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G = C9×D23order 414 = 2·32·23

Direct product of C9 and D23

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

Aliases: C9×D23, C23⋊C18, C69.C6, C2072C2, C3.(C3×D23), (C3×D23).C3, SmallGroup(414,2)

Series: Derived Chief Lower central Upper central

C1C23 — C9×D23
C1C23C69C207 — C9×D23
C23 — C9×D23
C1C9

Generators and relations for C9×D23
 G = < a,b,c | a9=b23=c2=1, ab=ba, ac=ca, cbc=b-1 >

23C2
23C6
23C18

Smallest permutation representation of C9×D23
On 207 points
Generators in S207
(1 196 128 64 176 96 28 148 89)(2 197 129 65 177 97 29 149 90)(3 198 130 66 178 98 30 150 91)(4 199 131 67 179 99 31 151 92)(5 200 132 68 180 100 32 152 70)(6 201 133 69 181 101 33 153 71)(7 202 134 47 182 102 34 154 72)(8 203 135 48 183 103 35 155 73)(9 204 136 49 184 104 36 156 74)(10 205 137 50 162 105 37 157 75)(11 206 138 51 163 106 38 158 76)(12 207 116 52 164 107 39 159 77)(13 185 117 53 165 108 40 160 78)(14 186 118 54 166 109 41 161 79)(15 187 119 55 167 110 42 139 80)(16 188 120 56 168 111 43 140 81)(17 189 121 57 169 112 44 141 82)(18 190 122 58 170 113 45 142 83)(19 191 123 59 171 114 46 143 84)(20 192 124 60 172 115 24 144 85)(21 193 125 61 173 93 25 145 86)(22 194 126 62 174 94 26 146 87)(23 195 127 63 175 95 27 147 88)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(24 31)(25 30)(26 29)(27 28)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(47 57)(48 56)(49 55)(50 54)(51 53)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(70 84)(71 83)(72 82)(73 81)(74 80)(75 79)(76 78)(85 92)(86 91)(87 90)(88 89)(93 98)(94 97)(95 96)(99 115)(100 114)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)(117 138)(118 137)(119 136)(120 135)(121 134)(122 133)(123 132)(124 131)(125 130)(126 129)(127 128)(139 156)(140 155)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)(157 161)(158 160)(162 166)(163 165)(167 184)(168 183)(169 182)(170 181)(171 180)(172 179)(173 178)(174 177)(175 176)(185 206)(186 205)(187 204)(188 203)(189 202)(190 201)(191 200)(192 199)(193 198)(194 197)(195 196)

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

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

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

117 conjugacy classes

class 1  2 3A3B6A6B9A···9F18A···18F23A···23K69A···69V207A···207BN
order1233669···918···1823···2369···69207···207
size1231123231···123···232···22···22···2

117 irreducible representations

dim111111222
type+++
imageC1C2C3C6C9C18D23C3×D23C9×D23
kernelC9×D23C207C3×D23C69D23C23C9C3C1
# reps112266112266

Matrix representation of C9×D23 in GL3(𝔽829) generated by

16600
01250
00125
,
100
01941
0778692
,
82800
0425283
0595404
G:=sub<GL(3,GF(829))| [166,0,0,0,125,0,0,0,125],[1,0,0,0,194,778,0,1,692],[828,0,0,0,425,595,0,283,404] >;

C9×D23 in GAP, Magma, Sage, TeX

C_9\times D_{23}
% in TeX

G:=Group("C9xD23");
// GroupNames label

G:=SmallGroup(414,2);
// by ID

G=gap.SmallGroup(414,2);
# by ID

G:=PCGroup([4,-2,-3,-3,-23,29,6339]);
// Polycyclic

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

Export

Subgroup lattice of C9×D23 in TeX

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