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C3×C7×(C5⋊D5)⋊ΓL(1,𝔽8) #36347948

order22050
automorphism?
homologyAbelianization (H1) = C2×C32×C7 · Schur multiplier (H2) = C3×C5

Aliases

direct productC3×C7×(C5⋊D5)⋊ΓL(1,𝔽8)
unambiguous semidirect product(C3×C5⋊D5)⋊(C7×ΓL(1,𝔽8))

Splittings

Fitting, socleC3×C52×C72 ⋊ C2×C3
derivedC52×C7 ⋊ C2×C32×C7
HallC3×C7×C7⋊(C52⋊C3) ⋊ C2
HallC72×C5⋊D5 ⋊ C32
HallC52×C72 ⋊ C2×C32
HallC5⋊D5 ⋊ C3×C7×ΓL(1,𝔽8)
SylowC72 ⋊ C3×(C5⋊D5)⋊C3
SylowC52 ⋊ C2×C3×C7×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC3×C7(C5⋊D5)⋊ΓL(1,𝔽8)
commutator, cosocle, hypocenterC52×C7C2×C32×C7
FrattiniC1G
Fitting, socleC3×C52×C72C2×C3
LCSG ▷ C52×C7
UCSC1 ◁ C3×C7

192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 315

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22050, 385);