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3−1+2×#546 #13154622

order1728
nilpotency class2
isoclinismisoclinic class of D42×H(3,𝔽3)
automorphism?
homologyAbelianization (H1) = C24×C32 · Schur multiplier (H2) = C25

Aliases

direct product3−1+2×#546
unambiguous semidirect product(C9×#546)⋊C3

Subgroups

subgroupG/H
center, commutator, Frattini, socle, cosocleC22×C3C24×C32
prime radicalC32×#546C3
LCSG ▷ C22×C3 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

1336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 4 generators on 21 points:

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

Conjugacy classes & character table

275 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofAGL(1,ℤ/9)×#546, #72268843
maximal subgroup of3−1+2×#3484, AGL(1,ℤ/9)×#546, #72268843
maximal normal subgroup of3−1+2×#3484, AGL(1,ℤ/9)×#546, #72268843

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 20685);