#72253301

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #1895

Subgroups

	subgroup	G/H
center	C₂²	D₅⋊(C₂²⋊AGL(1,𝔽₅))
commutator	C₂×C₄×C₅²	C₂²×C₄
Frattini	C₂×C₄	C₂×D₅⋊AGL(1,𝔽₅)
Fitting	C₅²×C₄⋊C₄	C₂×C₄
socle	C₂²×C₅²	C₂×C₂²⋊C₄
prime radical	#13105715	C₂
cosocle	C₂×C₄×C₅⋊D₅	C₂³
hypercenter	C₄⋊C₄	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#1895
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

938 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 101 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) = 160

Polycyclic presentation:

2^7,5^2|a^2=e,c^2=d,d^2=g,f^2=g,b^a=be,c^a=cf,c^b=cg,d^a=dg,f^a=fg,h^b=h^4,h^c=h^3,h^d=h^4,i^b=hi,i^c=i^3,i^d=i^4

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72253303, #72253345, #72253350
maximal normal subgroup of	#72253303, #72253345, #72253350

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