#72221599

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#127 ⋊ C₄
Sylow	C₅² ⋊ #1910

Subgroups

	subgroup	G/H
center	C₂	#13104665
commutator	C₂×C₄×C₅²	C₂²×C₄
Frattini	C₂×C₄	C₂²×#1042
Fitting	C₅²×#127	C₄
socle	C₂×C₅²	#386
prime radical	C₅×#66049	C₂
cosocle	C₂×C₄×C₅×D₅	C₂³
hypercenter	#127	#1042
hypocenter	C₅²	#1910
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #127

742 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 117 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=e,d^2=g,f^2=g,b^a=bf,c^b=ce,d^b=dg,e^a=eg,f^c=fg,h^a=h^3,h^d=h^4,i^a=h^2i^4,i^d=h^4i

Conjugacy classes & character table

143 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72221603
maximal normal subgroup of	#72221603

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