 115 120)(108 116 121)(127 144 149)(128 136 150)(129 137 151)(130 138 152)(131 139 153)(132 140 145)(133 141 146)(134 142 147)(135 143 148)(154 171 176)(155 163 177)(156 164 178)(157 165 179)(158 166 180)(159 167 172)(160 168 173)(161 169 174)(162 170 175)(181 198 203)(182 190 204)(183 191 205)(184 192 206)(185 193 207)(186 194 199)(187 195 200)(188 196 201)(189 197 202)
(1 132)(2 131)(3 130)(4 129)(5 128)(6 127)(7 135)(8 134)(9 133)(10 118)(11 126)(12 125)(13 124)(14 123)(15 122)(16 121)(17 120)(18 119)(19 115)(20 114)(21 113)(22 112)(23 111)(24 110)(25 109)(26 117)(27 116)(28 149)(29 148)(30 147)(31 146)(32 145)(33 153)(34 152)(35 151)(36 150)(37 140)(38 139)(39 138)(40 137)(41 136)(42 144)(43 143)(44 142)(45 141)(46 155)(47 154)(48 162)(49 161)(50 160)(51 159)(52 158)(53 157)(54 156)(55 176)(56 175)(57 174)(58 173)(59 172)(60 180)(61 179)(62 178)(63 177)(64 167)(65 166)(66 165)(67 164)(68 163)(69 171)(70 170)(71 169)(72 168)(73 182)(74 181)(75 189)(76 188)(77 187)(78 186)(79 185)(80 184)(81 183)(82 203)(83 202)(84 201)(85 200)(86 199)(87 207)(88 206)(89 205)(90 204)(91 194)(92 193)(93 192)(94 191)(95 190)(96 198)(97 197)(98 196)(99 195)(100 209)(101 208)(102 216)(103 215)(104 214)(105 213)(106 212)(107 211)(108 210)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6L 9A ··· 9I 12A ··· 12P 18A ··· 18AA 36A ··· 36AJ order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 27 27 27 27 2 2 2 2 1 1 1 1 27 27 27 27 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 S3 D6 D6 D6 D6 D9 C4×S3 C4×S3 D18 D18 C4×D9 kernel C2×C4×C9⋊S3 C4×C9⋊S3 C2×C9⋊Dic3 C6×C36 C22×C9⋊S3 C2×C9⋊S3 C2×C36 C6×C12 C36 C2×C18 C3×C12 C62 C2×C12 C18 C3×C6 C12 C2×C6 C6 # reps 1 4 1 1 1 8 3 1 6 3 2 1 9 12 4 18 9 36

Matrix representation of C2×C4×C9⋊S3 in GL4(𝔽37) generated by

 1 0 0 0 0 1 0 0 0 0 36 0 0 0 0 36
,
 6 0 0 0 0 6 0 0 0 0 6 0 0 0 0 6
,
 0 36 0 0 1 36 0 0 0 0 11 20 0 0 17 31
,
 0 36 0 0 1 36 0 0 0 0 0 36 0 0 1 36
,
 36 0 0 0 36 1 0 0 0 0 31 26 0 0 20 6
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[0,1,0,0,36,36,0,0,0,0,11,17,0,0,20,31],[0,1,0,0,36,36,0,0,0,0,0,1,0,0,36,36],[36,36,0,0,0,1,0,0,0,0,31,20,0,0,26,6] >;

C2×C4×C9⋊S3 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_9\rtimes S_3
% in TeX

G:=Group("C2xC4xC9:S3");
// GroupNames label

G:=SmallGroup(432,381);
// by ID

G=gap.SmallGroup(432,381);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,58,6164,662,4037,14118]);
// Polycyclic

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

׿
×
𝔽