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C11×C3⋊((C11⋊C5)⋊AGL(1,𝔽5)) #49137443

order36300
automorphism?
homologyAbelianization (H1) = C4×C5×C11 · Schur multiplier (H2) = C1

Aliases

direct productC11×C3⋊((C11⋊C5)⋊AGL(1,𝔽5))
unambiguous semidirect productC3⋊(C11×(C11⋊C5)⋊AGL(1,𝔽5))
unambiguous semidirect product(C11⋊C5)⋊(C11×ΓL(1,𝔽16))
unambiguous semidirect product(C3⋊(C11⋊AGL(1,𝔽5)))⋊(C5×C11)

Splittings

Fitting, socleC3×C5×C112 ⋊ C4×C5
derivedC3×C5×C11 ⋊ C4×C5×C11
HallC3×C5×C11×C11⋊C5 ⋊ C4
HallC5×C11×C11⋊C5 ⋊ Dic3
HallC3×C112 ⋊ C5×AGL(1,𝔽5)
SylowC112 ⋊ C5×ΓL(1,𝔽16)
SylowC3 ⋊ C11×(C11⋊C5)⋊AGL(1,𝔽5)

Subgroups

subgroupG/H
center, hypercenterC11C3⋊((C11⋊C5)⋊AGL(1,𝔽5))
commutator, hypocenterC3×C5×C11C4×C5×C11
FrattiniC1G
Fitting, socleC3×C5×C112C4×C5
prime radicalC3×D5×C11×C11⋊C5C2
cosocleC3×D5×C11C2×C5×C11
LCSG ▷ C3×C5×C11
UCSC1 ◁ C11

112 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 1650

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36300, 579);