Copied to
clipboard

## G = C3×C9⋊C9order 243 = 35

### Direct product of C3 and C9⋊C9

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C3×C9⋊C9, C33.39C32, C32.16C33, C32.113- 1+2, (C3×C9)⋊4C9, C92(C3×C9), C3.2(C32×C9), (C32×C9).10C3, C32.16(C3×C9), (C3×C9).19C32, C3.2(C3×3- 1+2), SmallGroup(243,33)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C3 — C3×C9⋊C9
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C3×C9⋊C9
 Lower central C1 — C3 — C3×C9⋊C9
 Upper central C1 — C33 — C3×C9⋊C9
 Jennings C1 — C32 — C32 — C3×C9⋊C9

Generators and relations for C3×C9⋊C9
G = < a,b,c | a3=b9=c9=1, ab=ba, ac=ca, cbc-1=b7 >

Subgroups: 126 in 90 conjugacy classes, 72 normal (6 characteristic)
C1, C3, C3, C9, C9, C32, C32, C3×C9, C3×C9, C33, C9⋊C9, C32×C9, C32×C9, C3×C9⋊C9
Quotients: C1, C3, C9, C32, C3×C9, 3- 1+2, C33, C9⋊C9, C32×C9, C3×3- 1+2, C3×C9⋊C9

Smallest permutation representation of C3×C9⋊C9
Regular action on 243 points
Generators in S243
(1 77 23)(2 78 24)(3 79 25)(4 80 26)(5 81 27)(6 73 19)(7 74 20)(8 75 21)(9 76 22)(10 124 31)(11 125 32)(12 126 33)(13 118 34)(14 119 35)(15 120 36)(16 121 28)(17 122 29)(18 123 30)(37 55 106)(38 56 107)(39 57 108)(40 58 100)(41 59 101)(42 60 102)(43 61 103)(44 62 104)(45 63 105)(46 210 228)(47 211 229)(48 212 230)(49 213 231)(50 214 232)(51 215 233)(52 216 234)(53 208 226)(54 209 227)(64 225 174)(65 217 175)(66 218 176)(67 219 177)(68 220 178)(69 221 179)(70 222 180)(71 223 172)(72 224 173)(82 109 154)(83 110 155)(84 111 156)(85 112 157)(86 113 158)(87 114 159)(88 115 160)(89 116 161)(90 117 162)(91 205 142)(92 206 143)(93 207 144)(94 199 136)(95 200 137)(96 201 138)(97 202 139)(98 203 140)(99 204 141)(127 145 187)(128 146 188)(129 147 189)(130 148 181)(131 149 182)(132 150 183)(133 151 184)(134 152 185)(135 153 186)(163 190 235)(164 191 236)(165 192 237)(166 193 238)(167 194 239)(168 195 240)(169 196 241)(170 197 242)(171 198 243)
(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)
(1 214 86 30 239 201 106 174 148)(2 209 84 31 243 199 107 178 146)(3 213 82 32 238 206 108 173 153)(4 208 89 33 242 204 100 177 151)(5 212 87 34 237 202 101 172 149)(6 216 85 35 241 200 102 176 147)(7 211 83 36 236 207 103 180 145)(8 215 90 28 240 205 104 175 152)(9 210 88 29 235 203 105 179 150)(10 171 136 38 68 188 78 227 111)(11 166 143 39 72 186 79 231 109)(12 170 141 40 67 184 80 226 116)(13 165 139 41 71 182 81 230 114)(14 169 137 42 66 189 73 234 112)(15 164 144 43 70 187 74 229 110)(16 168 142 44 65 185 75 233 117)(17 163 140 45 69 183 76 228 115)(18 167 138 37 64 181 77 232 113)(19 52 157 119 196 95 60 218 129)(20 47 155 120 191 93 61 222 127)(21 51 162 121 195 91 62 217 134)(22 46 160 122 190 98 63 221 132)(23 50 158 123 194 96 55 225 130)(24 54 156 124 198 94 56 220 128)(25 49 154 125 193 92 57 224 135)(26 53 161 126 197 99 58 219 133)(27 48 159 118 192 97 59 223 131)

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

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

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

C3×C9⋊C9 is a maximal subgroup of   C9⋊(S3×C9)

99 conjugacy classes

 class 1 3A ··· 3Z 9A ··· 9BT order 1 3 ··· 3 9 ··· 9 size 1 1 ··· 1 3 ··· 3

99 irreducible representations

 dim 1 1 1 1 3 type + image C1 C3 C3 C9 3- 1+2 kernel C3×C9⋊C9 C9⋊C9 C32×C9 C3×C9 C32 # reps 1 18 8 54 18

Matrix representation of C3×C9⋊C9 in GL5(𝔽19)

 11 0 0 0 0 0 11 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 11 0 0 0 0 0 1 0 0 0 0 0 0 7 0 0 0 0 0 7 0 0 1 0 0
,
 16 0 0 0 0 0 1 0 0 0 0 0 13 9 16 0 0 16 15 6 0 0 6 17 10

G:=sub<GL(5,GF(19))| [11,0,0,0,0,0,11,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[11,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,0,7,0],[16,0,0,0,0,0,1,0,0,0,0,0,13,16,6,0,0,9,15,17,0,0,16,6,10] >;

C3×C9⋊C9 in GAP, Magma, Sage, TeX

C_3\times C_9\rtimes C_9
% in TeX

G:=Group("C3xC9:C9");
// GroupNames label

G:=SmallGroup(243,33);
// by ID

G=gap.SmallGroup(243,33);
# by ID

G:=PCGroup([5,-3,3,3,-3,3,405,301,96]);
// Polycyclic

G:=Group<a,b,c|a^3=b^9=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^7>;
// generators/relations

׿
×
𝔽