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C3×C7⋊((C11⋊C5)⋊AGL(1,𝔽5)) #37528430

order23100
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC3×C7⋊((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, socleC3×C5×C7×C11 ⋊ C4×C5
derivedC5×C7×C11 ⋊ C3×C4×C5
HallC3×C5×C7×C11⋊C5 ⋊ C4
HallC5×C7×C11⋊C5 ⋊ C3×C4
HallC3×C5×C11⋊C5 ⋊ Dic7
HallC5×C11⋊C5 ⋊ C3×Dic7
HallC3×C7×C11 ⋊ C5×AGL(1,𝔽5)
HallC7×C11 ⋊ C3×C5×AGL(1,𝔽5)
HallC3×C11 ⋊ C5×C7⋊AGL(1,𝔽5)
HallC3×C7 ⋊ (C11⋊C5)⋊AGL(1,𝔽5)
SylowC11 ⋊ C3×C5×C7⋊AGL(1,𝔽5)
SylowC7 ⋊ C3×(C11⋊C5)⋊AGL(1,𝔽5)

Subgroups

subgroupG/H
center, hypercenterC3C7⋊((C11⋊C5)⋊AGL(1,𝔽5))
commutator, hypocenterC5×C7×C11C3×C4×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C11C4×C5
prime radicalC3×C7×D5×C11⋊C5C2
cosocleC7×D5×C11C2×C3×C5
LCSG ▷ C5×C7×C11
UCSC1 ◁ C3

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

normal subgroups only

This group has 104 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) = 1155

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 241);