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C5×C11×ΓL(1,𝔽8)⋊Dic7 #44568482

order32340
automorphism?
homologyAbelianization (H1) = C3×C4×C5×C11 · Schur multiplier (H2) = C1

Aliases

direct productC5×C11×ΓL(1,𝔽8)⋊Dic7
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×C11×Dic7)

Splittings

derived, SylowC72 ⋊ C3×C4×C5×C11
HallC5×C11×C7⋊Dic7 ⋊ C3
HallC5×C7×C11×ΓL(1,𝔽8) ⋊ C4
HallC5×C72×C11 ⋊ C3×C4
HallC11×C7⋊Dic7 ⋊ C3×C5
HallC7×C11×ΓL(1,𝔽8) ⋊ C4×C5
HallC5×C7⋊Dic7 ⋊ C3×C11
HallC5×C7×ΓL(1,𝔽8) ⋊ C4×C11
HallC72×C11 ⋊ C3×C4×C5
HallC5×C72 ⋊ C3×C4×C11
HallC7⋊Dic7 ⋊ C3×C5×C11
HallC7×ΓL(1,𝔽8) ⋊ C4×C5×C11

Subgroups

subgroupG/H
center, hypercenterC2×C5×C11ΓL(1,𝔽8)⋊D7
commutator, hypocenterC72C3×C4×C5×C11
FrattiniC2C5×C11×ΓL(1,𝔽8)⋊D7
Fitting, socleC2×C5×C72×C11C2×C3
prime radicalC2×C5×C7×C11×ΓL(1,𝔽8)C2
cosocleC2×C72C2×C3×C5×C11
LCSG ▷ C72
UCSC1 ◁ C2×C5×C11

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

normal subgroups only

This group has 120 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

2420 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 184);