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AGL(1,𝔽5)×ΓL(1,𝔽8)×C11⋊C5 #37528413

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

Aliases

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

Splittings

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

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC5×C7×C11C3×C4×C5
prime radicalD5×ΓL(1,𝔽8)×C11⋊C5C2
cosocleC7×D5×C11C2×C3×C5
LCSG ▷ C5×C7×C11
UCSC1

104 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 385

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 224);