Finite Group Databasebeta

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C2×C3×C5×D357 #36339581

order21420
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3×C5×D357
unambiguous semidirect product(C2×C3)⋊(C3×C5×D119)
unambiguous semidirect product(C2×C17)⋊(C3×C5×D21)

Splittings

Fitting, socleC2×C32×C5×C7×C17 ⋊ C2
derivedC3×C7×C17 ⋊ C22×C3×C5
HallC32×C5×C7×C17 ⋊ C22
HallC32×C7×C17 ⋊ C22×C5
HallC32×C5×C17 ⋊ C2×D7
HallC5×C7×C17 ⋊ C2×C3×D3
HallC32×C5×C7 ⋊ C2×D17
HallC32×C17 ⋊ C2×C5×D7
HallC7×C17 ⋊ C2×C3×C5×D3
HallC5×C17 ⋊ C2×C3×D21
HallC32×C7 ⋊ C2×C5×D17
HallC32×C5 ⋊ C2×D119
HallC5×C7 ⋊ C2×C3×D51
SylowC17 ⋊ C2×C3×C5×D21
SylowC32 ⋊ C2×C5×D119
SylowC7 ⋊ C2×C3×C5×D51

Subgroups

subgroupG/H
center, hypercenterC2×C3×C5D357
commutator, cosocle, hypocenterC3×C7×C17C22×C3×C5
FrattiniC1G
Fitting, socleC2×C32×C5×C7×C17C2
LCSG ▷ C3×C7×C17
UCSC1 ◁ C2×C3×C5

176 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 7140

Conjugacy classes & character table

5400 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 413);