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Cayley diagram

#3699⋊C3 #88716

order432
automorphismAut(G) = C2×D4×AGL(1,ℤ/9) (order 864) · Inn(G) = C22×AGL(1,ℤ/9) · Out(G) = C22
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

unambiguous semidirect product#3699⋊C3

Splittings

FittingD4×3−1+2 ⋊ C2
Sylow3−1+2 ⋊ #41

Subgroups

subgroupG/H
centerC2C22×AGL(1,ℤ/9)
commutator, cosocleC2×C9C23×C3
Frattini, socleC2×C3C22×C3×D3
FittingD4×3−1+2C2
hypercenterD4AGL(1,ℤ/9)
hypocenterC9C3×#41
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ D4

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 17 points:

(1,5,6,7,2,8,3,9,4)(11,15)(12,16);(10,12,13,16)(11,14,15,17);(2,4)(3,7)(5,6)(8,9)(10,11)(12,14)(13,15)(16,17)

Polycyclic presentation:

2^3,3,2,3^2|b^2=e,f^3=g^2,c^a=ce,f^a=f^2g,g^a=g^2,c^b=ce,f^d=fg

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#63473⋊C3, #63474⋊C3, #63477⋊C3, …
maximal normal subgroup of#63473⋊C3, #63474⋊C3, #63477⋊C3, …
maximal quotient of#63439⋊C3, #63440⋊C3, #63441⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 363);