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## G = C9⋊D27order 486 = 2·35

### The semidirect product of C9 and D27 acting via D27/C27=C2

Aliases: C9⋊D27, C27⋊D9, C92.4S3, (C9×C27)⋊3C2, C3.(C9⋊D9), (C3×C9).8D9, C9.1(C9⋊S3), (C3×C27).5S3, C3.1(C27⋊S3), C32.14(C9⋊S3), (C3×C9).17(C3⋊S3), SmallGroup(486,50)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9×C27 — C9⋊D27
 Chief series C1 — C3 — C32 — C3×C9 — C92 — C9×C27 — C9⋊D27
 Lower central C9×C27 — C9⋊D27
 Upper central C1

Generators and relations for C9⋊D27
G = < a,b,c | a9=b27=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 1084 in 72 conjugacy classes, 37 normal (7 characteristic)
C1, C2, C3, C3, S3, C9, C32, D9, C3⋊S3, C27, C3×C9, C3×C9, D27, C9⋊S3, C92, C3×C27, C9⋊D9, C27⋊S3, C9×C27, C9⋊D27
Quotients: C1, C2, S3, D9, C3⋊S3, D27, C9⋊S3, C9⋊D9, C27⋊S3, C9⋊D27

Smallest permutation representation of C9⋊D27
On 243 points
Generators in S243
(1 238 174 157 95 57 209 113 53)(2 239 175 158 96 58 210 114 54)(3 240 176 159 97 59 211 115 28)(4 241 177 160 98 60 212 116 29)(5 242 178 161 99 61 213 117 30)(6 243 179 162 100 62 214 118 31)(7 217 180 136 101 63 215 119 32)(8 218 181 137 102 64 216 120 33)(9 219 182 138 103 65 190 121 34)(10 220 183 139 104 66 191 122 35)(11 221 184 140 105 67 192 123 36)(12 222 185 141 106 68 193 124 37)(13 223 186 142 107 69 194 125 38)(14 224 187 143 108 70 195 126 39)(15 225 188 144 82 71 196 127 40)(16 226 189 145 83 72 197 128 41)(17 227 163 146 84 73 198 129 42)(18 228 164 147 85 74 199 130 43)(19 229 165 148 86 75 200 131 44)(20 230 166 149 87 76 201 132 45)(21 231 167 150 88 77 202 133 46)(22 232 168 151 89 78 203 134 47)(23 233 169 152 90 79 204 135 48)(24 234 170 153 91 80 205 109 49)(25 235 171 154 92 81 206 110 50)(26 236 172 155 93 55 207 111 51)(27 237 173 156 94 56 208 112 52)
(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)(217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243)
(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(28 236)(29 235)(30 234)(31 233)(32 232)(33 231)(34 230)(35 229)(36 228)(37 227)(38 226)(39 225)(40 224)(41 223)(42 222)(43 221)(44 220)(45 219)(46 218)(47 217)(48 243)(49 242)(50 241)(51 240)(52 239)(53 238)(54 237)(55 97)(56 96)(57 95)(58 94)(59 93)(60 92)(61 91)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 84)(69 83)(70 82)(71 108)(72 107)(73 106)(74 105)(75 104)(76 103)(77 102)(78 101)(79 100)(80 99)(81 98)(109 178)(110 177)(111 176)(112 175)(113 174)(114 173)(115 172)(116 171)(117 170)(118 169)(119 168)(120 167)(121 166)(122 165)(123 164)(124 163)(125 189)(126 188)(127 187)(128 186)(129 185)(130 184)(131 183)(132 182)(133 181)(134 180)(135 179)(136 203)(137 202)(138 201)(139 200)(140 199)(141 198)(142 197)(143 196)(144 195)(145 194)(146 193)(147 192)(148 191)(149 190)(150 216)(151 215)(152 214)(153 213)(154 212)(155 211)(156 210)(157 209)(158 208)(159 207)(160 206)(161 205)(162 204)

G:=sub<Sym(243)| (1,238,174,157,95,57,209,113,53)(2,239,175,158,96,58,210,114,54)(3,240,176,159,97,59,211,115,28)(4,241,177,160,98,60,212,116,29)(5,242,178,161,99,61,213,117,30)(6,243,179,162,100,62,214,118,31)(7,217,180,136,101,63,215,119,32)(8,218,181,137,102,64,216,120,33)(9,219,182,138,103,65,190,121,34)(10,220,183,139,104,66,191,122,35)(11,221,184,140,105,67,192,123,36)(12,222,185,141,106,68,193,124,37)(13,223,186,142,107,69,194,125,38)(14,224,187,143,108,70,195,126,39)(15,225,188,144,82,71,196,127,40)(16,226,189,145,83,72,197,128,41)(17,227,163,146,84,73,198,129,42)(18,228,164,147,85,74,199,130,43)(19,229,165,148,86,75,200,131,44)(20,230,166,149,87,76,201,132,45)(21,231,167,150,88,77,202,133,46)(22,232,168,151,89,78,203,134,47)(23,233,169,152,90,79,204,135,48)(24,234,170,153,91,80,205,109,49)(25,235,171,154,92,81,206,110,50)(26,236,172,155,93,55,207,111,51)(27,237,173,156,94,56,208,112,52), (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)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(28,236)(29,235)(30,234)(31,233)(32,232)(33,231)(34,230)(35,229)(36,228)(37,227)(38,226)(39,225)(40,224)(41,223)(42,222)(43,221)(44,220)(45,219)(46,218)(47,217)(48,243)(49,242)(50,241)(51,240)(52,239)(53,238)(54,237)(55,97)(56,96)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,108)(72,107)(73,106)(74,105)(75,104)(76,103)(77,102)(78,101)(79,100)(80,99)(81,98)(109,178)(110,177)(111,176)(112,175)(113,174)(114,173)(115,172)(116,171)(117,170)(118,169)(119,168)(120,167)(121,166)(122,165)(123,164)(124,163)(125,189)(126,188)(127,187)(128,186)(129,185)(130,184)(131,183)(132,182)(133,181)(134,180)(135,179)(136,203)(137,202)(138,201)(139,200)(140,199)(141,198)(142,197)(143,196)(144,195)(145,194)(146,193)(147,192)(148,191)(149,190)(150,216)(151,215)(152,214)(153,213)(154,212)(155,211)(156,210)(157,209)(158,208)(159,207)(160,206)(161,205)(162,204)>;

G:=Group( (1,238,174,157,95,57,209,113,53)(2,239,175,158,96,58,210,114,54)(3,240,176,159,97,59,211,115,28)(4,241,177,160,98,60,212,116,29)(5,242,178,161,99,61,213,117,30)(6,243,179,162,100,62,214,118,31)(7,217,180,136,101,63,215,119,32)(8,218,181,137,102,64,216,120,33)(9,219,182,138,103,65,190,121,34)(10,220,183,139,104,66,191,122,35)(11,221,184,140,105,67,192,123,36)(12,222,185,141,106,68,193,124,37)(13,223,186,142,107,69,194,125,38)(14,224,187,143,108,70,195,126,39)(15,225,188,144,82,71,196,127,40)(16,226,189,145,83,72,197,128,41)(17,227,163,146,84,73,198,129,42)(18,228,164,147,85,74,199,130,43)(19,229,165,148,86,75,200,131,44)(20,230,166,149,87,76,201,132,45)(21,231,167,150,88,77,202,133,46)(22,232,168,151,89,78,203,134,47)(23,233,169,152,90,79,204,135,48)(24,234,170,153,91,80,205,109,49)(25,235,171,154,92,81,206,110,50)(26,236,172,155,93,55,207,111,51)(27,237,173,156,94,56,208,112,52), (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)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(28,236)(29,235)(30,234)(31,233)(32,232)(33,231)(34,230)(35,229)(36,228)(37,227)(38,226)(39,225)(40,224)(41,223)(42,222)(43,221)(44,220)(45,219)(46,218)(47,217)(48,243)(49,242)(50,241)(51,240)(52,239)(53,238)(54,237)(55,97)(56,96)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,108)(72,107)(73,106)(74,105)(75,104)(76,103)(77,102)(78,101)(79,100)(80,99)(81,98)(109,178)(110,177)(111,176)(112,175)(113,174)(114,173)(115,172)(116,171)(117,170)(118,169)(119,168)(120,167)(121,166)(122,165)(123,164)(124,163)(125,189)(126,188)(127,187)(128,186)(129,185)(130,184)(131,183)(132,182)(133,181)(134,180)(135,179)(136,203)(137,202)(138,201)(139,200)(140,199)(141,198)(142,197)(143,196)(144,195)(145,194)(146,193)(147,192)(148,191)(149,190)(150,216)(151,215)(152,214)(153,213)(154,212)(155,211)(156,210)(157,209)(158,208)(159,207)(160,206)(161,205)(162,204) );

G=PermutationGroup([[(1,238,174,157,95,57,209,113,53),(2,239,175,158,96,58,210,114,54),(3,240,176,159,97,59,211,115,28),(4,241,177,160,98,60,212,116,29),(5,242,178,161,99,61,213,117,30),(6,243,179,162,100,62,214,118,31),(7,217,180,136,101,63,215,119,32),(8,218,181,137,102,64,216,120,33),(9,219,182,138,103,65,190,121,34),(10,220,183,139,104,66,191,122,35),(11,221,184,140,105,67,192,123,36),(12,222,185,141,106,68,193,124,37),(13,223,186,142,107,69,194,125,38),(14,224,187,143,108,70,195,126,39),(15,225,188,144,82,71,196,127,40),(16,226,189,145,83,72,197,128,41),(17,227,163,146,84,73,198,129,42),(18,228,164,147,85,74,199,130,43),(19,229,165,148,86,75,200,131,44),(20,230,166,149,87,76,201,132,45),(21,231,167,150,88,77,202,133,46),(22,232,168,151,89,78,203,134,47),(23,233,169,152,90,79,204,135,48),(24,234,170,153,91,80,205,109,49),(25,235,171,154,92,81,206,110,50),(26,236,172,155,93,55,207,111,51),(27,237,173,156,94,56,208,112,52)], [(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),(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243)], [(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(28,236),(29,235),(30,234),(31,233),(32,232),(33,231),(34,230),(35,229),(36,228),(37,227),(38,226),(39,225),(40,224),(41,223),(42,222),(43,221),(44,220),(45,219),(46,218),(47,217),(48,243),(49,242),(50,241),(51,240),(52,239),(53,238),(54,237),(55,97),(56,96),(57,95),(58,94),(59,93),(60,92),(61,91),(62,90),(63,89),(64,88),(65,87),(66,86),(67,85),(68,84),(69,83),(70,82),(71,108),(72,107),(73,106),(74,105),(75,104),(76,103),(77,102),(78,101),(79,100),(80,99),(81,98),(109,178),(110,177),(111,176),(112,175),(113,174),(114,173),(115,172),(116,171),(117,170),(118,169),(119,168),(120,167),(121,166),(122,165),(123,164),(124,163),(125,189),(126,188),(127,187),(128,186),(129,185),(130,184),(131,183),(132,182),(133,181),(134,180),(135,179),(136,203),(137,202),(138,201),(139,200),(140,199),(141,198),(142,197),(143,196),(144,195),(145,194),(146,193),(147,192),(148,191),(149,190),(150,216),(151,215),(152,214),(153,213),(154,212),(155,211),(156,210),(157,209),(158,208),(159,207),(160,206),(161,205),(162,204)]])

123 conjugacy classes

 class 1 2 3A 3B 3C 3D 9A ··· 9AJ 27A ··· 27CC order 1 2 3 3 3 3 9 ··· 9 27 ··· 27 size 1 243 2 2 2 2 2 ··· 2 2 ··· 2

123 irreducible representations

 dim 1 1 2 2 2 2 2 type + + + + + + + image C1 C2 S3 S3 D9 D9 D27 kernel C9⋊D27 C9×C27 C92 C3×C27 C27 C3×C9 C9 # reps 1 1 1 3 27 9 81

Matrix representation of C9⋊D27 in GL4(𝔽109) generated by

 1 0 0 0 0 1 0 0 0 0 77 50 0 0 59 27
,
 29 51 0 0 58 87 0 0 0 0 30 93 0 0 16 46
,
 1 0 0 0 108 108 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(109))| [1,0,0,0,0,1,0,0,0,0,77,59,0,0,50,27],[29,58,0,0,51,87,0,0,0,0,30,16,0,0,93,46],[1,108,0,0,0,108,0,0,0,0,0,1,0,0,1,0] >;

C9⋊D27 in GAP, Magma, Sage, TeX

C_9\rtimes D_{27}
% in TeX

G:=Group("C9:D27");
// GroupNames label

G:=SmallGroup(486,50);
// by ID

G=gap.SmallGroup(486,50);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,265,1195,1190,338,4755,453,3244,11669]);
// Polycyclic

G:=Group<a,b,c|a^9=b^27=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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