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Cayley diagram

C27⋊A4 #66351

order324
isoclinismisoclinic class of C9⋊A4
automorphismAut(G) = D3×C9×A4 (order 648) · Inn(G) = C3×A4 · Out(G) = C2×C9
homologyAbelianization (H1) = C3×C9 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC27⋊A4

Splittings

FittingC22×C27 ⋊ C3
SylowC22 ⋊ C27⋊C3

Subgroups

subgroupG/H
center, FrattiniC9C3×A4
commutator, socleC22×C3C3×C9
FittingC22×C27C3
prime radicalC3×A4C9
cosocleC22×C9C32
hypercenterC27A4
hypocenterC22C27⋊C3
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C9 ◁ C27

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:

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

Polycyclic presentation:

3^4,2^2|b^3=c,c^3=d,b^a=bd,e^a=f,f^a=ef

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD27⋊A4, C2×C27⋊A4, C3×C27⋊A4, …
maximal normal subgroup ofD27⋊A4, C2×C27⋊A4, C3×C27⋊A4, …
maximal quotient ofC27⋊SL(2,𝔽3), C2×C27⋊A4, (C22⋊C9)⋊C27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 43);