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C7×A4×AGL(1,𝔽5) #13132927

order1680
isoclinismisoclinic class of A4×AGL(1,𝔽5)
automorphism?
homologyAbelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×A4×AGL(1,𝔽5)
unambiguous semidirect product(C22×AGL(1,𝔽5))⋊(C3×C7)

Splittings

Fitting, socleC22×C5×C7 ⋊ C3×C4
derivedC22×C5 ⋊ C3×C4×C7
HallC22×C7×AGL(1,𝔽5) ⋊ C3
HallC22×AGL(1,𝔽5) ⋊ C3×C7
HallC5×C7 ⋊ C4×A4
SylowC5 ⋊ C4×C7×A4

Subgroups

subgroupG/H
center, hypercenterC7A4×AGL(1,𝔽5)
commutator, hypocenterC22×C5C3×C4×C7
FrattiniC1G
Fitting, socleC22×C5×C7C3×C4
prime radicalC7×D5×A4C2
cosocleC22×D5C2×C3×C7
LCSG ▷ C22×C5
UCSC1 ◁ C7

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 140

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofA4×AGL(1,𝔽5)×ΓL(1,𝔽8), A4×D7×AGL(1,𝔽5), C7×AGL(1,𝔽5)×S4, …
maximal normal subgroup ofA4×AGL(1,𝔽5)×ΓL(1,𝔽8), A4×D7×AGL(1,𝔽5), C7×AGL(1,𝔽5)×S4, …
maximal quotient ofC7×AGL(1,𝔽5)×SL(2,𝔽3)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 947);