#72258584

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₅² ⋊ C₂×C₄
Sylow	C₅² ⋊ D₄×C₂²⋊C₄

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅⋊AGL(1,𝔽₅)
commutator, socle	C₂²×C₅²	C₂³×C₄
Fitting	C₂⁴×C₅²	C₂×C₄
prime radical	C₂²×#87886	C₂
cosocle	C₂²×C₅⋊D₅	C₂⁴
hypercenter	C₂⁴	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	D₄×C₂²⋊C₄
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂⁴

3178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Polycyclic presentation:

2^7,5^2|c^2=e,c^b=cf,d^a=dg,h^a=h^4,h^c=h^3,h^e=h^4,i^a=h^3i,i^c=i^3,i^e=i^4

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

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