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ΓL(1,𝔽8)⋊((C11⋊C5)⋊AGL(1,𝔽5)) #37528410

order23100
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊((C11⋊C5)⋊AGL(1,𝔽5))
unambiguous semidirect product(C11⋊C5)⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))
unambiguous semidirect product(C11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5)))⋊C5
unambiguous semidirect product(C7⋊((C11⋊C5)⋊AGL(1,𝔽5)))⋊C3

Splittings

Fitting, socle, derivedC5×C7×C11 ⋊ C3×C4×C5
HallC7⋊((C11⋊C5)⋊AGL(1,𝔽5)) ⋊ C3
HallC5×ΓL(1,𝔽8)×C11⋊C5 ⋊ C4
HallC5×C7×C11⋊C5 ⋊ C3×C4
HallC5×C11⋊C5 ⋊ ΓL(1,𝔽8)⋊C4
HallC11×ΓL(1,𝔽8) ⋊ C5×AGL(1,𝔽5)
HallC7×C11 ⋊ C3×C5×AGL(1,𝔽5)
HallΓL(1,𝔽8) ⋊ (C11⋊C5)⋊AGL(1,𝔽5)
SylowC11 ⋊ C5×ΓL(1,𝔽8)⋊AGL(1,𝔽5)
SylowC7 ⋊ C3×(C11⋊C5)⋊AGL(1,𝔽5)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC5×C7×C11C3×C4×C5
prime radicalD5×ΓL(1,𝔽8)×C11⋊C5C2
cosocleC7×D5×C11C2×C3×C5
LCSG ▷ C5×C7×C11
UCSC1

104 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 31 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) = 385

Polycyclic presentation:

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

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 221);