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G = D9×C23order 414 = 2·32·23

Direct product of C23 and D9

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D9×C23, C9⋊C46, C2073C2, C69.2S3, C3.(S3×C23), SmallGroup(414,1)

Series: Derived Chief Lower central Upper central

C1C9 — D9×C23
C1C3C9C207 — D9×C23
C9 — D9×C23
C1C23

Generators and relations for D9×C23
 G = < a,b,c | a23=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

9C2
3S3
9C46
3S3×C23

Smallest permutation representation of D9×C23
On 207 points
Generators in S207
(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)
(1 193 125 26 95 82 50 163 146)(2 194 126 27 96 83 51 164 147)(3 195 127 28 97 84 52 165 148)(4 196 128 29 98 85 53 166 149)(5 197 129 30 99 86 54 167 150)(6 198 130 31 100 87 55 168 151)(7 199 131 32 101 88 56 169 152)(8 200 132 33 102 89 57 170 153)(9 201 133 34 103 90 58 171 154)(10 202 134 35 104 91 59 172 155)(11 203 135 36 105 92 60 173 156)(12 204 136 37 106 70 61 174 157)(13 205 137 38 107 71 62 175 158)(14 206 138 39 108 72 63 176 159)(15 207 116 40 109 73 64 177 160)(16 185 117 41 110 74 65 178 161)(17 186 118 42 111 75 66 179 139)(18 187 119 43 112 76 67 180 140)(19 188 120 44 113 77 68 181 141)(20 189 121 45 114 78 69 182 142)(21 190 122 46 115 79 47 183 143)(22 191 123 24 93 80 48 184 144)(23 192 124 25 94 81 49 162 145)
(1 146)(2 147)(3 148)(4 149)(5 150)(6 151)(7 152)(8 153)(9 154)(10 155)(11 156)(12 157)(13 158)(14 159)(15 160)(16 161)(17 139)(18 140)(19 141)(20 142)(21 143)(22 144)(23 145)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 122)(48 123)(49 124)(50 125)(51 126)(52 127)(53 128)(54 129)(55 130)(56 131)(57 132)(58 133)(59 134)(60 135)(61 136)(62 137)(63 138)(64 116)(65 117)(66 118)(67 119)(68 120)(69 121)(162 192)(163 193)(164 194)(165 195)(166 196)(167 197)(168 198)(169 199)(170 200)(171 201)(172 202)(173 203)(174 204)(175 205)(176 206)(177 207)(178 185)(179 186)(180 187)(181 188)(182 189)(183 190)(184 191)

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

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

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

138 conjugacy classes

class 1  2  3 9A9B9C23A···23V46A···46V69A···69V207A···207BN
order12399923···2346···4669···69207···207
size1922221···19···92···22···2

138 irreducible representations

dim11112222
type++++
imageC1C2C23C46S3D9S3×C23D9×C23
kernelD9×C23C207D9C9C69C23C3C1
# reps112222132266

Matrix representation of D9×C23 in GL2(𝔽829) generated by

6160
0616
,
46570
759535
,
759535
46570
G:=sub<GL(2,GF(829))| [616,0,0,616],[465,759,70,535],[759,465,535,70] >;

D9×C23 in GAP, Magma, Sage, TeX

D_9\times C_{23}
% in TeX

G:=Group("D9xC23");
// GroupNames label

G:=SmallGroup(414,1);
// by ID

G=gap.SmallGroup(414,1);
# by ID

G:=PCGroup([4,-2,-23,-3,-3,2762,82,4419]);
// Polycyclic

G:=Group<a,b,c|a^23=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D9×C23 in TeX

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