Finite Group Databasebeta

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C2×C5×C7×D165 #37528753

order23100
automorphism?
homologyAbelianization (H1) = C22×C5×C7 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5×C7×D165
unambiguous semidirect product(C2×C3)⋊(C5×C7×D55)
unambiguous semidirect product(C2×C5)⋊(C5×C7×D33)

Splittings

Fitting, socleC2×C3×C52×C7×C11 ⋊ C2
derivedC3×C5×C11 ⋊ C22×C5×C7
HallC3×C52×C7×C11 ⋊ C22
HallC52×C7×C11 ⋊ C2×D3
HallC3×C52×C11 ⋊ C22×C7
HallC3×C52×C7 ⋊ C2×D11
HallC52×C11 ⋊ C2×D3×C7
HallC3×C7×C11 ⋊ C2×C5×D5
HallC52×C7 ⋊ C2×D33
HallC7×C11 ⋊ C2×C5×D15
HallC3×C52 ⋊ C2×C7×D11
HallC3×C11 ⋊ C2×C5×C7×D5
SylowC52 ⋊ C2×C7×D33
HallC3×C7 ⋊ C2×C5×D55
SylowC11 ⋊ C2×C5×C7×D15
SylowC3 ⋊ C2×C5×C7×D55

Subgroups

subgroupG/H
center, hypercenterC2×C5×C7D165
commutator, cosocle, hypocenterC3×C5×C11C22×C5×C7
FrattiniC1G
Fitting, socleC2×C3×C52×C7×C11C2
LCSG ▷ C3×C5×C11
UCSC1 ◁ C2×C5×C7

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

normal subgroups only

This group has 192 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) = 4620

Conjugacy classes & character table

5880 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 564);