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Cayley diagram

C23×D27 #88580

order432
isoclinismisoclinic class of D27
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC23×D27
unambiguous semidirect productC27⋊C24
unambiguous semidirect product(C2×C27)⋊C23

Splittings

FittingC23×C27 ⋊ C2
derived, SylowC27 ⋊ C24

Subgroups

subgroupG/H
center, hypercenterC23D27
commutator, cosocle, hypocenterC27C24
FrattiniC9C23×D3
FittingC23×C27C2
socleC23×C3D9
LCSG ▷ C27
UCSC1 ◁ C23

268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 216

A minimal faithful permutation representation — 4 generators on 33 points:

(1,2)(3,4)(5,6)(8,11)(9,15)(10,18)(12,13)(14,24)(16,17)(19,26)(20,25)(21,31)(22,30)(23,29)(27,28)(32,33);(5,6)(7,26,20,27,31,29,9,8,30,10,33,24,16,13,12,17,14,32,18,22,11,15,23,21,28,25,19);(3,4);(1,2)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22⋊C4×D27, #11720131, C2×(C2×D27)⋊C4, …
maximal normal subgroup ofC22⋊C4×D27, #11720131, C2×(C2×D27)⋊C4, …
maximal quotient ofC22×Dic54, C22×C4×D27, C22×D108, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 227);