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## G = D9×C26order 468 = 22·32·13

### Direct product of C26 and D9

Aliases: D9×C26, C18⋊C26, C2343C2, C78.6S3, C39.3D6, C1174C22, C9⋊(C2×C26), C3.(S3×C26), C6.2(S3×C13), SmallGroup(468,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — D9×C26
 Chief series C1 — C3 — C9 — C117 — C13×D9 — D9×C26
 Lower central C9 — D9×C26
 Upper central C1 — C26

Generators and relations for D9×C26
G = < a,b,c | a26=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D9×C26
On 234 points
Generators in S234
(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)
(1 177 110 48 211 131 70 207 94)(2 178 111 49 212 132 71 208 95)(3 179 112 50 213 133 72 183 96)(4 180 113 51 214 134 73 184 97)(5 181 114 52 215 135 74 185 98)(6 182 115 27 216 136 75 186 99)(7 157 116 28 217 137 76 187 100)(8 158 117 29 218 138 77 188 101)(9 159 118 30 219 139 78 189 102)(10 160 119 31 220 140 53 190 103)(11 161 120 32 221 141 54 191 104)(12 162 121 33 222 142 55 192 79)(13 163 122 34 223 143 56 193 80)(14 164 123 35 224 144 57 194 81)(15 165 124 36 225 145 58 195 82)(16 166 125 37 226 146 59 196 83)(17 167 126 38 227 147 60 197 84)(18 168 127 39 228 148 61 198 85)(19 169 128 40 229 149 62 199 86)(20 170 129 41 230 150 63 200 87)(21 171 130 42 231 151 64 201 88)(22 172 105 43 232 152 65 202 89)(23 173 106 44 233 153 66 203 90)(24 174 107 45 234 154 67 204 91)(25 175 108 46 209 155 68 205 92)(26 176 109 47 210 156 69 206 93)
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 101)(9 102)(10 103)(11 104)(12 79)(13 80)(14 81)(15 82)(16 83)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 136)(28 137)(29 138)(30 139)(31 140)(32 141)(33 142)(34 143)(35 144)(36 145)(37 146)(38 147)(39 148)(40 149)(41 150)(42 151)(43 152)(44 153)(45 154)(46 155)(47 156)(48 131)(49 132)(50 133)(51 134)(52 135)(53 119)(54 120)(55 121)(56 122)(57 123)(58 124)(59 125)(60 126)(61 127)(62 128)(63 129)(64 130)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(157 187)(158 188)(159 189)(160 190)(161 191)(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)(178 208)(179 183)(180 184)(181 185)(182 186)

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

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

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

156 conjugacy classes

 class 1 2A 2B 2C 3 6 9A 9B 9C 13A ··· 13L 18A 18B 18C 26A ··· 26L 26M ··· 26AJ 39A ··· 39L 78A ··· 78L 117A ··· 117AJ 234A ··· 234AJ order 1 2 2 2 3 6 9 9 9 13 ··· 13 18 18 18 26 ··· 26 26 ··· 26 39 ··· 39 78 ··· 78 117 ··· 117 234 ··· 234 size 1 1 9 9 2 2 2 2 2 1 ··· 1 2 2 2 1 ··· 1 9 ··· 9 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

156 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C13 C26 C26 S3 D6 D9 D18 S3×C13 S3×C26 C13×D9 D9×C26 kernel D9×C26 C13×D9 C234 D18 D9 C18 C78 C39 C26 C13 C6 C3 C2 C1 # reps 1 2 1 12 24 12 1 1 3 3 12 12 36 36

Matrix representation of D9×C26 in GL2(𝔽937) generated by

 26 0 0 26
,
 262 472 465 734
,
 465 734 262 472
G:=sub<GL(2,GF(937))| [26,0,0,26],[262,465,472,734],[465,262,734,472] >;

D9×C26 in GAP, Magma, Sage, TeX

D_9\times C_{26}
% in TeX

G:=Group("D9xC26");
// GroupNames label

G:=SmallGroup(468,16);
// by ID

G=gap.SmallGroup(468,16);
# by ID

G:=PCGroup([5,-2,-2,-13,-3,-3,5203,138,7804]);
// Polycyclic

G:=Group<a,b,c|a^26=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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