Finite Group Databasebeta

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Cayley diagram

D92 #4404

order184
isoclinismisoclinic class of Dic46
automorphismAut(G) = D4×AGL(1,𝔽23) (order 4048) · Inn(G) = C2×D23 · Out(G) = C22×C11
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD92
unambiguous semidirect productC4⋊D23

Splittings

FittingC4×C23 ⋊ C2
SylowC23 ⋊ D4

Subgroups

subgroupG/H
center, FrattiniC2C2×D23
commutator, socle, cosocleC2×C23C22
FittingC4×C23C2
hypercenterC4D23
hypocenterC23D4
LCSG ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C4

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 92

A minimal faithful permutation representation — 2 generators on 27 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);(2,4)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)

Polycyclic presentation:

2^3,23|b^2=c,b^a=bc,d^a=d^22

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#67263, D184, #67272, …
maximal normal subgroup of#67263, D184, #67272, …
maximal quotient of#67263, D184, Dic92, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(184, 5);