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Cayley diagram

C2×C3×D27 #66373

order324
isoclinismisoclinic class of D27
automorphismAut(G) = C22×AGL(1,ℤ/27) (order 1944) · Inn(G) = D27 · Out(G) = C22×C9
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3×D27

Splittings

FittingC2×C3×C27 ⋊ C2
derivedC27 ⋊ C22×C3
SylowC3×C27 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2×C3D27
commutator, cosocle, hypocenterC27C22×C3
FrattiniC9C2×C3×D3
FittingC2×C3×C27C2
socleC2×C32D9
LCSG ▷ C27
UCSC1 ◁ C2×C3

46 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic3×D27, #10619542, #10619544, …
maximal normal subgroup ofDic3×D27, #10619542, #10619544, …
maximal quotient ofC3×Dic54, C3×C4×D27, C3×D108, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 65);