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D4×A4×AGL(1,𝔽7) #72244587

order4032
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C24×C3

Aliases

direct productD4×A4×AGL(1,𝔽7)

Splittings

FittingC22×C7×D4 ⋊ C2×C32
HallC22×D4×D7 ⋊ C32
SylowC7 ⋊ C2×C3×D4×A4

Subgroups

subgroupG/H
center, FrattiniC2C22×A4×AGL(1,𝔽7)
commutator, socle, cosocleC23×C7C23×C32
FittingC22×C7×D4C2×C32
hypercenterD4A4×AGL(1,𝔽7)
hypocenterC22×C7C2×C32×D4
LCSG ▷ C23×C7 ▷ C22×C7
UCSC1 ◁ C2 ◁ D4

1484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 84

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

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofA4×S4×AGL(1,𝔽7), D4×S4×AGL(1,𝔽7)
maximal normal subgroup ofD4×S4×AGL(1,𝔽7)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,5,6,4,7,3)(9,11)(13,14,15), (2,6,5,3,4,7)(12,15)(13,14), (8,9)(10,11));