#72242554

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#102 ⋊ C₂²
Sylow	C₅² ⋊ #3058

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₅²
commutator, cosocle	C₂×C₄×C₅²	C₂⁴
Frattini	C₂×C₄	C₂²×D₅²
Fitting	C₅²×#102	C₂²
socle	C₂²×C₅²	C₂²×D₄
hypercenter	#102	D₅²
hypocenter	C₅²	#3058
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

1178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 320

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72242561
maximal normal subgroup of	#72242561

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