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Cayley diagram

M16×ΓL(1,𝔽8) #66583

order336
isoclinismisoclinic class of D4×ΓL(1,𝔽8)
automorphismAut(G) = C2×D4×AGL(1,𝔽7) (order 672) · Inn(G) = C22×ΓL(1,𝔽8) · Out(G) = C23
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productM16×ΓL(1,𝔽8)
unambiguous semidirect product(C7×M16)⋊C3

Splittings

Fitting, HallC7×M16 ⋊ C3
SylowC7 ⋊ C3×M16

Subgroups

subgroupG/H
center, FrattiniC4C22×ΓL(1,𝔽8)
commutator, socleC2×C7C2×C3×C4
FittingC7×M16C3
prime radicalC22×ΓL(1,𝔽8)C4
cosocleC4×C7C22×C3
hypercenterM16ΓL(1,𝔽8)
hypocenterC7C3×M16
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ M16

40 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)(9,12)(13,15);(1,2,3,5,4,6,7)(8,9,10,13,11,12,14,15)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6396⋊C3, #6397⋊C3, #6398⋊C3, …
maximal normal subgroup of#6396⋊C3, #6397⋊C3, #6398⋊C3, …
maximal quotient ofΓL(1,𝔽8)×#97, ΓL(1,𝔽8)×C22⋊C8, ΓL(1,𝔽8)×C4⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 52);