Finite Group Databasebeta

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C2×C5×C5⋊#10587723 #41031874

order27300
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5×C5⋊#10587723
unambiguous semidirect product(C2×C5)⋊(C5×#10587723)

Splittings

Fitting, socleC2×C52×C7×C13 ⋊ C2×C3
derivedC5×C7×C13 ⋊ C22×C3×C5
HallC2×C5×D455 ⋊ C3
HallC52×#63271 ⋊ C22
HallC52×C7×C13 ⋊ C22×C3
HallC52×C13 ⋊ C2×AGL(1,𝔽7)
Hall#63271 ⋊ C2×C5×D5
HallC52×C7 ⋊ C2×D13⋊C3
HallC7×C13 ⋊ C2×C3×C5×D5
SylowC52 ⋊ C2×#10587723
SylowC13 ⋊ C2×C5×C5⋊AGL(1,𝔽7)
SylowC7 ⋊ C2×C5×C5⋊(D13⋊C3)

Subgroups

subgroupG/H
center, hypercenterC2×C5C5⋊#10587723
commutator, cosocle, hypocenterC5×C7×C13C22×C3×C5
FrattiniC1G
Fitting, socleC2×C52×C7×C13C2×C3
LCSG ▷ C5×C7×C13
UCSC1 ◁ C2×C5

192 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1820

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27300, 764);