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C3×ΓL(1,𝔽8)×#66772 #36339355

order21420
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productC3×ΓL(1,𝔽8)×#66772
unambiguous semidirect product(C7×#66772)⋊C32
unambiguous semidirect product(C3×C7×#66772)⋊C3

Splittings

Fitting, socleC3×C5×C7×C17 ⋊ C3×C4
derived, HallC5×C7×C17 ⋊ C32×C4
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C4
HallC7×#66772 ⋊ C32
HallC3×C17×ΓL(1,𝔽8) ⋊ Dic5
HallC3×C5×ΓL(1,𝔽8) ⋊ #595
HallC7×C17 ⋊ C32×Dic5
HallC5×C17 ⋊ C3×C4×ΓL(1,𝔽8)
HallC5×C7 ⋊ C32×#595
SylowC17 ⋊ C3×Dic5×ΓL(1,𝔽8)
SylowC7 ⋊ C32×#66772
SylowC5 ⋊ C3×ΓL(1,𝔽8)×#595

Subgroups

subgroupG/H
center, hypercenterC3ΓL(1,𝔽8)×#66772
commutator, hypocenterC5×C7×C17C32×C4
FrattiniC1G
Fitting, socleC3×C5×C7×C17C3×C4
prime radicalC3×C5×ΓL(1,𝔽8)×D17C2
cosocleC5×C7×D17C2×C32
LCSG ▷ C5×C7×C17
UCSC1 ◁ C3

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1785

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 187);