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## G = D8×C3×C9order 432 = 24·33

### Direct product of C3×C9 and D8

direct product, metacyclic, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — D8×C3×C9
 Chief series C1 — C2 — C6 — C12 — C3×C12 — C3×C36 — D4×C3×C9 — D8×C3×C9
 Lower central C1 — C2 — C4 — D8×C3×C9
 Upper central C1 — C3×C18 — C3×C36 — D8×C3×C9

Generators and relations for D8×C3×C9
G = < a,b,c,d | a3=b9=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 190 in 110 conjugacy classes, 70 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, D4, C9, C32, C12, C12, C2×C6, D8, C18, C18, C3×C6, C3×C6, C24, C24, C3×D4, C3×C9, C36, C2×C18, C3×C12, C62, C3×D8, C3×D8, C3×C18, C3×C18, C72, D4×C9, C3×C24, D4×C32, C3×C36, C6×C18, C9×D8, C32×D8, C3×C72, D4×C3×C9, D8×C3×C9
Quotients: C1, C2, C3, C22, C6, D4, C9, C32, C2×C6, D8, C18, C3×C6, C3×D4, C3×C9, C2×C18, C62, C3×D8, C3×C18, D4×C9, D4×C32, C6×C18, C9×D8, C32×D8, D4×C3×C9, D8×C3×C9

Smallest permutation representation of D8×C3×C9
On 216 points
Generators in S216
(1 98 113)(2 99 114)(3 91 115)(4 92 116)(5 93 117)(6 94 109)(7 95 110)(8 96 111)(9 97 112)(10 26 64)(11 27 65)(12 19 66)(13 20 67)(14 21 68)(15 22 69)(16 23 70)(17 24 71)(18 25 72)(28 149 84)(29 150 85)(30 151 86)(31 152 87)(32 153 88)(33 145 89)(34 146 90)(35 147 82)(36 148 83)(37 118 206)(38 119 207)(39 120 199)(40 121 200)(41 122 201)(42 123 202)(43 124 203)(44 125 204)(45 126 205)(46 59 103)(47 60 104)(48 61 105)(49 62 106)(50 63 107)(51 55 108)(52 56 100)(53 57 101)(54 58 102)(73 192 136)(74 193 137)(75 194 138)(76 195 139)(77 196 140)(78 197 141)(79 198 142)(80 190 143)(81 191 144)(127 174 181)(128 175 182)(129 176 183)(130 177 184)(131 178 185)(132 179 186)(133 180 187)(134 172 188)(135 173 189)(154 167 211)(155 168 212)(156 169 213)(157 170 214)(158 171 215)(159 163 216)(160 164 208)(161 165 209)(162 166 210)
(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)
(1 193 69 168 60 177 85 118)(2 194 70 169 61 178 86 119)(3 195 71 170 62 179 87 120)(4 196 72 171 63 180 88 121)(5 197 64 163 55 172 89 122)(6 198 65 164 56 173 90 123)(7 190 66 165 57 174 82 124)(8 191 67 166 58 175 83 125)(9 192 68 167 59 176 84 126)(10 216 108 188 33 201 93 141)(11 208 100 189 34 202 94 142)(12 209 101 181 35 203 95 143)(13 210 102 182 36 204 96 144)(14 211 103 183 28 205 97 136)(15 212 104 184 29 206 98 137)(16 213 105 185 30 207 99 138)(17 214 106 186 31 199 91 139)(18 215 107 187 32 200 92 140)(19 161 53 127 147 43 110 80)(20 162 54 128 148 44 111 81)(21 154 46 129 149 45 112 73)(22 155 47 130 150 37 113 74)(23 156 48 131 151 38 114 75)(24 157 49 132 152 39 115 76)(25 158 50 133 153 40 116 77)(26 159 51 134 145 41 117 78)(27 160 52 135 146 42 109 79)
(1 118)(2 119)(3 120)(4 121)(5 122)(6 123)(7 124)(8 125)(9 126)(10 188)(11 189)(12 181)(13 182)(14 183)(15 184)(16 185)(17 186)(18 187)(19 127)(20 128)(21 129)(22 130)(23 131)(24 132)(25 133)(26 134)(27 135)(28 136)(29 137)(30 138)(31 139)(32 140)(33 141)(34 142)(35 143)(36 144)(37 113)(38 114)(39 115)(40 116)(41 117)(42 109)(43 110)(44 111)(45 112)(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)(73 149)(74 150)(75 151)(76 152)(77 153)(78 145)(79 146)(80 147)(81 148)(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)

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

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

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

189 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4 6A ··· 6H 6I ··· 6X 8A 8B 9A ··· 9R 12A ··· 12H 18A ··· 18R 18S ··· 18BB 24A ··· 24P 36A ··· 36R 72A ··· 72AJ order 1 2 2 2 3 ··· 3 4 6 ··· 6 6 ··· 6 8 8 9 ··· 9 12 ··· 12 18 ··· 18 18 ··· 18 24 ··· 24 36 ··· 36 72 ··· 72 size 1 1 4 4 1 ··· 1 2 1 ··· 1 4 ··· 4 2 2 1 ··· 1 2 ··· 2 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2 2 ··· 2

189 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C3 C3 C6 C6 C6 C6 C9 C18 C18 D4 D8 C3×D4 C3×D4 C3×D8 C3×D8 D4×C9 C9×D8 kernel D8×C3×C9 C3×C72 D4×C3×C9 C9×D8 C32×D8 C72 D4×C9 C3×C24 D4×C32 C3×D8 C24 C3×D4 C3×C18 C3×C9 C18 C3×C6 C9 C32 C6 C3 # reps 1 1 2 6 2 6 12 2 4 18 18 36 1 2 6 2 12 4 18 36

Matrix representation of D8×C3×C9 in GL4(𝔽73) generated by

 1 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 37 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 0 72 0 0 0 0 57 16 0 0 57 57
,
 1 0 0 0 0 1 0 0 0 0 57 16 0 0 16 16
G:=sub<GL(4,GF(73))| [1,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[37,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,57,57,0,0,16,57],[1,0,0,0,0,1,0,0,0,0,57,16,0,0,16,16] >;

D8×C3×C9 in GAP, Magma, Sage, TeX

D_8\times C_3\times C_9
% in TeX

G:=Group("D8xC3xC9");
// GroupNames label

G:=SmallGroup(432,215);
// by ID

G=gap.SmallGroup(432,215);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,533,394,8824,4421,242]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^9=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽