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C7×C3⋊(ΓL(1,𝔽8)⋊D41) #49116358

order36162
automorphism?
homologyAbelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×C3⋊(ΓL(1,𝔽8)⋊D41)
unambiguous semidirect productC3⋊(C7×ΓL(1,𝔽8)⋊D41)
unambiguous semidirect productΓL(1,𝔽8)⋊(C7×D123)
unambiguous semidirect productC41⋊(C7×C3⋊AGL(1,𝔽7))
unambiguous semidirect productD861⋊(C3×C7)

Splittings

Fitting, socleC3×C72×C41 ⋊ C2×C3
derivedC3×C7×C41 ⋊ C2×C3×C7
HallC3×C7×ΓL(1,𝔽8)×C41 ⋊ C2
HallC72×C41 ⋊ C3×D3
HallC3×C7×ΓL(1,𝔽8) ⋊ D41
SylowC72 ⋊ C3×D123
SylowC41 ⋊ C7×C3⋊AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC7C3⋊(ΓL(1,𝔽8)⋊D41)
commutator, cosocle, hypocenterC3×C7×C41C2×C3×C7
FrattiniC1G
Fitting, socleC3×C72×C41C2×C3
LCSG ▷ C3×C7×C41
UCSC1 ◁ C7

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 5166

Conjugacy classes & character table

2184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36162, 100);