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Cayley diagram

C8×C22⋊A4 #85664

order384
isoclinismisoclinic class of C22⋊A4
automorphism?
homologyAbelianization (H1) = C3×C8 · Schur multiplier (H2) = C24

Aliases

direct productC8×C22⋊A4

Splittings

Fitting, SylowC24×C8 ⋊ C3
derivedC24 ⋊ C3×C8

Subgroups

subgroupG/H
center, hypercenterC8C22⋊A4
commutator, hypocenterC24C3×C8
FrattiniC4C2×C22⋊A4
FittingC24×C8C3
socleC25C3×C4
prime radicalC2×C22⋊A4C4
cosocleC24×C4C2×C3
LCSG ▷ C24
UCSC1 ◁ C8

376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#12679⋊C3, C8×#120⋊C3, #21440⋊C3, …
maximal normal subgroup of#12679⋊C3, C8×#120⋊C3, #21440⋊C3, …
maximal quotient ofC8×C22⋊SL(2,𝔽3), C8×#997, C16×C22⋊A4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18240);