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C3×ΓL(1,𝔽8)⋊C4×C137 #46828599

order34524
automorphism?
homologyAbelianization (H1) = C32×C4×C137 · Schur multiplier (H2) = C3

Aliases

direct productC3×ΓL(1,𝔽8)⋊C4×C137
unambiguous semidirect productΓL(1,𝔽8)⋊(C3×C4×C137)
unambiguous semidirect productDic7⋊(C32×C137)
unambiguous semidirect product(C3×Dic7)⋊(C3×C137)
unambiguous semidirect product(Dic7×C137)⋊C32
unambiguous semidirect product(C3×Dic7×C137)⋊C3

Splittings

derived, SylowC7 ⋊ C32×C4×C137
HallC3×ΓL(1,𝔽8)×C137 ⋊ C4
HallDic7×C137 ⋊ C32
HallC7×C137 ⋊ C32×C4
HallC3×ΓL(1,𝔽8) ⋊ C4×C137
HallDic7 ⋊ C32×C137

Subgroups

subgroupG/H
center, hypercenterC2×C3×C137AGL(1,𝔽7)
commutator, hypocenterC7C32×C4×C137
FrattiniC2C3×AGL(1,𝔽7)×C137
Fitting, socleC2×C3×C7×C137C2×C3
prime radicalC2×C3×ΓL(1,𝔽8)×C137C2
cosocleC2×C7C2×C32×C137
LCSG ▷ C7
UCSC1 ◁ C2×C3×C137

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 11508

Conjugacy classes & character table

5754 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34524, 42);