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Cayley diagram

Q16×ΓL(1,𝔽8) #66586

order336
isoclinismisoclinic class of D8×ΓL(1,𝔽8)
automorphismAut(G) = AGL(1,ℤ/8)×AGL(1,𝔽7) (order 1344) · Inn(G) = D4×ΓL(1,𝔽8) · Out(G) = C23
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productQ16×ΓL(1,𝔽8)
unambiguous semidirect product(C7×Q16)⋊C3

Splittings

Fitting, HallC7×Q16 ⋊ C3
SylowC7 ⋊ C3×Q16

Subgroups

subgroupG/H
centerC2D4×ΓL(1,𝔽8)
commutator, cosocleC4×C7C22×C3
FrattiniC4C22×ΓL(1,𝔽8)
FittingC7×Q16C3
socleC2×C7C3×D4
prime radicalC2×ΓL(1,𝔽8)D4
hypercenterQ16ΓL(1,𝔽8)
hypocenterC7C3×Q16
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C4 ◁ Q16

36 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)(8,10,12,18)(9,13,15,22)(11,20,21,17)(14,23,19,16);(1,2,3,5,4,6,7)(8,9,12,15)(10,16,18,23)(11,19,21,14)(13,20,22,17)

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6402⋊C3, #6403⋊C3, ΓL(1,𝔽8)×SD32, …
maximal normal subgroup of#6402⋊C3, #6403⋊C3, ΓL(1,𝔽8)×SD32, …
maximal quotient ofΓL(1,𝔽8)×#103, ΓL(1,𝔽8)×#107, C2×Q16×ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 55);