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ΓL(1,𝔽8)⋊(C7⋊(Dic11⋊C5)) #44568449

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

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊(C7⋊(Dic11⋊C5))
unambiguous semidirect product(C11⋊C5)⋊(ΓL(1,𝔽8)⋊Dic7)
unambiguous semidirect product(C11⋊(ΓL(1,𝔽8)⋊Dic7))⋊C5

Splittings

derived, HallC72×C11 ⋊ C3×C4×C5
HallC7⋊(C7⋊(Dic11⋊C5)) ⋊ C3
HallC7×ΓL(1,𝔽8)×C11⋊C5 ⋊ C4
HallC11⋊(ΓL(1,𝔽8)⋊Dic7) ⋊ C5
HallC72×C11⋊C5 ⋊ C3×C4
HallC7⋊Dic77 ⋊ C3×C5
HallC7×C11×ΓL(1,𝔽8) ⋊ C4×C5
HallC7×ΓL(1,𝔽8) ⋊ Dic11⋊C5
HallC11⋊C5 ⋊ ΓL(1,𝔽8)⋊Dic7
SylowC72 ⋊ C3×Dic11⋊C5
SylowC11 ⋊ C5×ΓL(1,𝔽8)⋊Dic7

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)⋊(C7⋊AGL(1,𝔽11))
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

120 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 40 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

Polycyclic presentation:

2,3,5,7^2,11,2|a^2=g,d^a=d^6,e^a=e^6,f^a=f^10,d^b=d^2,f^c=f^4

Conjugacy classes & character table

290 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 151);