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Cayley diagram

C3×#3773 #88928

order432
isoclinismisoclinic class of GO+(4,𝔽2)
automorphismAut(G) = C2×#10596417 (order 1152) · Inn(G) = GO+(4,𝔽2) · Out(G) = C2×D4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#3773

Splittings

SylowC33 ⋊ C4⋊C4

Subgroups

subgroupG/H
center, hypercenterC2×C3GO+(4,𝔽2)
commutatorC3⋊D3C2×C3×C4
FrattiniC2C3×GO+(4,𝔽2)
Fitting, socleC2×C33D4
prime radicalC2×C3×C3⋊D3C22
cosocleC2×C3⋊D3C22×C3
hypocenterC32C3×C4⋊C4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×PSU(3,𝔽2)⋊C4, C3×AGL(1,𝔽9)⋊C4, #11724099, …
maximal normal subgroup ofC3×PSU(3,𝔽2)⋊C4, C3×AGL(1,𝔽9)⋊C4, #11724099, …
maximal quotient ofC3×#63732, C3×#63733, C3×#63737, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 575);