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G = D119order 238 = 2·7·17

Dihedral group

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

Aliases: D119, C7⋊D17, C17⋊D7, C1191C2, sometimes denoted D238 or Dih119 or Dih238, SmallGroup(238,3)

Series: Derived Chief Lower central Upper central

C1C119 — D119
C1C17C119 — D119
C119 — D119
C1

Generators and relations for D119
 G = < a,b | a119=b2=1, bab=a-1 >

119C2
17D7
7D17

Smallest permutation representation of D119
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)
(1 119)(2 118)(3 117)(4 116)(5 115)(6 114)(7 113)(8 112)(9 111)(10 110)(11 109)(12 108)(13 107)(14 106)(15 105)(16 104)(17 103)(18 102)(19 101)(20 100)(21 99)(22 98)(23 97)(24 96)(25 95)(26 94)(27 93)(28 92)(29 91)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 84)(37 83)(38 82)(39 81)(40 80)(41 79)(42 78)(43 77)(44 76)(45 75)(46 74)(47 73)(48 72)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)

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

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

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

D119 is a maximal subgroup of   D7×D17
D119 is a maximal quotient of   Dic119

61 conjugacy classes

class 1  2 7A7B7C17A···17H119A···119AV
order1277717···17119···119
size11192222···22···2

61 irreducible representations

dim11222
type+++++
imageC1C2D7D17D119
kernelD119C119C17C7C1
# reps113848

Matrix representation of D119 in GL2(𝔽239) generated by

14119
220150
,
14119
19998
G:=sub<GL(2,GF(239))| [141,220,19,150],[141,199,19,98] >;

D119 in GAP, Magma, Sage, TeX

D_{119}
% in TeX

G:=Group("D119");
// GroupNames label

G:=SmallGroup(238,3);
// by ID

G=gap.SmallGroup(238,3);
# by ID

G:=PCGroup([3,-2,-7,-17,73,2018]);
// Polycyclic

G:=Group<a,b|a^119=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D119 in TeX

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