#72220969

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄×C₅

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#492 ⋊ C₂
Sylow	C₅² ⋊ #3151

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₄×C₅⋊D₅
commutator, cosocle	C₂×C₄×C₅²	C₂⁴
Frattini	C₂×C₄	C₂³×C₅⋊D₅
Fitting	C₅²×#492	C₂
socle	C₂²×C₅²	C₂²×D₄
hypercenter	#492	C₅⋊D₅
hypocenter	C₅²	#3151
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

2624 conjugacy classes of subgroups; only the 343 normal subgroups are shown.

normal subgroups only

This group has 2624 conjugacy classes of subgroups — too many to draw. Only its 343 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=f,c^2=fg,d^2=e,f^2=g,b^a=bf,c^a=cf,c^b=cg,d^a=de,f^a=fg,h^c=h^4,i^c=i^4

Conjugacy classes & character table

296 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅⋊D₅×#3151, #72223228, #72223239, …
maximal normal subgroup of	C₅⋊D₅×#3151, #72223228, #72223239, …
maximal quotient of	#72223228, #72223246

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