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C5×#20266354 #44568625

order32340
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×#20266354

Splittings

Fitting, socle, HallC5×C72×C11 ⋊ C2×D3
derived, HallC11×#3862 ⋊ C22×C5
HallC5×C11×#3862 ⋊ C22
HallC5×#3862 ⋊ C2×D11
HallC72×C11 ⋊ C2×C5×D3
HallC5×C72 ⋊ C2×D33
Hall#3862 ⋊ C2×C5×D11
HallC5×C11 ⋊ #64418⋊C2
SylowC72 ⋊ C2×C5×D33
SylowC11 ⋊ C5×#64418⋊C2

Subgroups

subgroupG/H
center, hypercenterC5#20266354
commutator, cosocle, hypocenterC11×#3862C22×C5
FrattiniC1G
Fitting, socleC5×C72×C11C2×D3
LCSG ▷ C11×#3862
UCSC1 ◁ C5

136 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 30
transitive degreeμt(G) = 770

A minimal faithful permutation representation — 2 generators on 30 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 327);