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#20266402⋊C5 #44568603

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

Aliases

unambiguous semidirect product#20266402⋊C5

Splittings

Fitting, socle, derived, HallC3×C72×C11 ⋊ C22×C5
HallC3×C72×C11⋊C5 ⋊ C22
Hall#20266402 ⋊ C5
HallC72×C11⋊C5 ⋊ C2×D3
HallC72×C11 ⋊ C2×C5×D3
HallC3×C11⋊C5 ⋊ D72
HallC3×C72 ⋊ C2×AGL(1,𝔽11)
HallC11⋊C5 ⋊ D21⋊D7
SylowC72 ⋊ C2×C3⋊AGL(1,𝔽11)
HallC3×C11 ⋊ C5×D72
SylowC11 ⋊ C5×D21⋊D7
SylowC3 ⋊ (C7⋊D7)⋊AGL(1,𝔽11)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C72×C11C22×C5
LCSG ▷ C3×C72×C11
UCSC1

208 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 28
transitive degreeμt(G) = 462

A minimal faithful permutation representation — 2 generators on 28 points:

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

Polycyclic presentation:

2^2,5,3,7^2,11|d^a=d^2,e^a=e^6,g^a=g^10,e^b=e^6,f^b=f^6,g^c=g^4

Conjugacy classes & character table

333 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 305);