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Cayley diagram

D108 #6169

order216
isoclinismisoclinic class of Dic54
automorphismAut(G) = D4×AGL(1,ℤ/27) (order 3888) · Inn(G) = C2×D27 · Out(G) = C22×C9
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD108
unambiguous semidirect productC4⋊D27

Splittings

FittingC4×C27 ⋊ C2
SylowC27 ⋊ D4

Subgroups

subgroupG/H
centerC2C2×D27
commutator, cosocleC2×C27C22
FrattiniC2×C9C2×D3
FittingC4×C27C2
socleC2×C3C2×D9
hypercenterC4D27
hypocenterC27D4
LCSG ▷ C2×C27 ▷ C27
UCSC1 ◁ C2 ◁ 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:

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

Polycyclic presentation:

2^3,3^3|b^2=c,d^3=e^2f,e^3=f^2,b^a=bc,d^a=d^2e,e^a=e^2f,f^a=f^2

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#88360, D216, #88369, …
maximal normal subgroup of#88360, D216, #88369, …
maximal quotient ofDic108, #88360, D216, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 6);