Finite Group Databasebeta

← Prev (#66656) Next (#66658) →

Cayley diagram

#1159⋊C3 #66657

order336
automorphismAut(G) = C2×D4×AGL(1,𝔽7) (order 672) · Inn(G) = C22×AGL(1,𝔽7) · Out(G) = C22
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

unambiguous semidirect product#1159⋊C3

Splittings

FittingC7×D4 ⋊ C2×C3
Hall#1159 ⋊ C3
HallΓL(1,𝔽8) ⋊ #41
SylowC7 ⋊ C3×#41

Subgroups

subgroupG/H
center, FrattiniC2C22×AGL(1,𝔽7)
commutator, socle, cosocleC2×C7C23×C3
FittingC7×D4C2×C3
hypercenterD4AGL(1,𝔽7)
hypocenterC7C3×#41
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ D4

80 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6474⋊C3, #6475⋊C3, #6478⋊C3, …
maximal normal subgroup of#6474⋊C3, #6475⋊C3, #6478⋊C3, …
maximal quotient of#6440⋊C3, #6441⋊C3, #6442⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 126);