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Cayley diagram

#1141⋊C3 #66549

order336
isoclinismisoclinic class of #1141⋊C3
automorphismAut(G) = C2×D4×AGL(1,𝔽7) (order 672) · Inn(G) = (C22⋊D7)⋊C3 · Out(G) = C22
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#1141⋊C3

Splittings

FittingC7×D4 ⋊ C2×C3
Hall#1141 ⋊ C3
HallΓL(1,𝔽8) ⋊ D8
SylowC7 ⋊ C3×D8

Subgroups

subgroupG/H
centerC2(C22⋊D7)⋊C3
commutator, cosocleC4×C7C22×C3
FrattiniC4C2×AGL(1,𝔽7)
FittingC7×D4C2×C3
socleC2×C7C3×D4
hypercenterD4AGL(1,𝔽7)
hypocenterC7C3×D8
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C4 ◁ D4

44 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:

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

Polycyclic presentation:

2^2,3,2^2,7|d^2=e,b^a=bd,d^a=de,f^a=f^6,d^b=de,f^c=f^4

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD8×AGL(1,𝔽7), #6474⋊C3, #6477⋊C3, …
maximal normal subgroup ofD8×AGL(1,𝔽7), #6474⋊C3, #6477⋊C3, …
maximal quotient of#6381⋊C3, #6383⋊C3, #6400⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 18);