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Cayley diagram

C2×Dic27 #6170

order216
isoclinismisoclinic class of D27
automorphismAut(G) = D4×AGL(1,ℤ/27) (order 3888) · Inn(G) = D27 · Out(G) = D4×C9
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×Dic27
unambiguous semidirect product(C2×C27)⋊C4

Splittings

derived, SylowC27 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC22D27
commutator, hypocenterC27C2×C4
FrattiniC2×C9C2×D3
FittingC22×C27C2
socle, prime radicalC22×C3D9
cosocleC2×C27C22
LCSG ▷ C27
UCSC1 ◁ C22

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 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);(1,2)(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)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×Dic27, Dic27⋊C4, C4⋊Dic27, …
maximal normal subgroup ofC4×Dic27, Dic27⋊C4, C4⋊Dic27, …
maximal quotient ofC2×C27⋊C8, #88363, C4×Dic27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 7);