Copied to
clipboard

G = D7×C26order 364 = 22·7·13

Direct product of C26 and D7

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

Aliases: D7×C26, C14⋊C26, C1823C2, C914C22, C7⋊(C2×C26), SmallGroup(364,9)

Series: Derived Chief Lower central Upper central

C1C7 — D7×C26
C1C7C91C13×D7 — D7×C26
C7 — D7×C26
C1C26

Generators and relations for D7×C26
 G = < a,b,c | a26=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C2
7C22
7C26
7C26
7C2×C26

Smallest permutation representation of D7×C26
On 182 points
Generators in S182
(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)
(1 92 120 147 49 177 63)(2 93 121 148 50 178 64)(3 94 122 149 51 179 65)(4 95 123 150 52 180 66)(5 96 124 151 27 181 67)(6 97 125 152 28 182 68)(7 98 126 153 29 157 69)(8 99 127 154 30 158 70)(9 100 128 155 31 159 71)(10 101 129 156 32 160 72)(11 102 130 131 33 161 73)(12 103 105 132 34 162 74)(13 104 106 133 35 163 75)(14 79 107 134 36 164 76)(15 80 108 135 37 165 77)(16 81 109 136 38 166 78)(17 82 110 137 39 167 53)(18 83 111 138 40 168 54)(19 84 112 139 41 169 55)(20 85 113 140 42 170 56)(21 86 114 141 43 171 57)(22 87 115 142 44 172 58)(23 88 116 143 45 173 59)(24 89 117 144 46 174 60)(25 90 118 145 47 175 61)(26 91 119 146 48 176 62)
(1 76)(2 77)(3 78)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 111)(28 112)(29 113)(30 114)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 105)(48 106)(49 107)(50 108)(51 109)(52 110)(79 177)(80 178)(81 179)(82 180)(83 181)(84 182)(85 157)(86 158)(87 159)(88 160)(89 161)(90 162)(91 163)(92 164)(93 165)(94 166)(95 167)(96 168)(97 169)(98 170)(99 171)(100 172)(101 173)(102 174)(103 175)(104 176)(131 144)(132 145)(133 146)(134 147)(135 148)(136 149)(137 150)(138 151)(139 152)(140 153)(141 154)(142 155)(143 156)

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

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

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

130 conjugacy classes

class 1 2A2B2C7A7B7C13A···13L14A14B14C26A···26L26M···26AJ91A···91AJ182A···182AJ
order122277713···1314141426···2626···2691···91182···182
size11772221···12221···17···72···22···2

130 irreducible representations

dim1111112222
type+++++
imageC1C2C2C13C26C26D7D14C13×D7D7×C26
kernelD7×C26C13×D7C182D14D7C14C26C13C2C1
# reps121122412333636

Matrix representation of D7×C26 in GL2(𝔽547) generated by

1970
0197
,
1791
5460
,
0546
5460
G:=sub<GL(2,GF(547))| [197,0,0,197],[179,546,1,0],[0,546,546,0] >;

D7×C26 in GAP, Magma, Sage, TeX

D_7\times C_{26}
% in TeX

G:=Group("D7xC26");
// GroupNames label

G:=SmallGroup(364,9);
// by ID

G=gap.SmallGroup(364,9);
# by ID

G:=PCGroup([4,-2,-2,-13,-7,4995]);
// Polycyclic

G:=Group<a,b,c|a^26=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×C26 in TeX

׿
×
𝔽