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ΓL(1,𝔽8)⋊#168 #11719701

order840
isoclinismisoclinic class of C5⋊AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C3×C8 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊#168
unambiguous semidirect product(C7⋊#168)⋊C3

Splittings

derived, HallC5×C7 ⋊ C3×C8
HallC7⋊#168 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C8
HallΓL(1,𝔽8) ⋊ #168
SylowC7 ⋊ C3×#168
SylowC5 ⋊ ΓL(1,𝔽8)⋊C8

Subgroups

subgroupG/H
center, Frattini, hypercenterC4C5⋊AGL(1,𝔽7)
commutator, hypocenterC5×C7C3×C8
Fitting, cosocleC4×C5×C7C2×C3
socleC2×C5×C7C3×C4
prime radicalC2×C5×ΓL(1,𝔽8)C4
LCSG ▷ C5×C7
UCSC1 ◁ C4

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 280

Polycyclic presentation:

2,3,2^2,5,7|a^2=c,c^2=d,e^a=e^4,f^a=f^6,f^b=f^2

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#168×AGL(1,𝔽7), D5×ΓL(1,𝔽8)⋊C8, #10587838⋊C3, …
maximal normal subgroup of#168×AGL(1,𝔽7), D5×ΓL(1,𝔽8)⋊C8, #10587838⋊C3, …
maximal quotient ofΓL(1,𝔽8)⋊#672, C2×ΓL(1,𝔽8)⋊#168, #72217145

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(840, 3);