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Cayley diagram

C4×D27 #6168

order216
isoclinismisoclinic class of D27
automorphismAut(G) = C22×AGL(1,ℤ/27) (order 1944) · Inn(G) = D27 · Out(G) = C22×C9
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productC4×D27

Splittings

FittingC4×C27 ⋊ C2
derived, SylowC27 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC4D27
commutator, hypocenterC27C2×C4
FrattiniC2×C9C2×D3
FittingC4×C27C2
socleC2×C3C2×D9
prime radicalC2×D27C2
cosocleC2×C27C22
LCSG ▷ C27
UCSC1 ◁ C4

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 31 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×D27, #88359, C2×C4×D27, …
maximal normal subgroup ofC8×D27, #88359, C2×C4×D27, …
maximal quotient ofC8×D27, #88359, C4×Dic27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 5);