#72223106

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #1335

Subgroups

	subgroup	G/H
center	C₈	#87770
commutator	C₅²×C₈	C₂×C₈
Frattini, hypercenter	C₄×C₈	D₅²
Fitting, cosocle	C₄×C₅²×C₈	C₂²
socle	C₂×C₅²	D₄⋊C₈
prime radical	C₅×#4050	C₄
hypocenter	C₅²	#1335
LCS	G ▷ C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

274 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 70 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=c,c^2=e,d^2=f,f^2=g,b^a=bd,c^a=cf,d^a=dfg,e^a=eg,f^a=fg,h^a=h^4,h^b=h^4,i^a=h^3i,i^b=i^4

Conjugacy classes & character table

284 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223108
maximal normal subgroup of	#72223108

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