Finite Group Databasebeta

← Prev (#64260) Next (#64262) →

Cayley diagram

C4⋊(C3⋊S4) #64261

order288
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22×C3

Aliases

unambiguous semidirect productC4⋊(C3⋊S4)

Splittings

FittingC22×C3×C4 ⋊ D3

Subgroups

subgroupG/H
center, FrattiniC2C2×C3⋊S4
commutator, cosocleC2×C3×A4C22
FittingC22×C3×C4D3
socleC23×C3C2×D3
hypercenterC4C3⋊S4
hypocenterC3×A4D4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2 ◁ C4

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 3 generators on 11 points:

(1,2,3,4)(5,6)(7,8)(9,10,11);(2,4)(6,7)(10,11);(6,7,8)

Polycyclic presentation:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10593178, #10593179, #10593184, …
maximal normal subgroup of#10593178, #10593179, #10593184, …
maximal quotient ofC4⋊(C3⋊#226), C4⋊(C3⋊GL(2,𝔽3)), #10593629, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 909);