Finite Group Databasebeta

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Cayley diagram

C2×C22⋊A4 #1022

order96
isoclinismisoclinic class of C22⋊A4
automorphismAut(G) = AΓL(2,𝔽4) (order 5760) · Inn(G) = C22⋊A4 · Out(G) = S5
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C24

Aliases

direct productC2×C22⋊A4

Splittings

Fitting, socle, SylowC25 ⋊ C3
derivedC24 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2C22⋊A4
commutator, cosocle, hypocenterC24C2×C3
FrattiniC1G
Fitting, socleC25C3
LCSG ▷ C24
UCSC1 ◁ C2

148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 12

A minimal faithful permutation representation — 3 generators on 10 points:

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

Conjugacy classes & character table

16 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72989, #72991, #73081, …
maximal subgroup of#379⋊C3, C2×#120⋊C3, #5936, …
maximal normal subgroup of#379⋊C3, C2×#120⋊C3, #5936, …
maximal quotient ofC4×C22⋊A4, C2×C22⋊SL(2,𝔽3), #5948, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 229);