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C5×C13×ΓL(1,𝔽8) #13062894

order1365
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C5×C13 · Schur multiplier (H2) = C1

Aliases

direct productC5×C13×ΓL(1,𝔽8)
unambiguous semidirect productC7⋊(C3×C5×C13)
unambiguous semidirect product(C5×C7)⋊(C3×C13)

Splittings

Fitting, socle, HallC5×C7×C13 ⋊ C3
derived, SylowC7 ⋊ C3×C5×C13
HallC7×C13 ⋊ C3×C5
HallC5×C7 ⋊ C3×C13

Subgroups

subgroupG/H
center, hypercenterC5×C13ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C3×C5×C13
FrattiniC1G
Fitting, socleC5×C7×C13C3
LCSG ▷ C7
UCSC1 ◁ C5×C13

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 455

A minimal faithful permutation representation — 2 generators on 25 points:

(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18)(20,21,22)(23,25,24);(19,20,21,23,22,24,25)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC2×C5×C13×ΓL(1,𝔽8), C5×ΓL(1,𝔽8)×D13, C5×C13×AGL(1,𝔽7), …
maximal normal subgroup ofC2×C5×C13×ΓL(1,𝔽8), C5×ΓL(1,𝔽8)×D13, C5×C13×AGL(1,𝔽7), …
maximal quotient ofC2×C5×C13×ΓL(1,𝔽8), C5×C13×C7⋊C9, C3×C5×C13×ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1365, 2);