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G = D117order 234 = 2·32·13

Dihedral group

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

Aliases: D117, C9⋊D13, C13⋊D9, C3.D39, C1171C2, C39.1S3, sometimes denoted D234 or Dih117 or Dih234, SmallGroup(234,5)

Series: Derived Chief Lower central Upper central

C1C117 — D117
C1C3C39C117 — D117
C117 — D117
C1

Generators and relations for D117
 G = < a,b | a117=b2=1, bab=a-1 >

117C2
39S3
9D13
13D9
3D39

Smallest permutation representation of D117
On 117 points
Generators in S117
(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)
(2 117)(3 116)(4 115)(5 114)(6 113)(7 112)(8 111)(9 110)(10 109)(11 108)(12 107)(13 106)(14 105)(15 104)(16 103)(17 102)(18 101)(19 100)(20 99)(21 98)(22 97)(23 96)(24 95)(25 94)(26 93)(27 92)(28 91)(29 90)(30 89)(31 88)(32 87)(33 86)(34 85)(35 84)(36 83)(37 82)(38 81)(39 80)(40 79)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 72)(48 71)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 60)

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

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

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

D117 is a maximal subgroup of   D9×D13
D117 is a maximal quotient of   Dic117

60 conjugacy classes

class 1  2  3 9A9B9C13A···13F39A···39L117A···117AJ
order12399913···1339···39117···117
size111722222···22···22···2

60 irreducible representations

dim1122222
type+++++++
imageC1C2S3D9D13D39D117
kernelD117C117C39C13C9C3C1
# reps111361236

Matrix representation of D117 in GL2(𝔽937) generated by

272599
338610
,
10
936936
G:=sub<GL(2,GF(937))| [272,338,599,610],[1,936,0,936] >;

D117 in GAP, Magma, Sage, TeX

D_{117}
% in TeX

G:=Group("D117");
// GroupNames label

G:=SmallGroup(234,5);
// by ID

G=gap.SmallGroup(234,5);
# by ID

G:=PCGroup([4,-2,-3,-13,-3,657,629,866,2499]);
// Polycyclic

G:=Group<a,b|a^117=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D117 in TeX

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