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C5×C7×C11⋊(ΓL(1,𝔽8)⋊C4) #44568472

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

Aliases

direct productC5×C7×C11⋊(ΓL(1,𝔽8)⋊C4)
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×C7×Dic11)
unambiguous semidirect product(C5×Dic77)⋊(C3×C7)

Splittings

derivedC7×C11 ⋊ C3×C4×C5×C7
HallC5×C7×Dic77 ⋊ C3
HallC5×C7×C11×ΓL(1,𝔽8) ⋊ C4
HallC5×C72×C11 ⋊ C3×C4
HallC7×Dic77 ⋊ C3×C5
HallC7×C11×ΓL(1,𝔽8) ⋊ C4×C5
HallC5×C7×ΓL(1,𝔽8) ⋊ Dic11
HallC72×C11 ⋊ C3×C4×C5
HallC5×C72 ⋊ C3×Dic11
HallC7×ΓL(1,𝔽8) ⋊ C5×Dic11
HallC5×C11 ⋊ C7×ΓL(1,𝔽8)⋊C4
SylowC72 ⋊ C3×C5×Dic11
SylowC11 ⋊ C5×C7×ΓL(1,𝔽8)⋊C4

Subgroups

subgroupG/H
center, hypercenterC2×C5×C7C11⋊AGL(1,𝔽7)
commutator, hypocenterC7×C11C3×C4×C5×C7
FrattiniC2C5×C7×C11⋊AGL(1,𝔽7)
Fitting, socleC2×C5×C72×C11C2×C3
prime radicalC2×C5×C7×C11×ΓL(1,𝔽8)C2
cosocleC2×C7×C11C2×C3×C5×C7
LCSG ▷ C7×C11
UCSC1 ◁ C2×C5×C7

104 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 4620

Conjugacy classes & character table

2240 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 174);