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Cayley diagram

C22×D27 #6186

order216
isoclinismisoclinic class of D27
automorphismAut(G) = S4×AGL(1,ℤ/27) (order 11664) · Inn(G) = D27 · Out(G) = C9×S4
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC22×D27
unambiguous semidirect productC27⋊C23
unambiguous semidirect product(C2×C27)⋊C22

Splittings

FittingC22×C27 ⋊ C2
derived, SylowC27 ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC22D27
commutator, cosocle, hypocenterC27C23
FrattiniC9C22×D3
FittingC22×C27C2
socleC22×C3D9
LCSG ▷ C27
UCSC1 ◁ C22

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 3 generators on 31 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×D27)⋊C4, C2×C4×D27, C2×D108, …
maximal normal subgroup of(C2×D27)⋊C4, C2×C4×D27, C2×D108, …
maximal quotient ofC2×Dic54, C2×C4×D27, C2×D108, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 23);