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Cayley diagram

C5×ΓL(1,𝔽8) #1069

order105
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphismAut(G) = C4×AGL(1,𝔽7) (order 168) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C2×C4
holomorphG ⋊ Aut(G) = AGL(1,𝔽5)×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
homologyAbelianization (H1) = C3×C5 · Schur multiplier (H2) = C1

Aliases

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

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC5ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C3×C5
FrattiniC1G
Fitting, socleC5×C7C3
LCSG ▷ C7
UCSC1 ◁ C5

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 35

A minimal faithful permutation representation — 2 generators on 12 points:

(1,2,3,4,5)(7,8,9)(10,12,11);(6,7,8,10,9,11,12)

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×AGL(1,𝔽7), D5×ΓL(1,𝔽8), C5⋊AGL(1,𝔽7), …
maximal normal subgroup ofC5×AGL(1,𝔽7), D5×ΓL(1,𝔽8), C5⋊AGL(1,𝔽7), …
maximal quotient ofC2×C5×ΓL(1,𝔽8), C5×C7⋊C9, C3×C5×ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(105, 1);