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Cayley diagram

C2×C4×#272 #88738

order432
isoclinismisoclinic class of #272
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×C4×#272

Splittings

FittingC2×C4×H(3,𝔽3) ⋊ C2
derived, SylowH(3,𝔽3) ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC2×C3×C4C3⋊D3
commutator, hypocenterH(3,𝔽3)C22×C4
FrattiniC2×C3C22×C3⋊D3
FittingC2×C4×H(3,𝔽3)C2
socleC22×C3C2×C3⋊D3
prime radicalC22×#272C2
cosocleC2×H(3,𝔽3)C23
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4

297 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 15 points:

(1,2)(8,10)(11,14)(13,15);(3,4,5,6)(7,8,10)(9,11,14)(12,15,13);(1,2)(8,11,15)(10,13,14)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11720323, #11720553, #11720784, …
maximal normal subgroup of#11720323, #11720553, #11720784, …
maximal quotient ofC4×#6229, C42×#272, #11721718, …
automorphism group ofC3×C5×C9, C2×C3×C5×C9, C16×3−1+2
outer automorphism group ofC3×C5×C9, C2×C3×C5×C9

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 385);