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3−1+2⋊C27 #10624448

order729
nilpotency class3
isoclinismisoclinic class of AΣL(1,𝔽27)
automorphism?
homologyAbelianization (H1) = C3×C27 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect product3−1+2⋊C27

Subgroups

subgroupG/H
centerC3×C9H(3,𝔽3)
commutatorC32C3×C27
Frattini, cosocleC32×C9C32
socleC32C32⋊C9
prime radicalC33C3×C9
LCSG ▷ C32 ▷ C3 ▷ C1
UCSC1 ◁ C3×C9 ◁ C32×C9 ◁ G
LECSG ▷ C32×C9 ▷ C32 ▷ C1

63 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 243

Polycyclic presentation:

3^6|a^3=d,b^3=e,d^3=f,b^a=bc,c^a=ce,c^b=ce

Conjugacy classes & character table

153 conjugacy classes — view the full character table.

Relationships

derived subgroup of(3−1+2⋊C27)⋊C2
Fitting subgroup of(3−1+2⋊C27)⋊C2
maximal subgroup ofC2×3−1+2⋊C27, (3−1+2⋊C27)⋊C2, C3×3−1+2⋊C27, …
maximal normal subgroup ofC2×3−1+2⋊C27, (3−1+2⋊C27)⋊C2, C3×3−1+2⋊C27, …
maximal quotient ofC2×3−1+2⋊C27, #14546144, #14546145, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(729, 8);