#72253137

order	3200
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×M₃₂ ⋊ C₄
Sylow	C₅² ⋊ #1387

Subgroups

	subgroup	G/H
center	C₂	#13100271
commutator	C₅²×C₈	C₄²
Frattini	C₂×C₈	C₂×#1043
Fitting	C₅²×M₃₂	C₄
socle	C₂×C₅²	#340
prime radical	C₂²×C₅⋊D₅	SD₁₆
cosocle	C₂×C₈×C₅⋊D₅	C₂²
hypercenter	M₃₂	#1043
hypocenter	C₅²	#1387
LCS	G ▷ C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

544 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 400

Polycyclic presentation:

2^7,5^2|a^2=c,b^2=cdfg,c^2=g,d^2=f,e^2=f,f^2=g,b^a=be,c^b=cg,d^a=df,d^b=dg,e^a=ef,e^b=eg,f^a=fg,h^a=h^3,h^c=h^4,h^d=h^4,i^a=i^3,i^c=i^4,i^d=i^4

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72253139, #361×#1043, #72253188
maximal normal subgroup of	#72253139, #361×#1043, #72253188
maximal quotient of	#72253138

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