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G = D118order 236 = 22·59

Dihedral group

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

Aliases: D118, C2×D59, C118⋊C2, C59⋊C22, sometimes denoted D236 or Dih118 or Dih236, SmallGroup(236,3)

Series: Derived Chief Lower central Upper central

C1C59 — D118
C1C59D59 — D118
C59 — D118
C1C2

Generators and relations for D118
 G = < a,b | a118=b2=1, bab=a-1 >

59C2
59C2
59C22

Smallest permutation representation of D118
On 118 points
Generators in S118
(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)
(1 118)(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(118)| (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), (1,118)(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( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118), (1,118)(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([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118)], [(1,118),(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)])

D118 is a maximal subgroup of   D236  C59⋊D4
D118 is a maximal quotient of   Dic118  D236  C59⋊D4

62 conjugacy classes

class 1 2A2B2C59A···59AC118A···118AC
order122259···59118···118
size1159592···22···2

62 irreducible representations

dim11122
type+++++
imageC1C2C2D59D118
kernelD118D59C118C2C1
# reps1212929

Matrix representation of D118 in GL3(𝔽709) generated by

70800
0412405
0304304
,
100
0412405
0626297
G:=sub<GL(3,GF(709))| [708,0,0,0,412,304,0,405,304],[1,0,0,0,412,626,0,405,297] >;

D118 in GAP, Magma, Sage, TeX

D_{118}
% in TeX

G:=Group("D118");
// GroupNames label

G:=SmallGroup(236,3);
// by ID

G=gap.SmallGroup(236,3);
# by ID

G:=PCGroup([3,-2,-2,-59,2090]);
// Polycyclic

G:=Group<a,b|a^118=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D118 in TeX

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