Finite Group Databasebeta

← Prev (#31801578) Next (#31801580) →

C5×ΓL(1,𝔽8)×C169 #31801579

order17745
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C5×C169 · Schur multiplier (H2) = C1

Aliases

direct productC5×ΓL(1,𝔽8)×C169
unambiguous semidirect productC7⋊(C3×C5×C169)
unambiguous semidirect product(C5×C7)⋊(C3×C169)

Splittings

Fitting, HallC5×C7×C169 ⋊ C3
derived, SylowC7 ⋊ C3×C5×C169
HallC7×C169 ⋊ C3×C5
HallC5×C7 ⋊ C3×C169

Subgroups

subgroupG/H
center, hypercenterC5×C169ΓL(1,𝔽8)
commutator, hypocenterC7C3×C5×C169
FrattiniC13C5×C13×ΓL(1,𝔽8)
FittingC5×C7×C169C3
socleC5×C7×C13C3×C13
prime radicalC5×C13×ΓL(1,𝔽8)C13
cosocleC7×C13C3×C5×C13
LCSG ▷ C7
UCSC1 ◁ C5×C169

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 5915

Conjugacy classes & character table

4225 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×C5×ΓL(1,𝔽8)×C169, C5×ΓL(1,𝔽8)×D169, C5×AGL(1,𝔽7)×C169, …
maximal normal subgroup ofC2×C5×ΓL(1,𝔽8)×C169, C5×ΓL(1,𝔽8)×D169, C5×AGL(1,𝔽7)×C169, …
maximal quotient ofC2×C5×ΓL(1,𝔽8)×C169

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17745, 3);