#72223211

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₅

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #1357

Subgroups

	subgroup	G/H
center	C₂	#13100395
commutator	C₄²×C₅²	C₂×C₄
Frattini	C₄²	C₂×#1044
Fitting	C₅²×#127	C₄
socle	C₂×C₅²	#331
prime radical	#13102935	C₂
cosocle	C₄×#6029	C₂²
hypercenter	#127	#1044
hypocenter	C₅²	#1357
LCS	G ▷ C₄²×C₅² ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #127

386 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 44 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=c,c^2=eg,d^2=ef,e^2=g,f^2=g,b^a=bd,c^b=ce,d^a=df,d^b=def,e^b=eg,f^a=fg,f^b=fg,h^a=h^2,h^c=h^4,i^a=h^3i^3,i^c=i^4

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223213
maximal normal subgroup of	#72223213

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