direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C4×C9⋊S3, C36⋊8D6, C12⋊8D18, C62.135D6, C6⋊2(C4×D9), C18⋊2(C4×S3), (C2×C36)⋊5S3, (C2×C12)⋊5D9, (C6×C36)⋊11C2, (C6×C12).51S3, (C2×C6).42D18, (C2×C18).42D6, (C3×C36)⋊12C22, (C3×C12).222D6, C6.40(C22×D9), (C3×C18).49C23, C18.40(C22×S3), (C6×C18).48C22, C9⋊Dic3⋊10C22, C9⋊3(S3×C2×C4), C3⋊3(C2×C4×D9), (C3×C18)⋊6(C2×C4), C6.11(C4×C3⋊S3), (C3×C9)⋊7(C22×C4), C32.6(S3×C2×C4), C12.72(C2×C3⋊S3), (C3×C6).74(C4×S3), C22.9(C2×C9⋊S3), C2.1(C22×C9⋊S3), (C2×C9⋊Dic3)⋊11C2, (C22×C9⋊S3).4C2, C6.29(C22×C3⋊S3), (C2×C12).17(C3⋊S3), (C2×C9⋊S3).15C22, (C3×C6).163(C22×S3), C3.(C2×C4×C3⋊S3), (C2×C6).35(C2×C3⋊S3), SmallGroup(432,381)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C2×C4×C9⋊S3 |
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
►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