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C7×C11⋊C5×ΓL(1,𝔽8)⋊C4 #44568439

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

Aliases

direct productC7×C11⋊C5×ΓL(1,𝔽8)⋊C4
unambiguous semidirect productΓL(1,𝔽8)⋊(C4×C7×C11⋊C5)
unambiguous semidirect productDic7⋊(C3×C7×C11⋊C5)
unambiguous semidirect product(C11×ΓL(1,𝔽8)⋊C4)⋊(C5×C7)
unambiguous semidirect product(Dic7×C11⋊C5)⋊(C3×C7)
unambiguous semidirect product(C7×C11×ΓL(1,𝔽8)⋊C4)⋊C5

Splittings

derivedC7×C11 ⋊ C3×C4×C5×C7
HallC7×Dic7×C11⋊C5 ⋊ C3
HallC7×ΓL(1,𝔽8)×C11⋊C5 ⋊ C4
HallC7×C11×ΓL(1,𝔽8)⋊C4 ⋊ C5
HallC72×C11⋊C5 ⋊ C3×C4
HallC7×C11×Dic7 ⋊ C3×C5
HallC7×C11×ΓL(1,𝔽8) ⋊ C4×C5
HallC72×C11 ⋊ C3×C4×C5
HallC7×Dic7 ⋊ C3×C11⋊C5
HallC7×ΓL(1,𝔽8) ⋊ C4×C11⋊C5
SylowC72 ⋊ C3×C4×C11⋊C5
SylowC11 ⋊ C5×C7×ΓL(1,𝔽8)⋊C4

Subgroups

subgroupG/H
center, hypercenterC2×C7AGL(1,𝔽7)×C11⋊C5
commutator, hypocenterC7×C11C3×C4×C5×C7
FrattiniC2C7×AGL(1,𝔽7)×C11⋊C5
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 48 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 48 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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 141);