Finite Group Databasebeta

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#72217241

order5280
automorphism?
homologyAbelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

FittingC2×C3×C4×C11 ⋊ C22×C5
Hall#11760923 ⋊ C5
HallC3×C11⋊C5 ⋊ C2×C22⋊C4
HallC11⋊C5 ⋊ D3×C22⋊C4
HallC3×C11 ⋊ C2×C5×C22⋊C4
SylowC11 ⋊ C5×D3×C22⋊C4
SylowC3 ⋊ C22⋊C4×AGL(1,𝔽11)

Subgroups

subgroupG/H
center, FrattiniC22C2×D3×AGL(1,𝔽11)
commutatorC2×C3×C11C22×C4×C5
FittingC2×C3×C4×C11C22×C5
socle, cosocleC22×C3×C11C23×C5
prime radicalC22×D3×AGL(1,𝔽11)C2
hypercenterC2×C4D3×AGL(1,𝔽11)
hypocenterC3×C11C2×C5×C22⋊C4
LCSG ▷ C2×C3×C11 ▷ C3×C11
UCSC1 ◁ C22 ◁ C2×C4

528 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,5,2^2,3,11|b^2=ef,b^a=be,g^c=g^2,h^a=h^10,h^c=h^10,h^d=h^9

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×C22⋊C4×AGL(1,𝔽11), #72217599, #72217616
maximal normal subgroup ofD3×C22⋊C4×AGL(1,𝔽11), #72217599, #72217616

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 2, 2, 3, 11]);;
gap> SetPower(coll, 2, F.5*F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^10);;
gap> SetConjugate(coll, 8, 3, F.8^10);;
gap> SetConjugate(coll, 8, 4, F.8^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);