Finite Group Databasebeta

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C2×C3×C5×D385 #37528743

order23100
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3×C5×D385
unambiguous semidirect product(C2×C5)⋊(C3×C5×D77)

Splittings

Fitting, socleC2×C3×C52×C7×C11 ⋊ C2
derivedC5×C7×C11 ⋊ C22×C3×C5
HallC3×C52×C7×C11 ⋊ C22
HallC52×C7×C11 ⋊ C22×C3
HallC3×C52×C11 ⋊ C2×D7
HallC3×C52×C7 ⋊ C2×D11
HallC52×C11 ⋊ C2×C3×D7
HallC3×C7×C11 ⋊ C2×C5×D5
HallC52×C7 ⋊ C2×C3×D11
HallC7×C11 ⋊ C2×C3×C5×D5
HallC3×C52 ⋊ C2×D77
HallC3×C11 ⋊ C2×C5×D35
SylowC52 ⋊ C2×C3×D77
HallC3×C7 ⋊ C2×C5×D55
SylowC11 ⋊ C2×C3×C5×D35
SylowC7 ⋊ C2×C3×C5×D55

Subgroups

subgroupG/H
center, hypercenterC2×C3×C5D385
commutator, cosocle, hypocenterC5×C7×C11C22×C3×C5
FrattiniC1G
Fitting, socleC2×C3×C52×C7×C11C2
LCSG ▷ C5×C7×C11
UCSC1 ◁ C2×C3×C5

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

5820 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 554);