#72220903

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₄×C₅²×C₈ ⋊ C₂²
Sylow	C₅² ⋊ C₂×#443

Subgroups

	subgroup	G/H
center	C₈	#87877
commutator	C₄×C₅²	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₅²
Fitting	C₄×C₅²×C₈	C₂²
socle	C₂×C₅²	C₂×C₄×D₄
prime radical	#13105775	C₂
cosocle	C₂×C₄×C₅²	C₂⁴
hypercenter	C₄×C₈	D₅²
hypocenter	C₅²	C₂×#443
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

1106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,5^2|c^2=e,d^2=f,f^2=g,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

296 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223022, #72223041, #72223043, …
maximal normal subgroup of	#72223022, #72223041, #72223043, …

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, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.7);;
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);