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C7×C7⋊H(3,𝔽3) #13050512

order1323
isoclinismisoclinic class of #4427
automorphism?
homologyAbelianization (H1) = C32×C7 · Schur multiplier (H2) = C32

Aliases

direct productC7×C7⋊H(3,𝔽3)
unambiguous semidirect productC7⋊(C7×H(3,𝔽3))
unambiguous semidirect productC32⋊(C7×ΓL(1,𝔽8))
unambiguous semidirect product(C3×ΓL(1,𝔽8))⋊(C3×C7)

Splittings

FittingC32×C72 ⋊ C3
SylowC72 ⋊ H(3,𝔽3)

Subgroups

subgroupG/H
centerC3×C7C3×ΓL(1,𝔽8)
commutator, cosocleC3×C7C32×C7
FrattiniC3C3×C7×ΓL(1,𝔽8)
FittingC32×C72C3
socleC3×C72C32
hypercenterC32×C7ΓL(1,𝔽8)
hypocenterC7C7×H(3,𝔽3)
LCSG ▷ C3×C7 ▷ C7
UCSC1 ◁ C3×C7 ◁ C32×C7

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 63

Conjugacy classes & character table

203 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72224636, #72224642, C7×(C3⋊D3)⋊ΓL(1,𝔽8), …
maximal normal subgroup of#72224636, #72224642, C7×(C3⋊D3)⋊ΓL(1,𝔽8), …
maximal quotient ofC2×C7×C7⋊H(3,𝔽3)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1323, 32);