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Cayley diagram

D116 #6602

order232
isoclinismisoclinic class of Dic58
automorphismAut(G) = D4×AGL(1,𝔽29) (order 6496) · Inn(G) = C2×D29 · Out(G) = C2×C4×C7
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD116
unambiguous semidirect productC4⋊D29

Splittings

FittingC4×C29 ⋊ C2
SylowC29 ⋊ D4

Subgroups

subgroupG/H
center, FrattiniC2C2×D29
commutator, socle, cosocleC2×C29C22
FittingC4×C29C2
hypercenterC4D29
hypocenterC29D4
LCSG ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C4

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 116

A minimal faithful permutation representation — 2 generators on 33 points:

(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);(2,4)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)

Polycyclic presentation:

2^3,29|b^2=c,b^a=bc,d^a=d^28

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#90773, D232, #90782, …
maximal normal subgroup of#90773, D232, #90782, …
maximal quotient of#90773, D232, Dic116, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(232, 6);