Finite Group Databasebeta

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C5×#64424⋊D11 #44568637

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

Aliases

direct productC5×#64424⋊D11
unambiguous semidirect product#64424⋊(C5×D11)

Splittings

Fitting, socle, HallC5×C72×C11 ⋊ C22×C3
derived, HallC72×C11 ⋊ C22×C3×C5
HallC5×(C7⋊D7)⋊D11 ⋊ C3
HallC5×C11×#3861 ⋊ C22
Hall(C7⋊D7)⋊D11 ⋊ C3×C5
HallC11×#3861 ⋊ C22×C5
HallC5×#3861 ⋊ C2×D11
HallC5×C72 ⋊ C2×C3×D11
Hall#3861 ⋊ C2×C5×D11
HallC5×C11 ⋊ #64423⋊C2
SylowC72 ⋊ C2×C3×C5×D11
SylowC11 ⋊ C5×#64423⋊C2

Subgroups

subgroupG/H
center, hypercenterC5#64424⋊D11
commutator, cosocle, hypocenterC72×C11C22×C3×C5
FrattiniC1G
Fitting, socleC5×C72×C11C22×C3
LCSG ▷ C72×C11
UCSC1 ◁ C5

208 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 44 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)(6,7,9,11,13,15,16,14,12,10,8)(17,18,19,23,20,22)(25,28,29,26,27,30);(7,8)(9,10)(11,12)(13,14)(15,16)(18,19)(20,21)(22,23)(24,25,27,30,29,28,26)

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, 339);