#72223190

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #1296

Subgroups

	subgroup	G/H
center	C₂²	#11716776
commutator	C₂×C₄×C₅²	C₄²
Frattini, hypercenter	C₄²	D₅⋊AGL(1,𝔽₅)
Fitting	C₄²×C₅²	C₂×C₄
socle	C₂²×C₅²	#95
prime radical	C₂²×C₅⋊D₅	C₂²⋊C₄
cosocle	C₄²×C₅⋊D₅	C₂²
hypocenter	C₅²	#1296
LCS	G ▷ C₂×C₄×C₅² ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

562 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 97 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=c,d^2=f,e^2=g,b^a=bdeg,d^b=df,e^a=eg,e^b=eg,h^a=h^4,h^b=h^3,h^c=h^4,i^a=h^2i,i^b=i^3,i^c=i^4

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72223191, #72223197

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