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Cayley diagram

C4×AGL(1,𝔽8) #6541

order224
isoclinismisoclinic class of AGL(1,𝔽8)
automorphismAut(G) = C2×AΓL(1,𝔽8) (order 336) · Inn(G) = AGL(1,𝔽8) · Out(G) = C2×C3
homologyAbelianization (H1) = C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC4×AGL(1,𝔽8)
unambiguous semidirect product(C23×C4)⋊C7

Splittings

Fitting, SylowC23×C4 ⋊ C7
derivedC23 ⋊ C4×C7

Subgroups

subgroupG/H
center, hypercenterC4AGL(1,𝔽8)
commutator, hypocenterC23C4×C7
FrattiniC2C2×AGL(1,𝔽8)
FittingC23×C4C7
socle, cosocleC24C2×C7
prime radicalC2×AGL(1,𝔽8)C2
LCSG ▷ C23
UCSC1 ◁ C4

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×AGL(1,𝔽8), C2×C4×AGL(1,𝔽8), D4×AGL(1,𝔽8), …
maximal normal subgroup ofC8×AGL(1,𝔽8), C2×C4×AGL(1,𝔽8), D4×AGL(1,𝔽8), …
maximal quotient ofC8×AGL(1,𝔽8), C2×C4×AGL(1,𝔽8), Dic3×AGL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 173);