Finite Group Databasebeta

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C3×C22⋊(C7⋊(C112⋊C3)) #44479252

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

Aliases

direct productC3×C22⋊(C7⋊(C112⋊C3))
unambiguous semidirect product(C22×C3)⋊(C7⋊(C112⋊C3))
unambiguous semidirect product(C3×C7)⋊(C22⋊(C112⋊C3))
unambiguous semidirect product(C3×C112)⋊(C22⋊ΓL(1,𝔽8))

Splittings

Fitting, socleC22×C3×C7×C112 ⋊ C3
derived, HallC22×C7×C112 ⋊ C32
HallC7×C112 ⋊ C3×A4
HallC22×C112 ⋊ C3×ΓL(1,𝔽8)
SylowC112 ⋊ C3×C22⋊ΓL(1,𝔽8)
HallC22×C7 ⋊ C3×C112⋊C3
SylowC7 ⋊ C3×C22⋊(C112⋊C3)
SylowC22 ⋊ C3×C7⋊(C112⋊C3)

Subgroups

subgroupG/H
center, hypercenterC3C22⋊(C7⋊(C112⋊C3))
commutator, cosocle, hypocenterC22×C7×C112C32
FrattiniC1G
Fitting, socleC22×C3×C7×C112C3
LCSG ▷ C22×C7×C112
UCSC1 ◁ C3

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1386

Conjugacy classes & character table

3396 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30492, 475);