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Cayley diagram

C2×C4⋊AGL(1,ℤ/9) #88707

order432
isoclinismisoclinic class of 3−1+2⋊Q8
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC2×C4⋊AGL(1,ℤ/9)
unambiguous semidirect product(C2×D36)⋊C3

Splittings

FittingC2×C4×3−1+2 ⋊ C2
Sylow3−1+2 ⋊ C2×D4

Subgroups

subgroupG/H
centerC22C2×AGL(1,ℤ/9)
commutator, cosocleC2×C9C23×C3
FrattiniC2×C3C22×C3×D3
FittingC2×C4×3−1+2C2
socleC22×C3C2×C3×D3
hypercenterC2×C4AGL(1,ℤ/9)
hypocenterC9C2×C3×D4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4

170 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:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#63369⋊C3, #63380⋊C3, #63383⋊C3, …
maximal normal subgroup of#63369⋊C3, #63380⋊C3, #63383⋊C3, …
maximal quotient of#63431⋊C3, C4×C4⋊AGL(1,ℤ/9), #11721394, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 354);