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G = D116order 232 = 23·29

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D116, C4⋊D29, C291D4, C1161C2, D581C2, C2.4D58, C58.3C22, sometimes denoted D232 or Dih116 or Dih232, SmallGroup(232,6)

Series: Derived Chief Lower central Upper central

C1C58 — D116
C1C29C58D58 — D116
C29C58 — D116
C1C2C4

Generators and relations for D116
 G = < a,b | a116=b2=1, bab=a-1 >

58C2
58C2
29C22
29C22
2D29
2D29
29D4

Smallest permutation representation of D116
On 116 points
Generators in S116
(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)
(1 116)(2 115)(3 114)(4 113)(5 112)(6 111)(7 110)(8 109)(9 108)(10 107)(11 106)(12 105)(13 104)(14 103)(15 102)(16 101)(17 100)(18 99)(19 98)(20 97)(21 96)(22 95)(23 94)(24 93)(25 92)(26 91)(27 90)(28 89)(29 88)(30 87)(31 86)(32 85)(33 84)(34 83)(35 82)(36 81)(37 80)(38 79)(39 78)(40 77)(41 76)(42 75)(43 74)(44 73)(45 72)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)

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

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

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

D116 is a maximal subgroup of   C232⋊C2  D232  D4⋊D29  Q8⋊D29  D1165C2  D4×D29  Q82D29
D116 is a maximal quotient of   C232⋊C2  D232  Dic116  C4⋊Dic29  D58⋊C4

61 conjugacy classes

class 1 2A2B2C 4 29A···29N58A···58N116A···116AB
order1222429···2958···58116···116
size11585822···22···22···2

61 irreducible representations

dim1112222
type+++++++
imageC1C2C2D4D29D58D116
kernelD116C116D58C29C4C2C1
# reps1121141428

Matrix representation of D116 in GL2(𝔽233) generated by

1123
230183
,
121230
220112
G:=sub<GL(2,GF(233))| [112,230,3,183],[121,220,230,112] >;

D116 in GAP, Magma, Sage, TeX

D_{116}
% in TeX

G:=Group("D116");
// GroupNames label

G:=SmallGroup(232,6);
// by ID

G=gap.SmallGroup(232,6);
# by ID

G:=PCGroup([4,-2,-2,-2,-29,49,21,3587]);
// Polycyclic

G:=Group<a,b|a^116=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D116 in TeX

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