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C5×C11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5)) #37528523

order23100
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×C11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×C11⋊AGL(1,𝔽5))
unambiguous semidirect product(C5×C7⋊(C11⋊AGL(1,𝔽5)))⋊C3

Splittings

Fitting, socle, HallC52×C7×C11 ⋊ C3×C4
derivedC5×C7×C11 ⋊ C3×C4×C5
HallC5×C7⋊(C11⋊AGL(1,𝔽5)) ⋊ C3
HallC52×C11×ΓL(1,𝔽8) ⋊ C4
HallC52×ΓL(1,𝔽8) ⋊ Dic11
HallC52×C11 ⋊ ΓL(1,𝔽8)⋊C4
HallC11×ΓL(1,𝔽8) ⋊ C5×AGL(1,𝔽5)
HallC52×C7 ⋊ C3×Dic11
HallC7×C11 ⋊ C3×C5×AGL(1,𝔽5)
SylowC52 ⋊ C11⋊(ΓL(1,𝔽8)⋊C4)
HallΓL(1,𝔽8) ⋊ C5×C11⋊AGL(1,𝔽5)
SylowC11 ⋊ C5×ΓL(1,𝔽8)⋊AGL(1,𝔽5)
SylowC7 ⋊ C3×C5×C11⋊AGL(1,𝔽5)

Subgroups

subgroupG/H
center, hypercenterC5C11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))
commutator, hypocenterC5×C7×C11C3×C4×C5
FrattiniC1G
Fitting, socleC52×C7×C11C3×C4
prime radicalC5×D5×C11×ΓL(1,𝔽8)C2
cosocleC7×D5×C11C2×C3×C5
LCSG ▷ C5×C7×C11
UCSC1 ◁ C5

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

normal subgroups only

This group has 104 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) = 1540

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 334);