#72222682

order	38720
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₅ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂×D₄×C₁₁² ⋊ C₂²×C₅
Hall	#72222664 ⋊ C₅
Hall	#10597243 ⋊ C₂×H(5,𝔽₂)
Sylow	C₁₁² ⋊ C₂×C₅×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂³×#11925892⋊C₂
commutator, socle, cosocle	C₂×C₁₁²	C₂⁵×C₅
Fitting	C₂×D₄×C₁₁²	C₂²×C₅
hypercenter	C₂×D₄	#11925892⋊C₂
hypocenter	C₁₁²	C₂×C₅×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₁² ▷ C₁₁²
UCS	C₁ ◁ C₂ ◁ C₂×D₄

4772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^5,5,2,11^2|b^a=bg,c^a=cg,d^b=dg,h^a=h^10,h^e=h^10,h^f=h^9,i^a=h^2i,i^e=i^10,i^f=i^9

Conjugacy classes & character table

254 conjugacy classes — view the full character table.

Relationships

holomorph of

C₄×D₁₁, C₂²⋊D₁₁

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 5, 2, 11, 11]);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^10);;
gap> SetConjugate(coll, 8, 5, F.8^10);;
gap> SetConjugate(coll, 8, 6, F.8^9);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 5, F.9^10);;
gap> SetConjugate(coll, 9, 6, F.9^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);