Finite Group Databasebeta

← Prev (#87585) Next (#87587) →

Cayley diagram

C24×S4 #87586

order384
isoclinismisoclinic class of S4
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C211

Aliases

direct productC24×S4
unambiguous semidirect productA4⋊C25
unambiguous semidirect product(C2×A4)⋊C24

Splittings

Fitting, socleC26 ⋊ D3
derivedA4 ⋊ C25

Subgroups

subgroupG/H
center, hypercenterC24S4
commutator, cosocle, hypocenterA4C25
FrattiniC1G
Fitting, socleC26D3
LCSG ▷ A4
UCSC1 ◁ C24

5127 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 5 generators on 12 points:

(9,10,12,11);(7,8)(10,11,12);(5,6);(3,4);(1,2)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11711694, C22×(C2×S4)⋊C4, C2×C22⋊C4×S4, …
maximal normal subgroup of#11711694, C22×(C2×S4)⋊C4, C2×C22⋊C4×S4, …
maximal quotient ofC24×SL(2,ℤ/4), C23×C4×S4, C23×GL(2,ℤ/4), …
automorphism group ofC4×#226, #5388, #5389, …
outer automorphism group ofC3×C8×Q8

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 20162);