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Cayley diagram

C23×C22⋊A4 #87590

order384
isoclinismisoclinic class of C22⋊A4
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C27

Aliases

direct productC23×C22⋊A4

Splittings

Fitting, socle, SylowC27 ⋊ C3
derivedC24 ⋊ C23×C3

Subgroups

subgroupG/H
center, hypercenterC23C22⋊A4
commutator, cosocle, hypocenterC24C23×C3
FrattiniC1G
Fitting, socleC27C3
LCSG ▷ C24
UCSC1 ◁ C23

9924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72341339, #72341365, #72341372, …
maximal subgroup of#14679⋊C3, C22×#379⋊C3, C2×AΣL(1,𝔽64), …
maximal normal subgroup of#14679⋊C3, C22×#379⋊C3, C2×AΣL(1,𝔽64), …
maximal quotient ofC22×C4×C22⋊A4, C2×D4×C22⋊A4, C2×Q8×C22⋊A4, …
automorphism group ofC3×#126

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 20166);