#72258486

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₅² ⋊ C₂×C₄
Sylow	C₅² ⋊ #1889

Subgroups

	subgroup	G/H
center	C₂	#13103835
commutator	C₂³×C₅²	C₂²×C₄
Frattini	C₂³	C₂×D₅×AGL(1,𝔽₅)
Fitting	C₂⁴×C₅²	C₂×C₄
socle	C₂×C₅²	#386
prime radical	D₅×#4099	C₂
cosocle	C₂³×C₅×D₅	C₂³
hypercenter	C₂⁴	D₅×AGL(1,𝔽₅)
hypocenter	C₅²	#1889
LCS	G ▷ C₂³×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁴

1221 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 1221 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Polycyclic presentation:

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

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258493, #72258517, D₅×#10604461
maximal normal subgroup of	D₅×#10604461

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