#72220975

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×#41 ⋊ C₄
Sylow	C₅² ⋊ #2297

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#1043
commutator, socle	C₂²×C₅²	C₂³×C₄
Frattini	C₂²	C₂³×#1043
Fitting	C₂×C₅²×#41	C₄
prime radical	C₂×#41×C₅⋊D₅	C₂
cosocle	C₂²×C₅⋊D₅	C₂⁴
hypercenter	C₂×#41	#1043
hypocenter	C₅²	#2297
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

3744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,5^2|a^2=g,b^2=e,d^2=g,b^a=bf,c^a=cg,d^a=dg,h^b=h^3,h^e=h^4,i^b=i^3,i^e=i^4

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72221132, #532×#1043, #525×#1043, …
maximal normal subgroup of	#72221132, #532×#1043, #525×#1043, …
maximal quotient of	#72223269

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