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Cayley diagram

C22×D23 #4410

order184
isoclinismisoclinic class of D23
automorphismAut(G) = S4×AGL(1,𝔽23) (order 12144) · Inn(G) = D23 · Out(G) = C11×S4
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC22×D23
unambiguous semidirect productC23⋊C23
unambiguous semidirect product(C2×C23)⋊C22

Splittings

Fitting, socleC22×C23 ⋊ C2
derived, SylowC23 ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC22D23
commutator, cosocle, hypocenterC23C23
FrattiniC1G
Fitting, socleC22×C23C2
LCSG ▷ C23
UCSC1 ◁ C22

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2)(3,4)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27);(3,4)(5,6,8,10,12,14,16,18,20,22,24,26,27,25,23,21,19,17,15,13,11,9,7);(1,2)

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×D23)⋊C4, C2×C4×D23, C2×D92, …
maximal normal subgroup of(C2×D23)⋊C4, C2×C4×D23, C2×D92, …
maximal quotient ofC2×Dic46, C2×C4×D23, C2×D92, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(184, 11);