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Cayley diagram

D56⋊C3 #66541

order336
isoclinismisoclinic class of #1132⋊C3
automorphismAut(G) = AGL(1,ℤ/8)×AGL(1,𝔽7) (order 1344) · Inn(G) = C4⋊AGL(1,𝔽7) · Out(G) = C23
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productD56⋊C3

Splittings

FittingC7×C8 ⋊ C2×C3
HallD56 ⋊ C3
HallΓL(1,𝔽8) ⋊ D8
SylowC7 ⋊ C3×D8

Subgroups

subgroupG/H
centerC2C4⋊AGL(1,𝔽7)
commutator, cosocleC4×C7C22×C3
FrattiniC4C2×AGL(1,𝔽7)
FittingC7×C8C2×C3
socleC2×C7C3×D4
hypercenterC8AGL(1,𝔽7)
hypocenterC7C3×D8
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C4 ◁ C8

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

Polycyclic presentation:

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

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD112⋊C3, #6374⋊C3, #6400⋊C3, …
maximal normal subgroup ofD112⋊C3, #6374⋊C3, #6400⋊C3, …
maximal quotient ofD112⋊C3, #6374⋊C3, Dic56⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 10);