#72221305

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×M₁₆ ⋊ C₂²
Sylow	C₅² ⋊ C₂×#413

Subgroups

	subgroup	G/H
center	C₄	#11717039
commutator	C₂²×C₅²	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₅²
Fitting	C₂×C₅²×M₁₆	C₂²
socle	C₂×C₅²	C₂²×C₂²⋊C₄
prime radical	C₂×#11717471	C₂
cosocle	C₂×C₄×C₅²	C₂⁴
hypercenter	C₂×M₁₆	D₅²
hypocenter	C₅²	C₂×#413
LCS	G ▷ C₂²×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

1810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,5^2|a^2=f,b^2=ef,f^2=g,b^a=be,c^a=cg,e^a=eg,e^b=eg,f^b=fg,h^a=h^4,h^d=h^4,i^a=h^3i,i^d=i^4

Conjugacy classes & character table

212 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223486, #72223500, #72223542, …
maximal normal subgroup of	#72223486, #72223500, #72223542, …

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