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Cayley diagram

(C27⋊C3)⋊C4 #66320

order324
isoclinismisoclinic class of D27⋊C3
automorphismAut(G) = C2×AGL(1,ℤ/27) (order 972) · Inn(G) = D27⋊C3 · Out(G) = C2×C3
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product(C27⋊C3)⋊C4
unambiguous semidirect productDic27⋊C3

Splittings

derivedC27 ⋊ C3×C4
SylowC27⋊C3 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2D27⋊C3
commutator, hypocenterC27C3×C4
FrattiniC2×C9C3×D3
FittingC2×C27⋊C3C2
socleC2×C3C3×D9
prime radicalC2×C32D9
cosocleC2×C27C2×C3
LCSG ▷ C27
UCSC1 ◁ C2

28 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C27⋊C3)⋊Q8, C4×D27⋊C3, C2×(C27⋊C3)⋊C4, …
maximal normal subgroup of(C27⋊C3)⋊Q8, C4×D27⋊C3, C2×(C27⋊C3)⋊C4, …
maximal quotient of(C27⋊C3)⋊C8, C2×(C27⋊C3)⋊C4, #6864⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 12);