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C4×C9×ΓL(1,𝔽8) #10625401

order756
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C4×C9 · Schur multiplier (H2) = C3

Aliases

direct productC4×C9×ΓL(1,𝔽8)

Splittings

FittingC4×C7×C9 ⋊ C3
derived, SylowC7 ⋊ C3×C4×C9
HallC4×C7 ⋊ C3×C9

Subgroups

subgroupG/H
center, hypercenterC4×C9ΓL(1,𝔽8)
commutator, hypocenterC7C3×C4×C9
FrattiniC2×C3C2×C3×ΓL(1,𝔽8)
FittingC4×C7×C9C3
socle, cosocleC2×C3×C7C2×C32
prime radicalC2×C3×ΓL(1,𝔽8)C2×C3
LCSG ▷ C7
UCSC1 ◁ C4×C9

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 252

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC9×ΓL(1,𝔽8)⋊C8, ΓL(1,𝔽8)×C9⋊C8, ΓL(1,𝔽8)⋊(C9⋊C8), …
maximal normal subgroup ofC9×ΓL(1,𝔽8)⋊C8, ΓL(1,𝔽8)×C9⋊C8, ΓL(1,𝔽8)⋊(C9⋊C8), …
maximal quotient ofC8×C9×ΓL(1,𝔽8), C2×C4×C9×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(756, 32);