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Cayley diagram

C22⋊D23 #4406

order184
isoclinismisoclinic class of Dic46
automorphismAut(G) = C22×AGL(1,𝔽23) (order 2024) · Inn(G) = C2×D23 · Out(G) = C2×C11
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC22⋊D23

Splittings

FittingC22×C23 ⋊ C2
SylowC23 ⋊ D4

Subgroups

subgroupG/H
center, FrattiniC2C2×D23
commutator, socle, cosocleC2×C23C22
FittingC22×C23C2
hypercenterC22D23
hypocenterC23D4
LCSG ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C22

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:

(1,2,4,6,8,10,12,14,16,18,20,22,23,21,19,17,15,13,11,9,7,5,3)(25,27);(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)

Polycyclic presentation:

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

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#67288, D4×D23, #67290, …
maximal normal subgroup of#67288, D4×D23, #67290, …
maximal quotient ofDic23⋊C4, (C2×D23)⋊C4, #67272, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(184, 7);