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Cayley diagram

C3×#3674 #88509

order432
isoclinismisoclinic class of #3672
automorphismAut(G) = C22×D4×AGL(1,ℤ/9) (order 1728) · Inn(G) = #611 · Out(G) = C23×C3
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×#3674

Splittings

SylowC3×C9 ⋊ Q16

Subgroups

subgroupG/H
centerC2×C3#611
commutator, cosocleC4×C9C22×C3
FrattiniC3×C4C2×C3×D3
FittingC3×Q8×C9C2
socle, prime radicalC2×C32#67
hypercenterC3×Q8D9
hypocenterC9C3×Q16
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×Q8

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 2 generators on 28 points:

(1,2,3)(4,5,8,10,12,11,6,9,7)(13,23,17,15)(14,27,20,18)(16,22,26,25)(19,21,24,28);(5,7)(6,10)(8,9)(11,12)(13,20,17,14)(15,28,23,21)(16,19,26,24)(18,22,27,25)

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#3674, #11721203, #11721208, …
maximal normal subgroup ofD3×#3674, #11721203, #11721208, …
maximal quotient ofC3×#63366, C3×#63368, C3×#63395, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 156);