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## G = C8×C9⋊S3order 432 = 24·33

### Direct product of C8 and C9⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C8×C9⋊S3
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C3×C36 — C4×C9⋊S3 — C8×C9⋊S3
 Lower central C3×C9 — C8×C9⋊S3
 Upper central C1 — C8

Generators and relations for C8×C9⋊S3
G = < a,b,c,d | a8=b9=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 572 in 110 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C9, C32, Dic3, C12, C12, D6, C2×C8, D9, C18, C3⋊S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C3×C9, Dic9, C36, D18, C3⋊Dic3, C3×C12, C2×C3⋊S3, S3×C8, C9⋊S3, C3×C18, C9⋊C8, C72, C4×D9, C324C8, C3×C24, C4×C3⋊S3, C9⋊Dic3, C3×C36, C2×C9⋊S3, C8×D9, C8×C3⋊S3, C36.S3, C3×C72, C4×C9⋊S3, C8×C9⋊S3
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D6, C2×C8, D9, C3⋊S3, C4×S3, D18, C2×C3⋊S3, S3×C8, C9⋊S3, C4×D9, C4×C3⋊S3, C2×C9⋊S3, C8×D9, C8×C3⋊S3, C4×C9⋊S3, C8×C9⋊S3

Smallest permutation representation of C8×C9⋊S3
On 216 points
Generators in S216
(1 13 116 170 62 197 89 143)(2 14 117 171 63 198 90 144)(3 15 109 163 55 190 82 136)(4 16 110 164 56 191 83 137)(5 17 111 165 57 192 84 138)(6 18 112 166 58 193 85 139)(7 10 113 167 59 194 86 140)(8 11 114 168 60 195 87 141)(9 12 115 169 61 196 88 142)(19 125 179 71 206 98 152 44)(20 126 180 72 207 99 153 45)(21 118 172 64 199 91 145 37)(22 119 173 65 200 92 146 38)(23 120 174 66 201 93 147 39)(24 121 175 67 202 94 148 40)(25 122 176 68 203 95 149 41)(26 123 177 69 204 96 150 42)(27 124 178 70 205 97 151 43)(28 214 106 160 52 187 79 133)(29 215 107 161 53 188 80 134)(30 216 108 162 54 189 81 135)(31 208 100 154 46 181 73 127)(32 209 101 155 47 182 74 128)(33 210 102 156 48 183 75 129)(34 211 103 157 49 184 76 130)(35 212 104 158 50 185 77 131)(36 213 105 159 51 186 78 132)
(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 33 37)(2 34 38)(3 35 39)(4 36 40)(5 28 41)(6 29 42)(7 30 43)(8 31 44)(9 32 45)(10 216 27)(11 208 19)(12 209 20)(13 210 21)(14 211 22)(15 212 23)(16 213 24)(17 214 25)(18 215 26)(46 71 60)(47 72 61)(48 64 62)(49 65 63)(50 66 55)(51 67 56)(52 68 57)(53 69 58)(54 70 59)(73 98 87)(74 99 88)(75 91 89)(76 92 90)(77 93 82)(78 94 83)(79 95 84)(80 96 85)(81 97 86)(100 125 114)(101 126 115)(102 118 116)(103 119 117)(104 120 109)(105 121 110)(106 122 111)(107 123 112)(108 124 113)(127 152 141)(128 153 142)(129 145 143)(130 146 144)(131 147 136)(132 148 137)(133 149 138)(134 150 139)(135 151 140)(154 179 168)(155 180 169)(156 172 170)(157 173 171)(158 174 163)(159 175 164)(160 176 165)(161 177 166)(162 178 167)(181 206 195)(182 207 196)(183 199 197)(184 200 198)(185 201 190)(186 202 191)(187 203 192)(188 204 193)(189 205 194)
(1 62)(2 61)(3 60)(4 59)(5 58)(6 57)(7 56)(8 55)(9 63)(10 191)(11 190)(12 198)(13 197)(14 196)(15 195)(16 194)(17 193)(18 192)(19 185)(20 184)(21 183)(22 182)(23 181)(24 189)(25 188)(26 187)(27 186)(28 69)(29 68)(30 67)(31 66)(32 65)(33 64)(34 72)(35 71)(36 70)(37 48)(38 47)(39 46)(40 54)(41 53)(42 52)(43 51)(44 50)(45 49)(73 120)(74 119)(75 118)(76 126)(77 125)(78 124)(79 123)(80 122)(81 121)(82 114)(83 113)(84 112)(85 111)(86 110)(87 109)(88 117)(89 116)(90 115)(91 102)(92 101)(93 100)(94 108)(95 107)(96 106)(97 105)(98 104)(99 103)(127 174)(128 173)(129 172)(130 180)(131 179)(132 178)(133 177)(134 176)(135 175)(136 168)(137 167)(138 166)(139 165)(140 164)(141 163)(142 171)(143 170)(144 169)(145 156)(146 155)(147 154)(148 162)(149 161)(150 160)(151 159)(152 158)(153 157)(199 210)(200 209)(201 208)(202 216)(203 215)(204 214)(205 213)(206 212)(207 211)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 8A 8B 8C 8D 8E 8F 8G 8H 9A ··· 9I 12A ··· 12H 18A ··· 18I 24A ··· 24P 36A ··· 36R 72A ··· 72AJ order 1 2 2 2 3 3 3 3 4 4 4 4 6 6 6 6 8 8 8 8 8 8 8 8 9 ··· 9 12 ··· 12 18 ··· 18 24 ··· 24 36 ··· 36 72 ··· 72 size 1 1 27 27 2 2 2 2 1 1 27 27 2 2 2 2 1 1 1 1 27 27 27 27 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 S3 S3 D6 D6 D9 C4×S3 C4×S3 D18 S3×C8 S3×C8 C4×D9 C8×D9 kernel C8×C9⋊S3 C36.S3 C3×C72 C4×C9⋊S3 C9⋊Dic3 C2×C9⋊S3 C9⋊S3 C72 C3×C24 C36 C3×C12 C24 C18 C3×C6 C12 C9 C32 C6 C3 # reps 1 1 1 1 2 2 8 3 1 3 1 9 6 2 9 12 4 18 36

Matrix representation of C8×C9⋊S3 in GL4(𝔽73) generated by

 22 0 0 0 0 22 0 0 0 0 51 0 0 0 0 51
,
 31 70 0 0 3 28 0 0 0 0 70 45 0 0 28 42
,
 1 0 0 0 0 1 0 0 0 0 0 72 0 0 1 72
,
 0 72 0 0 72 0 0 0 0 0 3 42 0 0 45 70
G:=sub<GL(4,GF(73))| [22,0,0,0,0,22,0,0,0,0,51,0,0,0,0,51],[31,3,0,0,70,28,0,0,0,0,70,28,0,0,45,42],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[0,72,0,0,72,0,0,0,0,0,3,45,0,0,42,70] >;

C8×C9⋊S3 in GAP, Magma, Sage, TeX

C_8\times C_9\rtimes S_3
% in TeX

G:=Group("C8xC9:S3");
// GroupNames label

G:=SmallGroup(432,169);
// by ID

G=gap.SmallGroup(432,169);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,36,58,6164,662,4037,14118]);
// Polycyclic

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

׿
×
𝔽