Finite Group Databasebeta

← Prev (#1076) Next (#1078) →

Cayley diagram

C2×D27 #1077

order108
isoclinismisoclinic class of D27
automorphismAut(G) = C2×AGL(1,ℤ/27) (order 972) · Inn(G) = D27 · Out(G) = C2×C9
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD54
direct productC2×D27
unambiguous semidirect productC27⋊C22
unambiguous semidirect product(C2×C27)⋊C2

Splittings

FittingC2×C27 ⋊ C2
derived, SylowC27 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2D27
commutator, cosocle, hypocenterC27C22
FrattiniC9C2×D3
FittingC2×C27C2
socleC2×C3D9
LCSG ▷ C27
UCSC1 ◁ C2

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 2 generators on 29 points:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×D27, D108, #6171, …
maximal normal subgroup ofC4×D27, D108, #6171, …
maximal quotient ofDic54, C4×D27, D108, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 4);