#72253147

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#135 ⋊ C₄
Sylow	C₅² ⋊ C₂×#444

Subgroups

	subgroup	G/H
center	C₄	D₄×#1043
commutator	C₄×C₅²	C₂³×C₄
Frattini	C₄	C₂³×#1043
Fitting	C₅²×#135	C₄
socle	C₂×C₅²	C₂×C₄×D₄
prime radical	#135×C₅⋊D₅	C₂
cosocle	C₄×C₅⋊D₅	C₂⁴
hypercenter	#135	#1043
hypocenter	C₅²	C₂×#444
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

1856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 400

Polycyclic presentation:

2^7,5^2|a^2=e,b^2=eg,d^2=f,e^2=g,f^2=g,d^a=df,d^b=dg,f^a=fg,h^a=h^3,h^b=h^3,h^c=h^4,h^e=h^4,i^a=i^3,i^b=i^3,i^c=i^4,i^e=i^4

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72253164, #444×#1043, #72253188, …
maximal normal subgroup of	#72253164, #444×#1043, #72253188, …
maximal quotient of	#72253148, C₂×#72253147

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