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Cayley diagram

C8×AGL(1,𝔽8) #90157

order448
isoclinismisoclinic class of AGL(1,𝔽8)
automorphismAut(G) = C22×AΓL(1,𝔽8) (order 672) · Inn(G) = AGL(1,𝔽8) · Out(G) = C22×C3
homologyAbelianization (H1) = C7×C8 · Schur multiplier (H2) = C1

Aliases

direct productC8×AGL(1,𝔽8)
unambiguous semidirect product(C23×C8)⋊C7

Splittings

Fitting, SylowC23×C8 ⋊ C7
derivedC23 ⋊ C7×C8

Subgroups

subgroupG/H
center, hypercenterC8AGL(1,𝔽8)
commutator, hypocenterC23C7×C8
FrattiniC4C2×AGL(1,𝔽8)
FittingC23×C8C7
socleC24C4×C7
prime radicalC2×AGL(1,𝔽8)C4
cosocleC23×C4C2×C7
LCSG ▷ C23
UCSC1 ◁ C8

39 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 56

A minimal faithful permutation representation — 2 generators on 16 points:

(1,2,3,5,4,6,7,8)(10,11,12,14,16,13,15);(9,13)(10,11)(12,16)(14,15)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC16×AGL(1,𝔽8), C2×C8×AGL(1,𝔽8), M16×AGL(1,𝔽8), …
maximal normal subgroup ofC16×AGL(1,𝔽8), C2×C8×AGL(1,𝔽8), M16×AGL(1,𝔽8), …
maximal quotient ofC16×AGL(1,𝔽8), C2×C8×AGL(1,𝔽8), C3⋊C8×AGL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 919);