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G = D7×C27order 378 = 2·33·7

Direct product of C27 and D7

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

Aliases: D7×C27, C73C54, C1893C2, C63.3C6, C21.3C18, C9.(C3×D7), C3.(C9×D7), (C9×D7).2C3, (C3×D7).2C9, SmallGroup(378,4)

Series: Derived Chief Lower central Upper central

C1C7 — D7×C27
C1C7C21C63C189 — D7×C27
C7 — D7×C27
C1C27

Generators and relations for D7×C27
 G = < a,b,c | a27=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C6
7C18
7C54

Smallest permutation representation of D7×C27
On 189 points
Generators in S189
(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)
(1 176 84 132 145 32 55)(2 177 85 133 146 33 56)(3 178 86 134 147 34 57)(4 179 87 135 148 35 58)(5 180 88 109 149 36 59)(6 181 89 110 150 37 60)(7 182 90 111 151 38 61)(8 183 91 112 152 39 62)(9 184 92 113 153 40 63)(10 185 93 114 154 41 64)(11 186 94 115 155 42 65)(12 187 95 116 156 43 66)(13 188 96 117 157 44 67)(14 189 97 118 158 45 68)(15 163 98 119 159 46 69)(16 164 99 120 160 47 70)(17 165 100 121 161 48 71)(18 166 101 122 162 49 72)(19 167 102 123 136 50 73)(20 168 103 124 137 51 74)(21 169 104 125 138 52 75)(22 170 105 126 139 53 76)(23 171 106 127 140 54 77)(24 172 107 128 141 28 78)(25 173 108 129 142 29 79)(26 174 82 130 143 30 80)(27 175 83 131 144 31 81)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 81)(28 172)(29 173)(30 174)(31 175)(32 176)(33 177)(34 178)(35 179)(36 180)(37 181)(38 182)(39 183)(40 184)(41 185)(42 186)(43 187)(44 188)(45 189)(46 163)(47 164)(48 165)(49 166)(50 167)(51 168)(52 169)(53 170)(54 171)(82 143)(83 144)(84 145)(85 146)(86 147)(87 148)(88 149)(89 150)(90 151)(91 152)(92 153)(93 154)(94 155)(95 156)(96 157)(97 158)(98 159)(99 160)(100 161)(101 162)(102 136)(103 137)(104 138)(105 139)(106 140)(107 141)(108 142)

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

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

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

135 conjugacy classes

class 1  2 3A3B6A6B7A7B7C9A···9F18A···18F21A···21F27A···27R54A···54R63A···63R189A···189BB
order1233667779···918···1821···2127···2754···5463···63189···189
size1711772221···17···72···21···17···72···22···2

135 irreducible representations

dim111111112222
type+++
imageC1C2C3C6C9C18C27C54D7C3×D7C9×D7D7×C27
kernelD7×C27C189C9×D7C63C3×D7C21D7C7C27C9C3C1
# reps1122661818361854

Matrix representation of D7×C27 in GL2(𝔽379) generated by

2940
0294
,
01
378205
,
01
10
G:=sub<GL(2,GF(379))| [294,0,0,294],[0,378,1,205],[0,1,1,0] >;

D7×C27 in GAP, Magma, Sage, TeX

D_7\times C_{27}
% in TeX

G:=Group("D7xC27");
// GroupNames label

G:=SmallGroup(378,4);
// by ID

G=gap.SmallGroup(378,4);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,36,57,8104]);
// Polycyclic

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

Export

Subgroup lattice of D7×C27 in TeX

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