Finite Group Databasebeta

← Prev (#36339303) Next (#36339305) →

C3×ΓL(1,𝔽8)×Dic85 #36339304

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

Aliases

direct productC3×ΓL(1,𝔽8)×Dic85
unambiguous semidirect product(C7×Dic85)⋊C32
unambiguous semidirect product(C3×C7×Dic85)⋊C3

Splittings

derived, HallC5×C7×C17 ⋊ C32×C4
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C4
HallC7×Dic85 ⋊ C32
HallC3×C17×ΓL(1,𝔽8) ⋊ Dic5
HallC3×C5×ΓL(1,𝔽8) ⋊ Dic17
HallC7×C17 ⋊ C32×Dic5
HallC5×C17 ⋊ C3×C4×ΓL(1,𝔽8)
HallC5×C7 ⋊ C32×Dic17
SylowC17 ⋊ C3×Dic5×ΓL(1,𝔽8)
SylowC7 ⋊ C32×Dic85
SylowC5 ⋊ C3×ΓL(1,𝔽8)×Dic17

Subgroups

subgroupG/H
center, hypercenterC2×C3ΓL(1,𝔽8)×D85
commutator, hypocenterC5×C7×C17C32×C4
FrattiniC2C3×ΓL(1,𝔽8)×D85
Fitting, socleC2×C3×C5×C7×C17C2×C3
prime radicalC2×C3×C5×C17×ΓL(1,𝔽8)C2
cosocleC2×C5×C7×C17C2×C32
LCSG ▷ C5×C7×C17
UCSC1 ◁ C2×C3

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 7140

Conjugacy classes & character table

1320 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 136);