Finite Group Databasebeta

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Cayley diagram

C2×A4⋊C8 #5408

order192
isoclinismisoclinic class of S4
automorphismAut(G) = C2×D4×S4 (order 384) · Inn(G) = S4 · Out(G) = C2×D4
homologyAbelianization (H1) = C2×C8 · Schur multiplier (H2) = C22

Aliases

direct productC2×A4⋊C8
unambiguous semidirect product(C2×A4)⋊C8

Splittings

derivedA4 ⋊ C2×C8

Subgroups

subgroupG/H
center, hypercenterC2×C4S4
commutator, hypocenterA4C2×C8
FrattiniC4C2×S4
FittingC23×C4D3
socleC24Dic3
prime radicalC22×A4C4
cosocleC4×A4C22
LCSG ▷ A4
UCSC1 ◁ C2×C4

89 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×A4⋊C8, #73044, C4⋊(A4⋊C8), …
maximal normal subgroup ofC4×A4⋊C8, #73044, C4⋊(A4⋊C8), …
maximal quotient ofC4×A4⋊C8, C4⋊(A4⋊C8), C2×SL(2,𝔽3)⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 967);