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Cayley diagram

C16×ΓL(1,𝔽8) #66533

order336
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphismAut(G) = C2×C4×AGL(1,𝔽7) (order 336) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C22×C4
homologyAbelianization (H1) = C3×C16 · Schur multiplier (H2) = C1

Aliases

direct productC16×ΓL(1,𝔽8)
unambiguous semidirect product(C7×C16)⋊C3

Splittings

Fitting, HallC7×C16 ⋊ C3
derived, SylowC7 ⋊ C3×C16

Subgroups

subgroupG/H
center, hypercenterC16ΓL(1,𝔽8)
commutator, hypocenterC7C3×C16
FrattiniC8C2×ΓL(1,𝔽8)
FittingC7×C16C3
socleC2×C7C3×C8
prime radicalC2×ΓL(1,𝔽8)C8
cosocleC7×C8C2×C3
LCSG ▷ C7
UCSC1 ◁ C16

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 2 generators on 23 points:

(2,3,4)(5,7,6)(8,9,10,13,11,14,16,19,12,15,17,20,18,21,22,23);(1,2,3,5,4,6,7)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)⋊C32, ΓL(1,𝔽8)×C32, C16×AGL(1,𝔽7), …
maximal normal subgroup ofΓL(1,𝔽8)⋊C32, ΓL(1,𝔽8)×C32, C16×AGL(1,𝔽7), …
maximal quotient ofΓL(1,𝔽8)×C32, C2×C16×ΓL(1,𝔽8), C16×C7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 2);