#72221708

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #1914

Subgroups

	subgroup	G/H
center	C₂	#13103802
commutator	C₂³×C₅²	C₂²×C₄
Frattini	C₂³	C₂×D₅×AGL(1,𝔽₅)
Fitting	C₂×C₅²×D₄	C₂×C₄
socle	C₂×C₅²	#386
prime radical	#13106295	C₂
cosocle	C₂²×C₅×Dic₅	C₂³
hypercenter	C₂×D₄	D₅×AGL(1,𝔽₅)
hypocenter	C₅²	#1914
LCS	G ▷ C₂³×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×D₄

745 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 745 conjugacy classes of subgroups — too many to draw. Only its 106 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,d^2=g,b^a=bf,c^b=ce,d^b=dg,e^a=eg,f^a=fg,f^c=fg,h^a=h^3,h^b=h^4,h^d=h^4,i^a=h^2i^4,i^b=i^4,i^d=h^4i

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72221709
maximal normal subgroup of	#72221709

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