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Cayley diagram

C2×C3×C22⋊A4 #64394

order288
isoclinismisoclinic class of C22⋊A4
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C24×C3

Aliases

direct productC2×C3×C22⋊A4

Splittings

Fitting, socleC25×C3 ⋊ C3
derivedC24 ⋊ C2×C32
SylowC25 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC2×C3C22⋊A4
commutator, cosocle, hypocenterC24C2×C32
FrattiniC1G
Fitting, socleC25×C3C3
LCSG ▷ C24
UCSC1 ◁ C2×C3

324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11920746, #11920749, #11920751, …
maximal subgroup ofC3⋊(#379⋊C3), C3×#379⋊C3, C2×C3⋊(#120⋊C3), …
maximal normal subgroup ofC3⋊(#379⋊C3), C3×#379⋊C3, C2×C3⋊(#120⋊C3), …
maximal quotient ofC3×C4×C22⋊A4, C2×C3×C22⋊SL(2,𝔽3), C3×#5948, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1042);