Finite Group Databasebeta

← Prev (#44568639) Next (#44568641) →

C5×#64422⋊D11 #44568640

order32340
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×#64422⋊D11
unambiguous semidirect product#64422⋊(C5×D11)

Splittings

Fitting, socle, HallC5×C72×C11 ⋊ C22×C3
derived, HallC72×C11 ⋊ C22×C3×C5
HallC5×D7×D77 ⋊ C3
HallC5×C11×#3862 ⋊ C22
HallD7×D77 ⋊ C3×C5
HallC11×#3862 ⋊ C22×C5
HallC5×#3862 ⋊ C2×D11
HallC5×C72 ⋊ C2×C3×D11
Hall#3862 ⋊ C2×C5×D11
HallC5×C11 ⋊ #64422⋊C2
SylowC72 ⋊ C2×C3×C5×D11
SylowC11 ⋊ C5×#64422⋊C2

Subgroups

subgroupG/H
center, hypercenterC5#64422⋊D11
commutator, cosocle, hypocenterC72×C11C22×C3×C5
FrattiniC1G
Fitting, socleC5×C72×C11C22×C3
LCSG ▷ C72×C11
UCSC1 ◁ C5

168 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 2310

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 342);