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## G = SD16×C3×C9order 432 = 24·33

### Direct product of C3×C9 and SD16

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — SD16×C3×C9
 Chief series C1 — C2 — C6 — C12 — C3×C12 — C3×C36 — Q8×C3×C9 — SD16×C3×C9
 Lower central C1 — C2 — C4 — SD16×C3×C9
 Upper central C1 — C3×C18 — C3×C36 — SD16×C3×C9

Generators and relations for SD16×C3×C9
G = < a,b,c,d | a3=b9=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 150 in 100 conjugacy classes, 70 normal (28 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C8, D4, Q8, C9, C32, C12, C12, C12, C2×C6, SD16, C18, C18, C3×C6, C3×C6, C24, C24, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×C9, C36, C36, C2×C18, C3×C12, C3×C12, C62, C3×SD16, C3×SD16, C3×C18, C3×C18, C72, D4×C9, Q8×C9, C3×C24, D4×C32, Q8×C32, C3×C36, C3×C36, C6×C18, C9×SD16, C32×SD16, C3×C72, D4×C3×C9, Q8×C3×C9, SD16×C3×C9
Quotients: C1, C2, C3, C22, C6, D4, C9, C32, C2×C6, SD16, C18, C3×C6, C3×D4, C3×C9, C2×C18, C62, C3×SD16, C3×C18, D4×C9, D4×C32, C6×C18, C9×SD16, C32×SD16, D4×C3×C9, SD16×C3×C9

Smallest permutation representation of SD16×C3×C9
On 216 points
Generators in S216
(1 98 111)(2 99 112)(3 91 113)(4 92 114)(5 93 115)(6 94 116)(7 95 117)(8 96 109)(9 97 110)(10 31 145)(11 32 146)(12 33 147)(13 34 148)(14 35 149)(15 36 150)(16 28 151)(17 29 152)(18 30 153)(19 68 87)(20 69 88)(21 70 89)(22 71 90)(23 72 82)(24 64 83)(25 65 84)(26 66 85)(27 67 86)(37 121 199)(38 122 200)(39 123 201)(40 124 202)(41 125 203)(42 126 204)(43 118 205)(44 119 206)(45 120 207)(46 56 100)(47 57 101)(48 58 102)(49 59 103)(50 60 104)(51 61 105)(52 62 106)(53 63 107)(54 55 108)(73 181 143)(74 182 144)(75 183 136)(76 184 137)(77 185 138)(78 186 139)(79 187 140)(80 188 141)(81 189 142)(127 176 195)(128 177 196)(129 178 197)(130 179 198)(131 180 190)(132 172 191)(133 173 192)(134 174 193)(135 175 194)(154 164 208)(155 165 209)(156 166 210)(157 167 211)(158 168 212)(159 169 213)(160 170 214)(161 171 215)(162 163 216)
(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)
(1 177 15 168 60 182 69 119)(2 178 16 169 61 183 70 120)(3 179 17 170 62 184 71 121)(4 180 18 171 63 185 72 122)(5 172 10 163 55 186 64 123)(6 173 11 164 56 187 65 124)(7 174 12 165 57 188 66 125)(8 175 13 166 58 189 67 126)(9 176 14 167 59 181 68 118)(19 43 110 127 149 157 49 73)(20 44 111 128 150 158 50 74)(21 45 112 129 151 159 51 75)(22 37 113 130 152 160 52 76)(23 38 114 131 153 161 53 77)(24 39 115 132 145 162 54 78)(25 40 116 133 146 154 46 79)(26 41 117 134 147 155 47 80)(27 42 109 135 148 156 48 81)(28 213 105 136 89 207 99 197)(29 214 106 137 90 199 91 198)(30 215 107 138 82 200 92 190)(31 216 108 139 83 201 93 191)(32 208 100 140 84 202 94 192)(33 209 101 141 85 203 95 193)(34 210 102 142 86 204 96 194)(35 211 103 143 87 205 97 195)(36 212 104 144 88 206 98 196)
(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 149)(20 150)(21 151)(22 152)(23 153)(24 145)(25 146)(26 147)(27 148)(28 89)(29 90)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 76)(38 77)(39 78)(40 79)(41 80)(42 81)(43 73)(44 74)(45 75)(118 181)(119 182)(120 183)(121 184)(122 185)(123 186)(124 187)(125 188)(126 189)(127 157)(128 158)(129 159)(130 160)(131 161)(132 162)(133 154)(134 155)(135 156)(136 207)(137 199)(138 200)(139 201)(140 202)(141 203)(142 204)(143 205)(144 206)(163 172)(164 173)(165 174)(166 175)(167 176)(168 177)(169 178)(170 179)(171 180)(190 215)(191 216)(192 208)(193 209)(194 210)(195 211)(196 212)(197 213)(198 214)

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

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

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

189 conjugacy classes

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

189 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 type + + + + + image C1 C2 C2 C2 C3 C3 C6 C6 C6 C6 C6 C6 C9 C18 C18 C18 D4 SD16 C3×D4 C3×D4 C3×SD16 C3×SD16 D4×C9 C9×SD16 kernel SD16×C3×C9 C3×C72 D4×C3×C9 Q8×C3×C9 C9×SD16 C32×SD16 C72 D4×C9 Q8×C9 C3×C24 D4×C32 Q8×C32 C3×SD16 C24 C3×D4 C3×Q8 C3×C18 C3×C9 C18 C3×C6 C9 C32 C6 C3 # reps 1 1 1 1 6 2 6 6 6 2 2 2 18 18 18 18 1 2 6 2 12 4 18 36

Matrix representation of SD16×C3×C9 in GL3(𝔽73) generated by

 8 0 0 0 1 0 0 0 1
,
 37 0 0 0 37 0 0 0 37
,
 1 0 0 0 0 61 0 6 61
,
 72 0 0 0 1 0 0 1 72
G:=sub<GL(3,GF(73))| [8,0,0,0,1,0,0,0,1],[37,0,0,0,37,0,0,0,37],[1,0,0,0,0,6,0,61,61],[72,0,0,0,1,1,0,0,72] >;

SD16×C3×C9 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times C_3\times C_9
% in TeX

G:=Group("SD16xC3xC9");
// GroupNames label

G:=SmallGroup(432,218);
// by ID

G=gap.SmallGroup(432,218);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,1512,533,394,8824,4421,242]);
// 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^3>;
// generators/relations

׿
×
𝔽