Finite Group Databasebeta

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C2×C3×C5×C7×D5×C11 #37528740

order23100
automorphism?
homologyAbelianization (H1) = C22×C3×C5×C7×C11 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3×C5×C7×D5×C11
unambiguous semidirect productC5⋊(C22×C3×C5×C7×C11)
unambiguous semidirect product(C2×C5)⋊(C2×C3×C5×C7×C11)

Splittings

Fitting, socleC2×C3×C52×C7×C11 ⋊ C2
derivedC5 ⋊ C22×C3×C5×C7×C11
HallC3×C52×C7×C11 ⋊ C22
HallC52×C7×C11 ⋊ C22×C3
HallC3×C52×C11 ⋊ C22×C7
HallC3×C52×C7 ⋊ C22×C11
HallC52×C11 ⋊ C22×C3×C7
HallC52×C7 ⋊ C22×C3×C11
HallC3×C52 ⋊ C22×C7×C11
SylowC52 ⋊ C22×C3×C7×C11

Subgroups

subgroupG/H
center, hypercenterC2×C3×C5×C7×C11D5
commutator, cosocle, hypocenterC5C22×C3×C5×C7×C11
FrattiniC1G
Fitting, socleC2×C3×C52×C7×C11C2
LCSG ▷ C5
UCSC1 ◁ C2×C3×C5×C7×C11

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

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 112 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

9240 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 551);