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C2×C3⋊((C11⋊C5)⋊AGL(1,𝔽11)) #49137358

order36300
automorphism?
homologyAbelianization (H1) = C22×C52 · Schur multiplier (H2) = C2×C5

Aliases

direct productC2×C3⋊((C11⋊C5)⋊AGL(1,𝔽11))
unambiguous semidirect product(C2×C3)⋊((C11⋊C5)⋊AGL(1,𝔽11))
unambiguous semidirect product(C2×C11⋊C5)⋊(C3⋊AGL(1,𝔽11))
unambiguous semidirect product(C2×C3⋊#11925892)⋊C5
unambiguous semidirect product(C2×C3⋊#11925893)⋊C5
unambiguous semidirect product(C2×C3⋊#11925894)⋊C5
unambiguous semidirect product(C2×C3⋊((C11⋊C5)⋊D11))⋊C5

Splittings

Fitting, socleC2×C3×C112 ⋊ C2×C52
derived, HallC3×C112 ⋊ C22×C52
HallC3×C11⋊C52 ⋊ C22
HallC11⋊C52 ⋊ C2×D3
HallC2×C3⋊(C11⋊D11) ⋊ C52
SylowC112 ⋊ C2×C52×D3
SylowC3 ⋊ C2×(C11⋊C5)⋊AGL(1,𝔽11)

Subgroups

subgroupG/H
center, hypercenterC2C3⋊((C11⋊C5)⋊AGL(1,𝔽11))
commutator, cosocle, hypocenterC3×C112C22×C52
FrattiniC1G
Fitting, socleC2×C3×C112C2×C52
LCSG ▷ C3×C112
UCSC1 ◁ C2

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

normal subgroups only

This group has 360 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) = 330

Conjugacy classes & character table

222 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36300, 494);