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C4×C7×AGL(1,𝔽7) #11924189

order1176
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C2×C3×C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC4×C7×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×C4×C7)
unambiguous semidirect product(C4×D7)⋊(C3×C7)

Splittings

FittingC4×C72 ⋊ C2×C3
derivedC7 ⋊ C2×C3×C4×C7
HallC4×C7×D7 ⋊ C3
HallC7×ΓL(1,𝔽8) ⋊ C2×C4
SylowC72 ⋊ C2×C3×C4

Subgroups

subgroupG/H
center, hypercenterC4×C7AGL(1,𝔽7)
commutator, hypocenterC7C2×C3×C4×C7
FrattiniC2C2×C7×AGL(1,𝔽7)
FittingC4×C72C2×C3
socleC2×C72C22×C3
prime radicalC2×C7×AGL(1,𝔽7)C2
cosocleC2×C7C22×C3×C7
LCSG ▷ C7
UCSC1 ◁ C4×C7

67 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC7×#1157⋊C3, C4×D7×AGL(1,𝔽7), #72245012, …
maximal normal subgroup ofC7×#1157⋊C3, C4×D7×AGL(1,𝔽7), #72245012, …
maximal quotient ofC7×(C2×AGL(1,𝔽7))⋊C4, C4×#72215772, C7×AGL(1,𝔽5)×AGL(1,𝔽7), …
automorphism group ofD7×C29

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1176, 104);