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Cayley diagram

C2×C4×AGL(1,𝔽8) #90600

order448
isoclinismisoclinic class of AGL(1,𝔽8)
automorphismAut(G) = D4×AΓL(1,𝔽8) (order 1344) · Inn(G) = AGL(1,𝔽8) · Out(G) = C3×D4
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC2×C4×AGL(1,𝔽8)
unambiguous semidirect product(C24×C4)⋊C7

Splittings

Fitting, SylowC24×C4 ⋊ C7
derivedC23 ⋊ C2×C4×C7

Subgroups

subgroupG/H
center, hypercenterC2×C4AGL(1,𝔽8)
commutator, hypocenterC23C2×C4×C7
FrattiniC2C22×AGL(1,𝔽8)
FittingC24×C4C7
socleC25C2×C7
prime radicalC22×AGL(1,𝔽8)C2
cosocleC24C22×C7
LCSG ▷ C23
UCSC1 ◁ C2×C4

127 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 56

A minimal faithful permutation representation — 2 generators on 14 points:

(3,4,5,6)(8,9,10,12,14,11,13);(1,2)(7,11)(8,9)(10,14)(12,13)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC42×AGL(1,𝔽8), C22⋊C4×AGL(1,𝔽8), C4⋊C4×AGL(1,𝔽8), …
maximal normal subgroup ofC42×AGL(1,𝔽8), C22⋊C4×AGL(1,𝔽8), C4⋊C4×AGL(1,𝔽8), …
maximal quotient ofC42×AGL(1,𝔽8), C22⋊C4×AGL(1,𝔽8), C4⋊C4×AGL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1362);