#72221025

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₄²×C₅² ⋊ C₂×C₄
Sylow	C₅² ⋊ #1860

Subgroups

	subgroup	G/H
center	C₂	#13104048
commutator	C₂×C₄×C₅²	C₂²×C₄
Frattini	C₂×C₄	C₂×D₅⋊AGL(1,𝔽₅)
Fitting	C₄²×C₅²	C₂×C₄
socle	C₂×C₅²	#385
prime radical	#11717471	C₂²
cosocle	C₂×C₄×C₅⋊D₅	C₂³
hypercenter	C₄²	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#1860
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄²

726 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 87 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:

2^7,5^2|b^2=d,c^2=eg,f^2=g,b^a=bf,c^a=cf,c^b=ce,e^a=eg,f^a=fg,f^b=fg,h^a=h^4,h^b=h^3,h^d=h^4,i^a=h^3i,i^b=i^3,i^d=i^4

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223093, #72223130, #72223184, …
maximal normal subgroup of	#72223093, #72223130, #72223184, …

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