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PSU(3,𝔽2)⋊C9 #10620015

order648
isoclinismisoclinic class of PSU(3,𝔽2)⋊C3
automorphism?
homologyAbelianization (H1) = C9 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productPSU(3,𝔽2)⋊C9

Splittings

derivedPSU(3,𝔽2) ⋊ C9

Subgroups

subgroupG/H
center, Frattini, hypercenterC3PSU(3,𝔽2)⋊C3
commutator, hypocenterPSU(3,𝔽2)C9
Fitting, socleC33SL(2,𝔽3)
prime radicalC3×C3⋊D3A4
cosocleC3×PSU(3,𝔽2)C3
LCSG ▷ PSU(3,𝔽2)
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 72

Polycyclic presentation:

3^2,2^3,3^2|a^3=b,c^2=e,d^2=e,c^a=cd,d^a=c,f^a=f^2g,g^a=f^2,d^c=de,f^c=g,g^c=f^2,f^d=f^2g^2,g^d=f^2g,f^e=f^2,g^e=g^2

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

derived subgroup of(PSU(3,𝔽2)⋊C9)⋊C2
maximal subgroup of(PSU(3,𝔽2)⋊C9)⋊C2, C2×PSU(3,𝔽2)⋊C9, C9×PSU(3,𝔽2)⋊C3, …
maximal normal subgroup of(PSU(3,𝔽2)⋊C9)⋊C2, C2×PSU(3,𝔽2)⋊C9, C9×PSU(3,𝔽2)⋊C3, …
maximal quotient ofC2×PSU(3,𝔽2)⋊C9, PSU(3,𝔽2)⋊C27, SU(3,𝔽2)⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(648, 534);