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C2×D5×ΓL(1,𝔽8)×C11⋊C5 #37528426

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

Aliases

direct productC2×D5×ΓL(1,𝔽8)×C11⋊C5
unambiguous semidirect productC5⋊(C22×ΓL(1,𝔽8)×C11⋊C5)
unambiguous semidirect product(C2×C5)⋊(C2×ΓL(1,𝔽8)×C11⋊C5)
unambiguous semidirect product(C2×C7×D5)⋊(C3×C11⋊C5)
unambiguous semidirect product(C2×D5×C11)⋊(C5×ΓL(1,𝔽8))
unambiguous semidirect product(C2×D5×C11×ΓL(1,𝔽8))⋊C5
unambiguous semidirect product(C2×C7×D5×C11⋊C5)⋊C3

Splittings

Fitting, socleC2×C5×C7×C11 ⋊ C2×C3×C5
derivedC5×C7×C11 ⋊ C22×C3×C5
HallC2×C7×D5×C11⋊C5 ⋊ C3
HallC5×ΓL(1,𝔽8)×C11⋊C5 ⋊ C22
HallC5×C7×C11⋊C5 ⋊ C22×C3
HallC5×C11⋊C5 ⋊ C22×ΓL(1,𝔽8)
HallC11×ΓL(1,𝔽8) ⋊ C2×C5×D5
HallC7×C11 ⋊ C2×C3×C5×D5
SylowC11 ⋊ C2×C5×D5×ΓL(1,𝔽8)
SylowC7 ⋊ C2×C3×D5×C11⋊C5

Subgroups

subgroupG/H
center, hypercenterC2D5×ΓL(1,𝔽8)×C11⋊C5
commutator, cosocle, hypocenterC5×C7×C11C22×C3×C5
FrattiniC1G
Fitting, socleC2×C5×C7×C11C2×C3×C5
LCSG ▷ C5×C7×C11
UCSC1 ◁ C2

192 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 63 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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 237);