#72260258

order	4800
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×C₄×C₅² ⋊ C₂×C₄
Hall	C₃×C₅² ⋊ C₄×C₂²⋊C₄
Sylow	C₅² ⋊ #4666
Sylow	C₃ ⋊ C₂²⋊C₄×#1043

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃×#1043
commutator	C₂×C₃×C₅²	C₂×C₄²
Fitting	C₂×C₃×C₄×C₅²	C₂×C₄
socle	C₂²×C₃×C₅²	C₂²×C₄
prime radical	C₂²×D₃×C₅⋊D₅	C₂²
cosocle	C₂²×C₃×C₅⋊D₅	C₂³
hypercenter	C₂×C₄	D₃×#1043
hypocenter	C₃×C₅²	C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

2064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 600

Polycyclic presentation:

2^6,3,5^2|a^2=e,c^2=d,b^a=bf,c^a=cf,g^b=g^2,h^c=h^3,h^d=h^4,i^c=i^3,i^d=i^4

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×C₂²⋊C₄×#1043
maximal normal subgroup of	D₃×C₂²⋊C₄×#1043

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