Finite Group Databasebeta

← Prev (#88594) Next (#88596) →

Cayley diagram

C9×SL(2,ℤ/4) #88595

order432
isoclinismisoclinic class of S4
automorphismAut(G) = C22×C3×S4 (order 288) · Inn(G) = S4 · Out(G) = C22×C3
homologyAbelianization (H1) = C4×C9 · Schur multiplier (H2) = C2

Aliases

direct productC9×SL(2,ℤ/4)
unambiguous semidirect productA4⋊(C4×C9)

Splittings

derivedA4 ⋊ C4×C9

Subgroups

subgroupG/H
center, hypercenterC2×C9S4
commutator, hypocenterA4C4×C9
FrattiniC2×C3C3×S4
FittingC23×C9D3
socleC23×C3C3×D3
prime radical, cosocleC2×C3×A4C2×C3
LCSG ▷ A4
UCSC1 ◁ C2×C9

65 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2,3,4)(5,6,8,7)(9,10,12,11,13,15,14,16,17);(6,7,8)

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD9×SL(2,ℤ/4), (C9⋊S4)⋊C4, #11722242, …
maximal normal subgroup ofD9×SL(2,ℤ/4), (C9⋊S4)⋊C4, #11722242, …
maximal quotient ofC9×A4⋊C8, C9×#859, C9×GU(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 242);