Finite Group Databasebeta

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C3×C3⋊(C7⋊#10596815) #49141568

order36540
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×C3⋊(C7⋊#10596815)
unambiguous semidirect productC3⋊(C3×C7⋊#10596815)

Splittings

Fitting, socle, HallC32×C5×C7×C29 ⋊ C4
derivedC3×C5×C7×C29 ⋊ C3×C4
HallC32×C7×C29 ⋊ AGL(1,𝔽5)
HallC32×C5×C29 ⋊ Dic7
HallC5×C7×C29 ⋊ C3×Dic3
HallC32×C5×C7 ⋊ #1181
HallC32×C29 ⋊ C7⋊AGL(1,𝔽5)
HallC7×C29 ⋊ C3×ΓL(1,𝔽16)
HallC5×C29 ⋊ C3×Dic21
HallC32×C7 ⋊ #10596815
HallC32×C5 ⋊ C7⋊#1181
HallC5×C7 ⋊ C3×C3⋊#1181
SylowC29 ⋊ C3×C3⋊(C7⋊AGL(1,𝔽5))
SylowC32 ⋊ C7⋊#10596815
SylowC7 ⋊ C3×C3⋊#10596815
SylowC5 ⋊ C3×C3⋊(C7⋊#1181)

Subgroups

subgroupG/H
center, hypercenterC3C3⋊(C7⋊#10596815)
commutator, hypocenterC3×C5×C7×C29C3×C4
FrattiniC1G
Fitting, socleC32×C5×C7×C29C4
prime radicalC32×C7×D145C2
cosocleC3×C7×D145C2×C3
LCSG ▷ C3×C5×C7×C29
UCSC1 ◁ C3

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

normal subgroups only

This group has 112 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) = 6090

Conjugacy classes & character table

2340 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 410);