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Cayley diagram

C2×C32×C4×D3 #89054

order432
isoclinismisoclinic class of D3
automorphism?
homologyAbelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3

Aliases

direct productC2×C32×C4×D3

Splittings

FittingC2×C33×C4 ⋊ C2
derivedC3 ⋊ C22×C32×C4
SylowC33 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC2×C32×C4D3
commutator, hypocenterC3C22×C32×C4
FrattiniC2C22×C32×D3
FittingC2×C33×C4C2
socleC22×C33C22
prime radicalC22×C32×D3C2
cosocleC2×C3C23×C32
LCSG ▷ C3
UCSC1 ◁ C2×C32×C4

388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 144

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

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

Conjugacy classes & character table

216 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×(C2×D3)⋊(C3⋊C8), (C2×D3)⋊(C3⋊(C3⋊C8)), C32×(C2×D3)⋊C8, …
maximal normal subgroup ofC3×(C2×D3)⋊(C3⋊C8), (C2×D3)⋊(C3⋊(C3⋊C8)), C32×(C2×D3)⋊C8, …
maximal quotient ofC32×C4×Dic6, C32×C42×D3, C32×#872, …
automorphism group ofC22×C7×C13, C22×C9×C13
outer automorphism group ofC22×C7×C13, C22×C9×C13

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 701);