#72247204

order	5808
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃×C₁₁² ⋊ C₂²
Hall	C₃×C₁₁² ⋊ C₂×D₄
Sylow	C₁₁² ⋊ D₃×D₄
Sylow	C₃ ⋊ #14515540

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₁×D₃₃
commutator, socle, cosocle	C₂×C₃×C₁₁²	C₂³
Fitting	C₂²×C₃×C₁₁²	C₂²
hypercenter	C₂²	D₁₁×D₃₃
hypocenter	C₃×C₁₁²	C₂×D₄
LCS	G ▷ C₂×C₃×C₁₁² ▷ C₃×C₁₁²
UCS	C₁ ◁ C₂ ◁ C₂²

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

normal subgroups only

This group has 376 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

transitive degree

μ_t(G) = 132

Polycyclic presentation:

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

Conjugacy classes & character table

435 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#14515540, #72247317
maximal normal subgroup of	D₃×#14515540, #72247317

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 11, 11]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 6, 3, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 3, F.7^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);