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C2×C5⋊(C5⋊(C11⋊AGL(1,𝔽7))) #37528595

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

Aliases

direct productC2×C5⋊(C5⋊(C11⋊AGL(1,𝔽7)))
unambiguous semidirect product(C2×C5)⋊(C5⋊(C11⋊AGL(1,𝔽7)))
unambiguous semidirect product(C2×C11)⋊(C5⋊(C5⋊AGL(1,𝔽7)))
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C5⋊D55)

Splittings

Fitting, socleC2×C52×C7×C11 ⋊ C2×C3
derived, HallC52×C7×C11 ⋊ C22×C3
HallC2×C5⋊D385 ⋊ C3
HallC52×C11×ΓL(1,𝔽8) ⋊ C22
HallC52×ΓL(1,𝔽8) ⋊ C2×D11
HallC52×C11 ⋊ C2×AGL(1,𝔽7)
HallC11×ΓL(1,𝔽8) ⋊ C2×C5⋊D5
HallC52×C7 ⋊ C2×C3×D11
HallC7×C11 ⋊ C2×C3×C5⋊D5
SylowC52 ⋊ C2×C11⋊AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ C2×C5⋊D55
SylowC11 ⋊ C2×C5⋊(C5⋊AGL(1,𝔽7))
SylowC7 ⋊ C2×C3×C5⋊D55

Subgroups

subgroupG/H
center, hypercenterC2C5⋊(C5⋊(C11⋊AGL(1,𝔽7)))
commutator, cosocle, hypocenterC52×C7×C11C22×C3
FrattiniC1G
Fitting, socleC2×C52×C7×C11C2×C3
LCSG ▷ C52×C7×C11
UCSC1 ◁ C2

320 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 3850

Conjugacy classes & character table

1384 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 406);