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Cayley diagram

C2×A4⋊C16 #73103

order384
isoclinismisoclinic class of S4
automorphism?
homologyAbelianization (H1) = C2×C16 · Schur multiplier (H2) = C22

Aliases

direct productC2×A4⋊C16
unambiguous semidirect product(C2×A4)⋊C16

Splittings

derivedA4 ⋊ C2×C16

Subgroups

subgroupG/H
center, hypercenterC2×C8S4
commutator, hypocenterA4C2×C16
FrattiniC8C2×S4
FittingC23×C8D3
socleC24C3⋊C8
prime radicalC22×A4C8
cosocleC8×A4C22
LCSG ▷ A4
UCSC1 ◁ C2×C8

115 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 22 points:

(3,4,6,5)(7,8,9,12,10,13,15,18,11,14,16,19,17,20,21,22);(1,2)(4,5,6)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×A4⋊C16, #11711481, C16×SL(2,ℤ/4), …
maximal normal subgroup ofC4×A4⋊C16, #11711481, C16×SL(2,ℤ/4), …
maximal quotient ofC2×SL(2,𝔽3)⋊C16, #11711349, #11711351, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5679);