#72242181

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ C₂×#322

Subgroups

	subgroup	G/H
center, hypercenter	C₄²	D₅⋊AGL(1,𝔽₅)
commutator	C₂×C₅²	C₂×C₄×C₈
Frattini	C₂×C₄	C₂×D₅⋊AGL(1,𝔽₅)
Fitting	C₄²×C₅²	C₂×C₄
socle	C₂²×C₅²	C₂×C₄²
prime radical	C₂²×C₅⋊D₅	C₄²
cosocle	C₄×#1039	C₂³
hypocenter	C₅²	C₂×#322
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄²

658 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 167 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) = 320

Polycyclic presentation:

2^7,5^2|a^2=d,b^2=e,d^2=f,e^2=g,b^a=bg,h^a=h^3,h^b=h^4,h^c=h^4,h^d=h^4,i^a=i^3,i^b=h^4i,i^c=i^4,i^d=i^4

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72242211, #72242313
maximal normal subgroup of	#72242211, #72242313
maximal quotient of	#72242183

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