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G = D7×C19order 266 = 2·7·19

Direct product of C19 and D7

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

Aliases: D7×C19, C7⋊C38, C1333C2, SmallGroup(266,1)

Series: Derived Chief Lower central Upper central

C1C7 — D7×C19
C1C7C133 — D7×C19
C7 — D7×C19
C1C19

Generators and relations for D7×C19
 G = < a,b,c | a19=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C38

Smallest permutation representation of D7×C19
On 133 points
Generators in S133
(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)
(1 58 30 56 92 123 105)(2 59 31 57 93 124 106)(3 60 32 39 94 125 107)(4 61 33 40 95 126 108)(5 62 34 41 77 127 109)(6 63 35 42 78 128 110)(7 64 36 43 79 129 111)(8 65 37 44 80 130 112)(9 66 38 45 81 131 113)(10 67 20 46 82 132 114)(11 68 21 47 83 133 96)(12 69 22 48 84 115 97)(13 70 23 49 85 116 98)(14 71 24 50 86 117 99)(15 72 25 51 87 118 100)(16 73 26 52 88 119 101)(17 74 27 53 89 120 102)(18 75 28 54 90 121 103)(19 76 29 55 91 122 104)
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 113)(10 114)(11 96)(12 97)(13 98)(14 99)(15 100)(16 101)(17 102)(18 103)(19 104)(20 82)(21 83)(22 84)(23 85)(24 86)(25 87)(26 88)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 95)(34 77)(35 78)(36 79)(37 80)(38 81)(58 123)(59 124)(60 125)(61 126)(62 127)(63 128)(64 129)(65 130)(66 131)(67 132)(68 133)(69 115)(70 116)(71 117)(72 118)(73 119)(74 120)(75 121)(76 122)

G:=sub<Sym(133)| (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), (1,58,30,56,92,123,105)(2,59,31,57,93,124,106)(3,60,32,39,94,125,107)(4,61,33,40,95,126,108)(5,62,34,41,77,127,109)(6,63,35,42,78,128,110)(7,64,36,43,79,129,111)(8,65,37,44,80,130,112)(9,66,38,45,81,131,113)(10,67,20,46,82,132,114)(11,68,21,47,83,133,96)(12,69,22,48,84,115,97)(13,70,23,49,85,116,98)(14,71,24,50,86,117,99)(15,72,25,51,87,118,100)(16,73,26,52,88,119,101)(17,74,27,53,89,120,102)(18,75,28,54,90,121,103)(19,76,29,55,91,122,104), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,113)(10,114)(11,96)(12,97)(13,98)(14,99)(15,100)(16,101)(17,102)(18,103)(19,104)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,77)(35,78)(36,79)(37,80)(38,81)(58,123)(59,124)(60,125)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,115)(70,116)(71,117)(72,118)(73,119)(74,120)(75,121)(76,122)>;

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), (1,58,30,56,92,123,105)(2,59,31,57,93,124,106)(3,60,32,39,94,125,107)(4,61,33,40,95,126,108)(5,62,34,41,77,127,109)(6,63,35,42,78,128,110)(7,64,36,43,79,129,111)(8,65,37,44,80,130,112)(9,66,38,45,81,131,113)(10,67,20,46,82,132,114)(11,68,21,47,83,133,96)(12,69,22,48,84,115,97)(13,70,23,49,85,116,98)(14,71,24,50,86,117,99)(15,72,25,51,87,118,100)(16,73,26,52,88,119,101)(17,74,27,53,89,120,102)(18,75,28,54,90,121,103)(19,76,29,55,91,122,104), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,113)(10,114)(11,96)(12,97)(13,98)(14,99)(15,100)(16,101)(17,102)(18,103)(19,104)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,77)(35,78)(36,79)(37,80)(38,81)(58,123)(59,124)(60,125)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,115)(70,116)(71,117)(72,118)(73,119)(74,120)(75,121)(76,122) );

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)], [(1,58,30,56,92,123,105),(2,59,31,57,93,124,106),(3,60,32,39,94,125,107),(4,61,33,40,95,126,108),(5,62,34,41,77,127,109),(6,63,35,42,78,128,110),(7,64,36,43,79,129,111),(8,65,37,44,80,130,112),(9,66,38,45,81,131,113),(10,67,20,46,82,132,114),(11,68,21,47,83,133,96),(12,69,22,48,84,115,97),(13,70,23,49,85,116,98),(14,71,24,50,86,117,99),(15,72,25,51,87,118,100),(16,73,26,52,88,119,101),(17,74,27,53,89,120,102),(18,75,28,54,90,121,103),(19,76,29,55,91,122,104)], [(1,105),(2,106),(3,107),(4,108),(5,109),(6,110),(7,111),(8,112),(9,113),(10,114),(11,96),(12,97),(13,98),(14,99),(15,100),(16,101),(17,102),(18,103),(19,104),(20,82),(21,83),(22,84),(23,85),(24,86),(25,87),(26,88),(27,89),(28,90),(29,91),(30,92),(31,93),(32,94),(33,95),(34,77),(35,78),(36,79),(37,80),(38,81),(58,123),(59,124),(60,125),(61,126),(62,127),(63,128),(64,129),(65,130),(66,131),(67,132),(68,133),(69,115),(70,116),(71,117),(72,118),(73,119),(74,120),(75,121),(76,122)])

95 conjugacy classes

class 1  2 7A7B7C19A···19R38A···38R133A···133BB
order1277719···1938···38133···133
size172221···17···72···2

95 irreducible representations

dim111122
type+++
imageC1C2C19C38D7D7×C19
kernelD7×C19C133D7C7C19C1
# reps111818354

Matrix representation of D7×C19 in GL2(𝔽1597) generated by

5900
0590
,
01
15961321
,
01
10
G:=sub<GL(2,GF(1597))| [590,0,0,590],[0,1596,1,1321],[0,1,1,0] >;

D7×C19 in GAP, Magma, Sage, TeX

D_7\times C_{19}
% in TeX

G:=Group("D7xC19");
// GroupNames label

G:=SmallGroup(266,1);
// by ID

G=gap.SmallGroup(266,1);
# by ID

G:=PCGroup([3,-2,-19,-7,2054]);
// Polycyclic

G:=Group<a,b,c|a^19=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×C19 in TeX

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