Finite Group Databasebeta

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C3×C7×C3⋊#10596815 #49141564

order36540
automorphism?
homologyAbelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C1

Aliases

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

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC3×C7C3⋊#10596815
commutator, hypocenterC3×C5×C29C3×C4×C7
FrattiniC1G
Fitting, socleC32×C5×C7×C29C4
prime radicalC32×C7×D145C2
cosocleC3×D145C2×C3×C7
LCSG ▷ C3×C5×C29
UCSC1 ◁ C3×C7

112 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 44 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

2394 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 406);