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C7×ΓL(1,𝔽8)⋊(Dic11⋊C5) #44568438

order32340
automorphism?
homologyAbelianization (H1) = C3×C4×C5×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×ΓL(1,𝔽8)⋊(Dic11⋊C5)
unambiguous semidirect productΓL(1,𝔽8)⋊(C7×Dic11⋊C5)
unambiguous semidirect product(C11⋊C5)⋊(C7×ΓL(1,𝔽8)⋊C4)
unambiguous semidirect product(C11⋊(ΓL(1,𝔽8)⋊C4))⋊(C5×C7)
unambiguous semidirect product(C7⋊(Dic11⋊C5))⋊(C3×C7)
unambiguous semidirect product(C7×C11⋊(ΓL(1,𝔽8)⋊C4))⋊C5

Splittings

derivedC7×C11 ⋊ C3×C4×C5×C7
HallC7×C7⋊(Dic11⋊C5) ⋊ C3
HallC7×ΓL(1,𝔽8)×C11⋊C5 ⋊ C4
HallC7×C11⋊(ΓL(1,𝔽8)⋊C4) ⋊ C5
HallC72×C11⋊C5 ⋊ C3×C4
HallC7×Dic77 ⋊ C3×C5
HallC7×C11×ΓL(1,𝔽8) ⋊ C4×C5
HallC72×C11 ⋊ C3×C4×C5
HallC7×ΓL(1,𝔽8) ⋊ Dic11⋊C5
HallC11⋊C5 ⋊ C7×ΓL(1,𝔽8)⋊C4
SylowC72 ⋊ C3×Dic11⋊C5
SylowC11 ⋊ C5×C7×ΓL(1,𝔽8)⋊C4

Subgroups

subgroupG/H
center, hypercenterC2×C7ΓL(1,𝔽8)⋊AGL(1,𝔽11)
commutator, hypocenterC7×C11C3×C4×C5×C7
FrattiniC2C7×ΓL(1,𝔽8)⋊AGL(1,𝔽11)
Fitting, socleC2×C72×C11C2×C3×C5
prime radicalC2×C7×ΓL(1,𝔽8)×C11⋊C5C2
cosocleC2×C7×C11C2×C3×C5×C7
LCSG ▷ C7×C11
UCSC1 ◁ C2×C7

104 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 924

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 140);