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C3⋊(C7⋊((C11⋊C5)⋊AGL(1,𝔽5))) #37528446

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

Aliases

unambiguous semidirect productC3⋊(C7⋊((C11⋊C5)⋊AGL(1,𝔽5)))
unambiguous semidirect productC7⋊(C3⋊((C11⋊C5)⋊AGL(1,𝔽5)))
unambiguous semidirect product(C11⋊C5)⋊(C3⋊(C7⋊AGL(1,𝔽5)))
unambiguous semidirect product(C3⋊(C7⋊(C11⋊AGL(1,𝔽5))))⋊C5

Splittings

Fitting, socle, derivedC3×C5×C7×C11 ⋊ C4×C5
HallC3×C5×C7×C11⋊C5 ⋊ C4
HallC5×C7×C11⋊C5 ⋊ Dic3
HallC3×C5×C11⋊C5 ⋊ Dic7
HallC5×C11⋊C5 ⋊ Dic21
HallC3×C7×C11 ⋊ C5×AGL(1,𝔽5)
HallC7×C11 ⋊ C5×ΓL(1,𝔽16)
HallC3×C11 ⋊ C5×C7⋊AGL(1,𝔽5)
HallC3×C7 ⋊ (C11⋊C5)⋊AGL(1,𝔽5)
SylowC11 ⋊ C5×C3⋊(C7⋊AGL(1,𝔽5))
SylowC7 ⋊ C3⋊((C11⋊C5)⋊AGL(1,𝔽5))
SylowC3 ⋊ C7⋊((C11⋊C5)⋊AGL(1,𝔽5))

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C7×C11C4×C5
prime radicalC3×C7×D5×C11⋊C5C2
cosocleC3×C7×D5×C11C2×C5
LCSG ▷ C3×C5×C7×C11
UCSC1

104 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 1155

Polycyclic presentation:

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

Conjugacy classes & character table

309 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 257);