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C2×C3×C112×AGL(1,𝔽7) #44479146

order30492
automorphism?
homologyAbelianization (H1) = C22×C32×C112 · Schur multiplier (H2) = C2×C3×C11

Aliases

direct productC2×C3×C112×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C2×C32×C112)
unambiguous semidirect productΓL(1,𝔽8)⋊(C22×C3×C112)
unambiguous semidirect product(C2×D7)⋊(C32×C112)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C2×C3×C112)
unambiguous semidirect product(C2×C3×D7)⋊(C3×C112)
unambiguous semidirect product(C2×C11×D7)⋊(C32×C11)
unambiguous semidirect product(C2×C3×C11×D7)⋊(C3×C11)

Splittings

Fitting, socleC2×C3×C7×C112 ⋊ C2×C3
derived, SylowC7 ⋊ C22×C32×C112
HallC3×C112×ΓL(1,𝔽8) ⋊ C22
HallC2×C112×D7 ⋊ C32
HallC7×C112 ⋊ C22×C32
HallC3×ΓL(1,𝔽8) ⋊ C22×C112
HallC2×D7 ⋊ C32×C112

Subgroups

subgroupG/H
center, hypercenterC2×C3×C112AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C22×C32×C112
FrattiniC1G
Fitting, socleC2×C3×C7×C112C2×C3
LCSG ▷ C7
UCSC1 ◁ C2×C3×C112

840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 5082

Conjugacy classes & character table

5082 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30492, 369);