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Cayley diagram

Q8×AGL(1,𝔽7) #66658

order336
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productQ8×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×Q8)

Splittings

FittingC7×Q8 ⋊ C2×C3
HallQ8×D7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×Q8
SylowC7 ⋊ C2×C3×Q8

Subgroups

subgroupG/H
center, FrattiniC2C22×AGL(1,𝔽7)
commutator, socle, cosocleC2×C7C23×C3
FittingC7×Q8C2×C3
prime radicalC2×AGL(1,𝔽7)C22
hypercenterQ8AGL(1,𝔽7)
hypocenterC7C2×C3×Q8
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ Q8

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofSD16×AGL(1,𝔽7), #6478⋊C3, Q16×AGL(1,𝔽7), …
maximal normal subgroup ofSD16×AGL(1,𝔽7), #6478⋊C3, Q16×AGL(1,𝔽7), …
maximal quotient of#6450⋊C3, #6451⋊C3, #6452⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 127);