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D5×C11×C3⋊(C7⋊D7) #44568913

order32340
automorphism?
homologyAbelianization (H1) = C22×C11 · Schur multiplier (H2) = C2×C7

Aliases

direct productD5×C11×C3⋊(C7⋊D7)
unambiguous semidirect product(C3×D5)⋊(C11×C7⋊D7)
unambiguous semidirect product(C7×D5)⋊(C11×D21)

Splittings

Fitting, socle, HallC3×C5×C72×C11 ⋊ C22
derived, HallC3×C5×C72 ⋊ C22×C11
HallC5×C72×C11 ⋊ C2×D3
HallC3×C72×C11 ⋊ C2×D5
HallC72×C11 ⋊ D3×D5
HallC5×C72 ⋊ C2×D3×C11
HallC3×C5×C11 ⋊ C2×C7⋊D7
HallC3×C72 ⋊ C2×D5×C11
HallC5×C11 ⋊ C2×C3⋊(C7⋊D7)
SylowC72 ⋊ D3×D5×C11
HallC3×C11 ⋊ D5×C7⋊D7
HallC3×C5 ⋊ C2×C11×C7⋊D7
SylowC5 ⋊ C2×C11×C3⋊(C7⋊D7)
SylowC3 ⋊ D5×C11×C7⋊D7

Subgroups

subgroupG/H
center, hypercenterC11D5×C3⋊(C7⋊D7)
commutator, cosocle, hypocenterC3×C5×C72C22×C11
FrattiniC1G
Fitting, socleC3×C5×C72×C11C22
LCSG ▷ C3×C5×C72
UCSC1 ◁ C11

400 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 33
transitive degreeμt(G) = 8085

A minimal faithful permutation representation — 3 generators on 33 points:

(2,3)(4,5,7,8,6)(10,11)(12,13)(14,15)(17,18)(19,20)(21,22)(23,24,25,26,27,28,29,30,31,32,33);(1,2,3)(5,6)(7,8)(16,17,19,21,22,20,18);(9,10,12,14,15,13,11)

Conjugacy classes & character table

3300 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 615);