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Cayley diagram

C22×A4 #247

order48
isoclinismisoclinic class of A4
automorphismAut(G) = D3×S4 (order 144) · Inn(G) = A4 · Out(G) = C2×D3
holomorphG ⋊ Aut(G) = S4×A4⋊S4
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C22

Aliases

direct productC22×A4

Splittings

Fitting, socle, SylowC24 ⋊ C3
derivedC22 ⋊ C22×C3

Subgroups

subgroupG/H
center, hypercenterC22A4
commutator, cosocle, hypocenterC22C22×C3
FrattiniC1G
Fitting, socleC24C3
LCSG ▷ C22
UCSC1 ◁ C22

39 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 8
transitive degreeμt(G) = 12

A minimal faithful permutation representation — 2 generators on 8 points:

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

Conjugacy classes & character table

16 conjugacy classes — view the full character table.

Relationships

derived subgroup ofA4×S4, #73049, #73115, …
maximal subgroup ofC2×SL(2,ℤ/4), GL(2,ℤ/4), C2×C4×A4, …
maximal normal subgroup ofC2×SL(2,ℤ/4), GL(2,ℤ/4), C2×C4×A4, …
maximal quotient ofC2×C4×A4, D4×A4, C22×SL(2,𝔽3), …
outer automorphism group of#876, C3×#126, #2069, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 49);