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(C5⋊AGL(1,𝔽7))⋊AGL(1,𝔽11) #37528417

order23100
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product(C5⋊AGL(1,𝔽7))⋊AGL(1,𝔽11)
unambiguous semidirect product(C5⋊AGL(1,𝔽11))⋊AGL(1,𝔽7)
unambiguous semidirect product(ΓL(1,𝔽8)⋊AGL(1,𝔽11))⋊D5
unambiguous semidirect product(D55⋊AGL(1,𝔽7))⋊C5
unambiguous semidirect product(D35⋊AGL(1,𝔽11))⋊C3

Splittings

Fitting, socle, derivedC5×C7×C11 ⋊ C22×C3×C5
HallD35⋊AGL(1,𝔽11) ⋊ C3
HallC5×ΓL(1,𝔽8)×C11⋊C5 ⋊ C22
HallC5×C7×C11⋊C5 ⋊ C22×C3
HallC5×C11⋊C5 ⋊ C2×AGL(1,𝔽7)
HallC11×ΓL(1,𝔽8) ⋊ C2×C5×D5
HallC7×C11 ⋊ C2×C3×C5×D5
HallΓL(1,𝔽8) ⋊ D5×AGL(1,𝔽11)
SylowC11 ⋊ C5×D5×AGL(1,𝔽7)
SylowC7 ⋊ C3×D5×AGL(1,𝔽11)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C7×C11C22×C3×C5
LCSG ▷ C5×C7×C11
UCSC1

192 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 42 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:

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

Conjugacy classes & character table

157 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 228);