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C7×AGL(1,𝔽7)×D31 #32912136

order18228
automorphism?
homologyAbelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×AGL(1,𝔽7)×D31
unambiguous semidirect productD7⋊(C3×C7×D31)
unambiguous semidirect product(ΓL(1,𝔽8)×D31)⋊(C2×C7)
unambiguous semidirect product(C31×AGL(1,𝔽7))⋊(C2×C7)
unambiguous semidirect product(ΓL(1,𝔽8)⋊D31)⋊(C2×C7)

Splittings

Fitting, socle, HallC72×C31 ⋊ C22×C3
derivedC7×C31 ⋊ C22×C3×C7
HallC7×D7×D31 ⋊ C3
HallC7×ΓL(1,𝔽8)×C31 ⋊ C22
HallC7×ΓL(1,𝔽8) ⋊ C2×D31
SylowC72 ⋊ C2×C3×D31
SylowC31 ⋊ C2×C7×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC7AGL(1,𝔽7)×D31
commutator, cosocle, hypocenterC7×C31C22×C3×C7
FrattiniC1G
Fitting, socleC72×C31C22×C3
LCSG ▷ C7×C31
UCSC1 ◁ C7

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1302

Conjugacy classes & character table

833 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18228, 144);