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G = C3×C81order 243 = 35

Abelian group of type [3,81]

direct product, p-group, abelian, monomial

Aliases: C3×C81, SmallGroup(243,23)

Series: Derived Chief Lower central Upper central Jennings

C1 — C3×C81
C1C3C9C27C3×C27 — C3×C81
C1 — C3×C81
C1 — C3×C81
C1C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C9C9C9C9C9C9C27C27 — C3×C81

Generators and relations for C3×C81
 G = < a,b | a3=b81=1, ab=ba >


Smallest permutation representation of C3×C81
Regular action on 243 points
Generators in S243
(1 95 163)(2 96 164)(3 97 165)(4 98 166)(5 99 167)(6 100 168)(7 101 169)(8 102 170)(9 103 171)(10 104 172)(11 105 173)(12 106 174)(13 107 175)(14 108 176)(15 109 177)(16 110 178)(17 111 179)(18 112 180)(19 113 181)(20 114 182)(21 115 183)(22 116 184)(23 117 185)(24 118 186)(25 119 187)(26 120 188)(27 121 189)(28 122 190)(29 123 191)(30 124 192)(31 125 193)(32 126 194)(33 127 195)(34 128 196)(35 129 197)(36 130 198)(37 131 199)(38 132 200)(39 133 201)(40 134 202)(41 135 203)(42 136 204)(43 137 205)(44 138 206)(45 139 207)(46 140 208)(47 141 209)(48 142 210)(49 143 211)(50 144 212)(51 145 213)(52 146 214)(53 147 215)(54 148 216)(55 149 217)(56 150 218)(57 151 219)(58 152 220)(59 153 221)(60 154 222)(61 155 223)(62 156 224)(63 157 225)(64 158 226)(65 159 227)(66 160 228)(67 161 229)(68 162 230)(69 82 231)(70 83 232)(71 84 233)(72 85 234)(73 86 235)(74 87 236)(75 88 237)(76 89 238)(77 90 239)(78 91 240)(79 92 241)(80 93 242)(81 94 243)
(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)

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

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

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

C3×C81 is a maximal subgroup of   C81⋊S3

243 conjugacy classes

class 1 3A···3H9A···9R27A···27BB81A···81FF
order13···39···927···2781···81
size11···11···11···11···1

243 irreducible representations

dim11111111
type+
imageC1C3C3C9C9C27C27C81
kernelC3×C81C81C3×C27C27C3×C9C9C32C3
# reps1621263618162

Matrix representation of C3×C81 in GL2(𝔽163) generated by

580
0104
,
490
01
G:=sub<GL(2,GF(163))| [58,0,0,104],[49,0,0,1] >;

C3×C81 in GAP, Magma, Sage, TeX

C_3\times C_{81}
% in TeX

G:=Group("C3xC81");
// GroupNames label

G:=SmallGroup(243,23);
// by ID

G=gap.SmallGroup(243,23);
# by ID

G:=PCGroup([5,-3,3,-3,-3,-3,45,57,78]);
// Polycyclic

G:=Group<a,b|a^3=b^81=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C81 in TeX

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