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Cayley diagram

C5×AGL(1,𝔽8) #63328

order280
isoclinismisoclinic class of AGL(1,𝔽8)
automorphismAut(G) = C4×AΓL(1,𝔽8) (order 672) · Inn(G) = AGL(1,𝔽8) · Out(G) = C3×C4
homologyAbelianization (H1) = C5×C7 · Schur multiplier (H2) = C1

Aliases

direct productC5×AGL(1,𝔽8)
unambiguous semidirect productC23⋊(C5×C7)
unambiguous semidirect product(C23×C5)⋊C7

Splittings

Fitting, socle, HallC23×C5 ⋊ C7
derived, SylowC23 ⋊ C5×C7

Subgroups

subgroupG/H
center, hypercenterC5AGL(1,𝔽8)
commutator, cosocle, hypocenterC23C5×C7
FrattiniC1G
Fitting, socleC23×C5C7
LCSG ▷ C23
UCSC1 ◁ C5

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 40

A minimal faithful permutation representation — 2 generators on 13 points:

(2,3,4,6,8,5,7)(9,10,11,12,13);(1,5)(2,3)(4,8)(6,7)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup ofD5×AΓL(1,𝔽8), C2×D5×AΓL(1,𝔽8), C2×AGL(1,𝔽5)×AΓL(1,𝔽8)
maximal subgroup ofD5×AGL(1,𝔽8), C2×C5×AGL(1,𝔽8), C5×AΓL(1,𝔽8), …
maximal normal subgroup ofD5×AGL(1,𝔽8), C2×C5×AGL(1,𝔽8), C5×AΓL(1,𝔽8), …
maximal quotient ofC2×C5×AGL(1,𝔽8), C3×C5×AGL(1,𝔽8), C25×AGL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(280, 33);