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C5×D21⋊D77 #44568851

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

Aliases

direct productC5×D21⋊D77
unambiguous semidirect productD21⋊(C5×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 ⋊ D21⋊D7
SylowC72 ⋊ C5×D3×D11
HallC3×C11 ⋊ C5×D72
HallC3×C5 ⋊ D7×D77
SylowC11 ⋊ C5×D21⋊D7
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 50 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Conjugacy classes & character table

2355 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 553);