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C7×D11×C3⋊(C5⋊D5) #37528700

order23100
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2×C5

Aliases

direct productC7×D11×C3⋊(C5⋊D5)
unambiguous semidirect product(C3×D11)⋊(C7×C5⋊D5)
unambiguous semidirect product(C5×D11)⋊(C7×D15)

Splittings

Fitting, socle, HallC3×C52×C7×C11 ⋊ C22
derived, HallC3×C52×C11 ⋊ C22×C7
HallC52×C7×C11 ⋊ C2×D3
HallC3×C52×C7 ⋊ C2×D11
HallC52×C11 ⋊ C2×D3×C7
HallC3×C7×C11 ⋊ C2×C5⋊D5
HallC52×C7 ⋊ D3×D11
HallC7×C11 ⋊ C2×C3⋊(C5⋊D5)
HallC3×C52 ⋊ C2×C7×D11
HallC3×C11 ⋊ C2×C7×C5⋊D5
SylowC52 ⋊ D3×C7×D11
HallC3×C7 ⋊ D11×C5⋊D5
SylowC11 ⋊ C2×C7×C3⋊(C5⋊D5)
SylowC3 ⋊ C7×D11×C5⋊D5

Subgroups

subgroupG/H
center, hypercenterC7D11×C3⋊(C5⋊D5)
commutator, cosocle, hypocenterC3×C52×C11C22×C7
FrattiniC1G
Fitting, socleC3×C52×C7×C11C22
LCSG ▷ C3×C52×C11
UCSC1 ◁ C7

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

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 5775

A minimal faithful permutation representation — 3 generators on 31 points:

(2,3)(4,5,7,9,11,13,14,12,10,8,6)(16,17)(18,19)(21,22)(23,24)(25,26,27,28,29,30,31);(1,2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(20,21,23,24,22);(15,16,18,19,17)

Conjugacy classes & character table

1911 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 511);