Finite Group Databasebeta

← Prev (#66655) Next (#66657) →

Cayley diagram

D4×AGL(1,𝔽7) #66656

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) = C23

Aliases

direct productD4×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×D4)
unambiguous semidirect product(D4×D7)⋊C3

Splittings

FittingC7×D4 ⋊ C2×C3
HallD4×D7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×D4
SylowC7 ⋊ C2×C3×D4

Subgroups

subgroupG/H
center, FrattiniC2C22×AGL(1,𝔽7)
commutator, socle, cosocleC2×C7C23×C3
FittingC7×D4C2×C3
hypercenterD4AGL(1,𝔽7)
hypocenterC7C2×C3×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ D4

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 28

A minimal faithful permutation representation — 3 generators on 11 points:

(2,5,6,3,4,7);(1,2,4,7,6,5,3)(9,11);(8,9)(10,11)

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD8×AGL(1,𝔽7), #6474⋊C3, SD16×AGL(1,𝔽7), …
maximal normal subgroup ofD8×AGL(1,𝔽7), #6474⋊C3, SD16×AGL(1,𝔽7), …
maximal quotient of#6441⋊C3, C22⋊C4×AGL(1,𝔽7), ((C22⋊D7)⋊C3)⋊C4, …
automorphism group ofDic14, D28, C2×Dic7, …
holomorph ofC4×C7

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 125);