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D55⋊(C5⋊AGL(1,𝔽7)) #37528533

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

Aliases

unambiguous semidirect productD55⋊(C5⋊AGL(1,𝔽7))
unambiguous semidirect product(D55⋊D35)⋊C3

Splittings

Fitting, socle, derived, HallC52×C7×C11 ⋊ C22×C3
HallD55⋊D35 ⋊ C3
HallC52×C11×ΓL(1,𝔽8) ⋊ C22
HallC52×ΓL(1,𝔽8) ⋊ C2×D11
HallC52×C11 ⋊ C2×AGL(1,𝔽7)
HallC11×ΓL(1,𝔽8) ⋊ D52
HallC52×C7 ⋊ C2×C3×D11
HallC7×C11 ⋊ C3×D52
SylowC52 ⋊ D11×AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ (C5⋊D5)⋊D11
SylowC11 ⋊ D5×C5⋊AGL(1,𝔽7)
SylowC7 ⋊ C3×(C5⋊D5)⋊D11

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC52×C7×C11C22×C3
LCSG ▷ C52×C7×C11
UCSC1

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

normal subgroups only

This group has 192 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) = 770

Polycyclic presentation:

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

Conjugacy classes & character table

403 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 344);