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C2×C3⋊(C5⋊(C7⋊AGL(1,𝔽11))) #37528567

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

Aliases

direct productC2×C3⋊(C5⋊(C7⋊AGL(1,𝔽11)))
unambiguous semidirect product(C2×C3)⋊(C5⋊(C7⋊AGL(1,𝔽11)))
unambiguous semidirect product(C2×C5)⋊(C3⋊(C7⋊AGL(1,𝔽11)))
unambiguous semidirect product(C2×C7)⋊(C3⋊(C5⋊AGL(1,𝔽11)))
unambiguous semidirect product(C2×C11⋊C5)⋊D105
unambiguous semidirect product(C2×D1155)⋊C5

Splittings

Fitting, socleC2×C3×C5×C7×C11 ⋊ C2×C5
derivedC3×C5×C7×C11 ⋊ C22×C5
HallC3×C5×C7×C11⋊C5 ⋊ C22
HallC5×C7×C11⋊C5 ⋊ C2×D3
HallC3×C5×C11⋊C5 ⋊ C2×D7
HallC5×C11⋊C5 ⋊ C2×D21
HallC3×C7×C11 ⋊ C2×C5×D5
HallC7×C11 ⋊ C2×C5×D15
HallC3×C11 ⋊ C2×C5×D35
HallC3×C7 ⋊ C2×C5⋊AGL(1,𝔽11)
SylowC11 ⋊ C2×C5×D105
SylowC7 ⋊ C2×C3⋊(C5⋊AGL(1,𝔽11))
SylowC3 ⋊ C2×C5⋊(C7⋊AGL(1,𝔽11))

Subgroups

subgroupG/H
center, hypercenterC2C3⋊(C5⋊(C7⋊AGL(1,𝔽11)))
commutator, cosocle, hypocenterC3×C5×C7×C11C22×C5
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C11C2×C5
LCSG ▷ C3×C5×C7×C11
UCSC1 ◁ C2

192 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 2310

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 378);