Finite Group Databasebeta

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C5×C3⋊(C7⋊#63121) #41031999

order27300
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×C3⋊(C7⋊#63121)
unambiguous semidirect productC3⋊(C5×C7⋊#63121)
unambiguous semidirect productC7⋊(C5×C3⋊#63121)

Splittings

Fitting, socle, HallC3×C52×C7×C13 ⋊ C4
derivedC3×C5×C7×C13 ⋊ C4×C5
HallC52×C7×C13 ⋊ Dic3
HallC3×C52×C13 ⋊ Dic7
HallC3×C52×C7 ⋊ #261
HallC52×C13 ⋊ Dic21
HallC3×C7×C13 ⋊ C5×AGL(1,𝔽5)
HallC52×C7 ⋊ C3⋊#261
HallC7×C13 ⋊ C5×ΓL(1,𝔽16)
HallC3×C52 ⋊ C7⋊#261
HallC3×C13 ⋊ C5×C7⋊AGL(1,𝔽5)
SylowC52 ⋊ C3⋊(C7⋊#261)
HallC3×C7 ⋊ C5×#63121
SylowC13 ⋊ C5×C3⋊(C7⋊AGL(1,𝔽5))
SylowC7 ⋊ C5×C3⋊#63121
SylowC3 ⋊ C5×C7⋊#63121

Subgroups

subgroupG/H
center, hypercenterC5C3⋊(C7⋊#63121)
commutator, hypocenterC3×C5×C7×C13C4×C5
FrattiniC1G
Fitting, socleC3×C52×C7×C13C4
prime radicalC3×C5×C7×D65C2
cosocleC3×C7×D65C2×C5
LCSG ▷ C3×C5×C7×C13
UCSC1 ◁ C5

104 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 5460

Conjugacy classes & character table

1800 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27300, 889);