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G = D7xC17order 238 = 2·7·17

Direct product of C17 and D7

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

Aliases: D7xC17, C7:C34, C119:3C2, SmallGroup(238,1)

Series: Derived Chief Lower central Upper central

C1C7 — D7xC17
C1C7C119 — D7xC17
C7 — D7xC17
C1C17

Generators and relations for D7xC17
 G = < a,b,c | a17=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 20 in 8 conjugacy classes, 6 normal (all characteristic)
Quotients: C1, C2, D7, C17, C34, D7xC17
7C2
7C34

Smallest permutation representation of D7xC17
On 119 points
Generators in S119
(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)
(1 117 50 64 99 82 27)(2 118 51 65 100 83 28)(3 119 35 66 101 84 29)(4 103 36 67 102 85 30)(5 104 37 68 86 69 31)(6 105 38 52 87 70 32)(7 106 39 53 88 71 33)(8 107 40 54 89 72 34)(9 108 41 55 90 73 18)(10 109 42 56 91 74 19)(11 110 43 57 92 75 20)(12 111 44 58 93 76 21)(13 112 45 59 94 77 22)(14 113 46 60 95 78 23)(15 114 47 61 96 79 24)(16 115 48 62 97 80 25)(17 116 49 63 98 81 26)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(35 101)(36 102)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 99)(51 100)(69 104)(70 105)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 113)(79 114)(80 115)(81 116)(82 117)(83 118)(84 119)(85 103)

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

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

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

85 conjugacy classes

class 1  2 7A7B7C17A···17P34A···34P119A···119AV
order1277717···1734···34119···119
size172221···17···72···2

85 irreducible representations

dim111122
type+++
imageC1C2C17C34D7D7xC17
kernelD7xC17C119D7C7C17C1
# reps111616348

Matrix representation of D7xC17 in GL2(F239) generated by

750
075
,
23249
23841
,
23248
2387
G:=sub<GL(2,GF(239))| [75,0,0,75],[232,238,49,41],[232,238,48,7] >;

D7xC17 in GAP, Magma, Sage, TeX

D_7\times C_{17}
% in TeX

G:=Group("D7xC17");
// GroupNames label

G:=SmallGroup(238,1);
// by ID

G=gap.SmallGroup(238,1);
# by ID

G:=PCGroup([3,-2,-17,-7,1838]);
// Polycyclic

G:=Group<a,b,c|a^17=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D7xC17 in TeX

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