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Cayley diagram

AGL(1,𝔽5)×ΓL(1,𝔽8) #88289

order420
isoclinismisoclinic class of AGL(1,𝔽5)×ΓL(1,𝔽8)
automorphismAut(G) = AGL(1,𝔽5)×AGL(1,𝔽7) (order 840) · Inn(G) = G · Out(G) = C2
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productAGL(1,𝔽5)×ΓL(1,𝔽8)
unambiguous semidirect product(C7×AGL(1,𝔽5))⋊C3

Splittings

Fitting, socle, derived, HallC5×C7 ⋊ C3×C4
HallC7×AGL(1,𝔽5) ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C4
SylowC7 ⋊ C3×AGL(1,𝔽5)
SylowC5 ⋊ C4×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC5×C7C3×C4
prime radicalD5×ΓL(1,𝔽8)C2
cosocleC7×D5C2×C3
LCSG ▷ C5×C7
UCSC1

24 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:

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

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×AGL(1,𝔽7), C2×AGL(1,𝔽5)×ΓL(1,𝔽8), C3×AGL(1,𝔽5)×ΓL(1,𝔽8), …
maximal normal subgroup ofAGL(1,𝔽5)×AGL(1,𝔽7), C2×AGL(1,𝔽5)×ΓL(1,𝔽8), C3×AGL(1,𝔽5)×ΓL(1,𝔽8), …
maximal quotient ofΓL(1,𝔽8)×#170, C2×AGL(1,𝔽5)×ΓL(1,𝔽8), AGL(1,𝔽5)×C7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 14);