Finite Group Databasebeta

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C5×D21×D77 #44568847

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

Aliases

direct productC5×D21×D77

Splittings

Fitting, socle, HallC3×C5×C72×C11 ⋊ C22
derived, HallC3×C72×C11 ⋊ C22×C5
HallC5×C72×C11 ⋊ C2×D3
HallC3×C5×C72 ⋊ C2×D11
HallC72×C11 ⋊ C2×C5×D3
HallC5×C72 ⋊ D3×D11
HallC3×C5×C11 ⋊ D72
HallC3×C72 ⋊ C2×C5×D11
HallC5×C11 ⋊ D7×D21
SylowC72 ⋊ C5×D3×D11
HallC3×C11 ⋊ C5×D72
HallC3×C5 ⋊ D7×D77
SylowC11 ⋊ C5×D7×D21
SylowC3 ⋊ C5×D7×D77

Subgroups

subgroupG/H
center, hypercenterC5D21×D77
commutator, cosocle, hypocenterC3×C72×C11C22×C5
FrattiniC1G
Fitting, socleC3×C5×C72×C11C22
LCSG ▷ C3×C72×C11
UCSC1 ◁ C5

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

normal subgroups only

This group has 208 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) = 33
transitive degreeμt(G) = 2310

A minimal faithful permutation representation — 2 generators on 33 points:

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

Conjugacy classes & character table

2400 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 549);