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3−1+2×#486 #13145218

order1728
nilpotency class3
isoclinismisoclinic class of H(3,𝔽3)×#465
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C2×C4

Aliases

direct product3−1+2×#486
unambiguous semidirect product(C9×#486)⋊C3

Subgroups

subgroupG/H
center, socleC22×C3C2×C32×D4
commutator, Frattini, cosocleC2×C3×C4C23×C32
prime radicalC32×#127C2×C3
LCSG ▷ C2×C3×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C42×3−1+2 ◁ G

472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 288

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

242 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofAGL(1,ℤ/9)×#486, #72252773, #72252795, …
maximal subgroup of3−1+2×#1715, AGL(1,ℤ/9)×#486, #72252773, …
maximal normal subgroup of3−1+2×#1715, AGL(1,ℤ/9)×#486, #72252773, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 11281);