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GU(3,𝔽2)⋊C2 #13049173

order1296
isoclinismisoclinic class of GU(3,𝔽2)⋊C2
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productGU(3,𝔽2)⋊C2

Splittings

FittingH(3,𝔽3) ⋊ GL(2,𝔽3)
derivedGU(3,𝔽2) ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterGU(3,𝔽2)C2
Frattini, socleC3AGL(2,𝔽3)
FittingH(3,𝔽3)GL(2,𝔽3)
LCSG ▷ GU(3,𝔽2)
UCSC1

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 27

Polycyclic presentation:

2,3,2^3,3^3|c^2=e,d^2=e,b^a=b^2,c^a=cd,d^a=de,f^a=f^2gh,g^a=gh,h^a=h^2,c^b=d,d^b=cde,f^b=fg,d^c=de,f^c=g^2,g^c=fh^2,f^d=f^2gh^2,g^d=fgh^2,f^e=f^2h,g^e=g^2h^2,g^f=gh^2

Conjugacy classes & character table

18 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72210405, #72214540, C2×GU(3,𝔽2)⋊C2, …
maximal normal subgroup ofC2×GU(3,𝔽2)⋊C2
maximal quotient ofC2×GU(3,𝔽2)⋊C2

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 2891);