#72221768

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₄×C₅² ⋊ C₄²
Sylow	C₅² ⋊ C₄×#99

Subgroups

	subgroup	G/H
center	C₂	#13107468
commutator	C₂²×C₅²	C₂×C₄²
Frattini	C₂²	C₂×AGL(1,𝔽₅)²
Fitting	C₂×C₄×C₅²	C₄²
socle	C₂×C₅²	C₄×C₂²⋊C₄
prime radical	C₂×#87877	C₂²
cosocle	C₂²×D₅²	C₂³
hypercenter	C₂×C₄	AGL(1,𝔽₅)²
hypocenter	C₅²	C₄×#99
LCS	G ▷ C₂²×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄

878 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 106 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|a^2=d,b^2=g,c^2=e,b^a=bf,d^b=dg,f^a=fg,h^a=h^2,h^c=h^3,h^d=h^4,h^e=h^4,i^a=h^4i,i^c=i^3,i^d=h^2i,i^e=i^4

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72221770
maximal normal subgroup of	#72221770

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