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C7×C11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5)) #44568652

order32340
automorphism?
homologyAbelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×C11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))
unambiguous semidirect productC11⋊(C7×ΓL(1,𝔽8)⋊AGL(1,𝔽5))
unambiguous semidirect productΓL(1,𝔽8)⋊(C7×C11⋊AGL(1,𝔽5))
unambiguous semidirect product(C7⋊(C11⋊AGL(1,𝔽5)))⋊(C3×C7)

Splittings

Fitting, socle, HallC5×C72×C11 ⋊ C3×C4
derivedC5×C7×C11 ⋊ C3×C4×C7
HallC7×C7⋊(C11⋊AGL(1,𝔽5)) ⋊ C3
HallC5×C7×C11×ΓL(1,𝔽8) ⋊ C4
HallC7×C11×ΓL(1,𝔽8) ⋊ AGL(1,𝔽5)
HallC5×C7×ΓL(1,𝔽8) ⋊ Dic11
HallC72×C11 ⋊ C3×AGL(1,𝔽5)
HallC5×C72 ⋊ C3×Dic11
HallC7×ΓL(1,𝔽8) ⋊ C11⋊AGL(1,𝔽5)
HallC5×C11 ⋊ C7×ΓL(1,𝔽8)⋊C4
SylowC72 ⋊ C3×C11⋊AGL(1,𝔽5)
SylowC11 ⋊ C7×ΓL(1,𝔽8)⋊AGL(1,𝔽5)
SylowC5 ⋊ C7×C11⋊(ΓL(1,𝔽8)⋊C4)

Subgroups

subgroupG/H
center, hypercenterC7C11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))
commutator, hypocenterC5×C7×C11C3×C4×C7
FrattiniC1G
Fitting, socleC5×C72×C11C3×C4
prime radicalC7×D5×C11×ΓL(1,𝔽8)C2
cosocleC7×D5×C11C2×C3×C7
LCSG ▷ C5×C7×C11
UCSC1 ◁ C7

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) = 2310

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 354);