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C13×ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C4 #37524644

order22932
automorphism?
homologyAbelianization (H1) = C32×C4×C13 · Schur multiplier (H2) = C3

Aliases

direct productC13×ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C4
unambiguous semidirect productΓL(1,𝔽8)⋊(C4×C13×ΓL(1,𝔽8))
unambiguous semidirect product(C7×ΓL(1,𝔽8)⋊C4)⋊(C3×C13)
unambiguous semidirect product(ΓL(1,𝔽8)×Dic7)⋊(C3×C13)
unambiguous semidirect product#10596872⋊(C3×C13)
unambiguous semidirect product#10596873⋊(C3×C13)

Splittings

derived, SylowC72 ⋊ C32×C4×C13
HallC13×ΓL(1,𝔽8)2 ⋊ C4
HallC7×C13×Dic7 ⋊ C32
HallC72×C13 ⋊ C32×C4
HallΓL(1,𝔽8)2 ⋊ C4×C13
HallC7×Dic7 ⋊ C32×C13

Subgroups

subgroupG/H
center, hypercenterC2×C13ΓL(1,𝔽8)×AGL(1,𝔽7)
commutator, hypocenterC72C32×C4×C13
FrattiniC2C13×ΓL(1,𝔽8)×AGL(1,𝔽7)
Fitting, socleC2×C72×C13C2×C32
prime radicalC2×C13×ΓL(1,𝔽8)2C2
cosocleC2×C72C2×C32×C13
LCSG ▷ C72
UCSC1 ◁ C2×C13

152 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 1092

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22932, 589);