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C4×AΓL(1,𝔽8) #10621467

order672
isoclinismisoclinic class of AΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC4×AΓL(1,𝔽8)
unambiguous semidirect productAGL(1,𝔽8)⋊(C3×C4)
unambiguous semidirect product(C4×AGL(1,𝔽8))⋊C3

Splittings

Fitting, SylowC23×C4 ⋊ ΓL(1,𝔽8)
derivedAGL(1,𝔽8) ⋊ C3×C4
HallC4×AGL(1,𝔽8) ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC4AΓL(1,𝔽8)
commutator, hypocenterAGL(1,𝔽8)C3×C4
FrattiniC2C2×AΓL(1,𝔽8)
FittingC23×C4ΓL(1,𝔽8)
socleC24C2×ΓL(1,𝔽8)
prime radicalC2×AΓL(1,𝔽8)C2
cosocleC2×AGL(1,𝔽8)C2×C3
LCSG ▷ AGL(1,𝔽8)
UCSC1 ◁ C4

46 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 28

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×AΓL(1,𝔽8), C2×C4×AΓL(1,𝔽8), D4×AΓL(1,𝔽8), …
maximal normal subgroup ofC8×AΓL(1,𝔽8), C2×C4×AΓL(1,𝔽8), D4×AΓL(1,𝔽8), …
maximal quotient ofC8×AΓL(1,𝔽8), C2×C4×AΓL(1,𝔽8), AGL(1,𝔽5)×AΓL(1,𝔽8), …
automorphism group ofC5×AGL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 1049);