Finite Group Databasebeta

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C3×C7×C3⋊#66772 #36339468

order21420
automorphism?
homologyAbelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC3×C7×C3⋊#66772
unambiguous semidirect productC3⋊(C3×C7×#66772)

Splittings

Fitting, socle, HallC32×C5×C7×C17 ⋊ C4
derivedC3×C5×C17 ⋊ C3×C4×C7
HallC32×C7×C17 ⋊ Dic5
HallC32×C5×C17 ⋊ C4×C7
HallC5×C7×C17 ⋊ C3×Dic3
HallC32×C5×C7 ⋊ #595
HallC32×C17 ⋊ C7×Dic5
HallC7×C17 ⋊ C3×Dic15
HallC5×C17 ⋊ C3×C7×Dic3
HallC32×C7 ⋊ #66772
HallC32×C5 ⋊ C7×#595
HallC5×C7 ⋊ C3×C3⋊#595
SylowC17 ⋊ C3×C7×Dic15
SylowC32 ⋊ C7×#66772
SylowC5 ⋊ C3×C7×C3⋊#595

Subgroups

subgroupG/H
center, hypercenterC3×C7C3⋊#66772
commutator, hypocenterC3×C5×C17C3×C4×C7
FrattiniC1G
Fitting, socleC32×C5×C7×C17C4
prime radicalC32×C5×C7×D17C2
cosocleC3×C5×D17C2×C3×C7
LCSG ▷ C3×C5×C17
UCSC1 ◁ C3×C7

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

normal subgroups only

This group has 112 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) = 3570

Conjugacy classes & character table

1638 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 300);