#72220919

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₅

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×D₈ ⋊ C₄
Sylow	C₅² ⋊ #2948

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄×#1044
commutator	C₄×C₅²	C₂³×C₄
Frattini	C₄	C₂³×#1044
Fitting	C₂×C₅²×D₈	C₄
socle	C₂×C₅²	C₂×C₄×D₄
prime radical	C₂×D₈×C₅⋊D₅	C₂
cosocle	C₄×C₅⋊D₅	C₂⁴
hypercenter	C₂×D₈	#1044
hypocenter	C₅²	#2948
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₈

1394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223002, #72223004, #72223011, …
maximal normal subgroup of	#72223002, #72223004, #72223011, …

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