Finite Group Databasebeta

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C2×C5×C11⋊#64425 #44568778

order32340
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2×C7

Aliases

direct productC2×C5×C11⋊#64425

Splittings

Fitting, socleC2×C5×C72×C11 ⋊ C2×C3
derived, HallC72×C11 ⋊ C22×C3×C5
HallC2×C5×C7⋊D77 ⋊ C3
HallC5×C11×#3862 ⋊ C22
HallC5×C72×C11 ⋊ C22×C3
HallC2×C7⋊D77 ⋊ C3×C5
HallC11×#3862 ⋊ C22×C5
HallC5×#3862 ⋊ C2×D11
HallC5×C72 ⋊ C2×C3×D11
Hall#3862 ⋊ C2×C5×D11
HallC5×C11 ⋊ C2×#64425
SylowC72 ⋊ C2×C3×C5×D11
SylowC11 ⋊ C2×C5×#64425

Subgroups

subgroupG/H
center, hypercenterC2×C5C11⋊#64425
commutator, cosocle, hypocenterC72×C11C22×C3×C5
FrattiniC1G
Fitting, socleC2×C5×C72×C11C2×C3
LCSG ▷ C72×C11
UCSC1 ◁ C2×C5

200 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 32
transitive degreeμt(G) = 2310

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

1090 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 480);