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C22⋊(PSU(3,𝔽2)⋊C3) #11724699

order864
isoclinismisoclinic class of C22⋊(PSU(3,𝔽2)⋊C3)
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C23×C3

Aliases

unambiguous semidirect productC22⋊(PSU(3,𝔽2)⋊C3)
unambiguous semidirect productPSU(3,𝔽2)⋊A4

Splittings

Fitting, socleC22×C32 ⋊ SL(2,𝔽3)
derivedC22×PSU(3,𝔽2) ⋊ C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterC22×PSU(3,𝔽2)C3
Fitting, socleC22×C32SL(2,𝔽3)
LCSG ▷ C22×PSU(3,𝔽2)
UCSC1

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 36

Polycyclic presentation:

3,2^5,3^2|b^2=f,c^2=f,b^a=cf,c^a=bcf,d^a=e,e^a=de,g^a=h^2,h^a=gh^2,c^b=cf,g^b=gh^2,h^b=g^2h^2,g^c=h,h^c=g^2,g^f=g^2,h^f=h^2

Conjugacy classes & character table

16 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13181783, C2×#13181783
maximal subgroup of#10593570⋊C3, #13181783, C2×C22⋊(PSU(3,𝔽2)⋊C3), …
maximal normal subgroup of#10593570⋊C3, #13181783, C2×C22⋊(PSU(3,𝔽2)⋊C3), …
maximal quotient ofQ8⋊(PSU(3,𝔽2)⋊C3), #10596550⋊C3, C2×C22⋊(PSU(3,𝔽2)⋊C3)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 4666);