Finite Group Databasebeta

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ΓL(1,𝔽8)×Dic7×C11⋊C5 #44568454

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

Aliases

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

Splittings

derived, HallC72×C11 ⋊ C3×C4×C5
HallC7×Dic7×C11⋊C5 ⋊ C3
HallC7×ΓL(1,𝔽8)×C11⋊C5 ⋊ C4
HallC11×ΓL(1,𝔽8)×Dic7 ⋊ C5
HallC72×C11⋊C5 ⋊ C3×C4
HallC7×C11×Dic7 ⋊ C3×C5
HallC7×C11×ΓL(1,𝔽8) ⋊ C4×C5
HallC7×Dic7 ⋊ C3×C11⋊C5
HallC7×ΓL(1,𝔽8) ⋊ C4×C11⋊C5
SylowC72 ⋊ C3×C4×C11⋊C5
SylowC11 ⋊ C5×ΓL(1,𝔽8)×Dic7

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D7×ΓL(1,𝔽8)×C11⋊C5
commutator, hypocenterC72×C11C3×C4×C5
Fitting, socle, cosocleC2×C72×C11C2×C3×C5
prime radicalC2×C7×ΓL(1,𝔽8)×C11⋊C5C2
LCSG ▷ C72×C11
UCSC1 ◁ C2

104 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 45 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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 156);