#72223175

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #2165

Subgroups

	subgroup	G/H
center	C₄	#11717075
commutator	C₂×C₄×C₅²	C₂²×C₄
Frattini	C₂×C₈	C₂×D₅²
Fitting	C₂×C₅²×M₁₆	C₂²
socle	C₂×C₅²	#432
prime radical	C₂³×C₅²	M₁₆
cosocle	C₂×C₅²×C₈	C₂³
hypercenter	C₂×M₁₆	D₅²
hypocenter	C₅²	#2165
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

398 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 86 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,c^2=efg,d^2=f,f^2=g,b^a=be,c^a=cef,c^b=cg,d^b=dg,e^a=eg,h^a=h^4,h^c=h^4,i^a=h^3i,i^c=i^4

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223180
maximal normal subgroup of	#72223180

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