#72222614

order	19360
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×D₄×C₁₁² ⋊ C₂×C₅
Hall	C₁₁×#67037 ⋊ C₅
Hall	#10597243 ⋊ H(5,𝔽₂)
Sylow	C₁₁² ⋊ C₅×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂³×#11925895
commutator, socle, cosocle	C₂×C₁₁²	C₂⁴×C₅
Fitting	C₂×D₄×C₁₁²	C₂×C₅
hypercenter	C₂×D₄	#11925895
hypocenter	C₁₁²	C₅×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₁² ▷ C₁₁²
UCS	C₁ ◁ C₂ ◁ C₂×D₄

934 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 238 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:

2^4,5,2,11^2|b^a=bf,c^a=cf,d^b=df,g^a=g^10,g^e=g^9,h^a=g^10h,h^e=h^9

Conjugacy classes & character table

229 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222682
maximal normal subgroup of	#72222682

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