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## G = C7×D27order 378 = 2·33·7

### Direct product of C7 and D27

Aliases: C7×D27, C27⋊C14, C1892C2, C63.2S3, C21.2D9, C9.(S3×C7), C3.(C7×D9), SmallGroup(378,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C27 — C7×D27
 Chief series C1 — C3 — C9 — C27 — C189 — C7×D27
 Lower central C27 — C7×D27
 Upper central C1 — C7

Generators and relations for C7×D27
G = < a,b,c | a7=b27=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C7×D27
On 189 points
Generators in S189
(1 176 139 119 92 55 28)(2 177 140 120 93 56 29)(3 178 141 121 94 57 30)(4 179 142 122 95 58 31)(5 180 143 123 96 59 32)(6 181 144 124 97 60 33)(7 182 145 125 98 61 34)(8 183 146 126 99 62 35)(9 184 147 127 100 63 36)(10 185 148 128 101 64 37)(11 186 149 129 102 65 38)(12 187 150 130 103 66 39)(13 188 151 131 104 67 40)(14 189 152 132 105 68 41)(15 163 153 133 106 69 42)(16 164 154 134 107 70 43)(17 165 155 135 108 71 44)(18 166 156 109 82 72 45)(19 167 157 110 83 73 46)(20 168 158 111 84 74 47)(21 169 159 112 85 75 48)(22 170 160 113 86 76 49)(23 171 161 114 87 77 50)(24 172 162 115 88 78 51)(25 173 136 116 89 79 52)(26 174 137 117 90 80 53)(27 175 138 118 91 81 54)
(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)
(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(56 81)(57 80)(58 79)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(103 108)(104 107)(105 106)(109 129)(110 128)(111 127)(112 126)(113 125)(114 124)(115 123)(116 122)(117 121)(118 120)(130 135)(131 134)(132 133)(136 142)(137 141)(138 140)(143 162)(144 161)(145 160)(146 159)(147 158)(148 157)(149 156)(150 155)(151 154)(152 153)(163 189)(164 188)(165 187)(166 186)(167 185)(168 184)(169 183)(170 182)(171 181)(172 180)(173 179)(174 178)(175 177)

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

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

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

105 conjugacy classes

 class 1 2 3 7A ··· 7F 9A 9B 9C 14A ··· 14F 21A ··· 21F 27A ··· 27I 63A ··· 63R 189A ··· 189BB order 1 2 3 7 ··· 7 9 9 9 14 ··· 14 21 ··· 21 27 ··· 27 63 ··· 63 189 ··· 189 size 1 27 2 1 ··· 1 2 2 2 27 ··· 27 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

105 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C7 C14 S3 D9 S3×C7 D27 C7×D9 C7×D27 kernel C7×D27 C189 D27 C27 C63 C21 C9 C7 C3 C1 # reps 1 1 6 6 1 3 6 9 18 54

Matrix representation of C7×D27 in GL2(𝔽379) generated by

 125 0 0 125
,
 30 25 354 55
,
 183 153 336 196
G:=sub<GL(2,GF(379))| [125,0,0,125],[30,354,25,55],[183,336,153,196] >;

C7×D27 in GAP, Magma, Sage, TeX

C_7\times D_{27}
% in TeX

G:=Group("C7xD27");
// GroupNames label

G:=SmallGroup(378,3);
// by ID

G=gap.SmallGroup(378,3);
# by ID

G:=PCGroup([5,-2,-7,-3,-3,-3,1052,237,4203,138,6304]);
// Polycyclic

G:=Group<a,b,c|a^7=b^27=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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