#72253069

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³×C₅

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #1580

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄×C₅⋊D₅
commutator	C₄×C₅²	C₂²×C₈
Frattini	C₄²	C₂²×C₅⋊D₅
Fitting	C₄×C₅²×D₈	C₂
socle	C₂²×C₅²	C₄×D₄
prime radical	C₂×C₅²×D₈	C₄
cosocle	C₄²×C₅²	C₂³
hypercenter	C₄×D₈	C₅⋊D₅
hypocenter	C₅²	#1580
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

768 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 309 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) = 1600

Polycyclic presentation:

2^7,5^2|a^2=d,b^2=df,c^2=e,d^2=f,e^2=g,c^a=ce,c^b=cg,e^a=eg,h^a=h^4,h^b=h^4,i^a=i^4,i^b=i^4

Conjugacy classes & character table

380 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅⋊D₅×#1580, #72253103, #72253108
maximal normal subgroup of	C₅⋊D₅×#1580, #72253103, #72253108
maximal quotient of	#72253108

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