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Cayley diagram

C8×ΓL(1,𝔽8) #4232

order168
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphismAut(G) = C22×AGL(1,𝔽7) (order 168) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C23
holomorphG ⋊ Aut(G) = AGL(1,ℤ/8)×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
homologyAbelianization (H1) = C3×C8 · Schur multiplier (H2) = C1

Aliases

direct productC8×ΓL(1,𝔽8)
unambiguous semidirect product(C7×C8)⋊C3

Splittings

Fitting, HallC7×C8 ⋊ C3
derived, SylowC7 ⋊ C3×C8

Subgroups

subgroupG/H
center, hypercenterC8ΓL(1,𝔽8)
commutator, hypocenterC7C3×C8
FrattiniC4C2×ΓL(1,𝔽8)
FittingC7×C8C3
socleC2×C7C3×C4
prime radicalC2×ΓL(1,𝔽8)C4
cosocleC4×C7C2×C3
LCSG ▷ C7
UCSC1 ◁ C8

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 56

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)⋊C16, C16×ΓL(1,𝔽8), C8×AGL(1,𝔽7), …
maximal normal subgroup ofΓL(1,𝔽8)⋊C16, C16×ΓL(1,𝔽8), C8×AGL(1,𝔽7), …
maximal quotient ofC16×ΓL(1,𝔽8), C2×C8×ΓL(1,𝔽8), C8×C7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 2);