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Cayley diagram

C22×C22⋊A4 #5981

order192
isoclinismisoclinic class of C22⋊A4
automorphismAut(G) = D3×AΓL(2,𝔽4) (order 34560) · Inn(G) = C22⋊A4 · Out(G) = D3×S5
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C25

Aliases

direct productC22×C22⋊A4

Splittings

Fitting, socle, SylowC26 ⋊ C3
derivedC24 ⋊ C22×C3

Subgroups

subgroupG/H
center, hypercenterC22C22⋊A4
commutator, cosocle, hypocenterC24C22×C3
FrattiniC1G
Fitting, socleC26C3
LCSG ▷ C24
UCSC1 ◁ C22

1000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 12 points:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

derived subgroup ofA4×#1020, #1020⋊A4, C2×#1020⋊A4, …
maximal subgroup ofC2×#379⋊C3, AΣL(1,𝔽64), C22×#120⋊C3, …
maximal normal subgroup ofC2×#379⋊C3, AΣL(1,𝔽64), C22×#120⋊C3, …
maximal quotient ofC2×C4×C22⋊A4, D4×C22⋊A4, Q8×C22⋊A4, …
automorphism group of#126
outer automorphism group of#396

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1540);