#72275751

order	4000
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	SD₁₆×5₋¹⁺² ⋊ C₂
Sylow	5₋¹⁺² ⋊ AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂	D₄×D₂₅⋊C₅
commutator, cosocle	C₄×C₂₅	C₂³×C₅
Frattini	C₄×C₅	C₂²×C₅×D₅
Fitting	SD₁₆×5₋¹⁺²	C₂
socle	C₂×C₅	C₅×D₄×D₅
hypercenter	SD₁₆	D₂₅⋊C₅
hypocenter	C₂₅	C₅×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

224 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 200

Polycyclic presentation:

2^3,5,2^2,5^2|a^2=e,e^2=f,g^5=h,b^a=bef,c^a=cf,e^b=ef,g^c=g^4h^4,g^d=gh^3,h^c=h^4

Conjugacy classes & character table

139 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×D₂₅⋊C₅
maximal normal subgroup of	AGL(1,ℤ/8)×D₂₅⋊C₅

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