#72231021

order	4800
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×C₅²×D₄ ⋊ C₄
Hall	C₃×C₅² ⋊ #390
Sylow	C₅² ⋊ #5221
Sylow	C₃ ⋊ #13104792

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊#1043
commutator, socle	C₂²×C₃×C₅²	C₂²×C₄
Fitting	C₂×C₃×C₅²×D₄	C₄
prime radical	C₂×C₃×D₄×C₅⋊D₅	C₂
cosocle	C₂²×C₃×C₅⋊D₅	C₂³
hypercenter	C₂×D₄	C₃⋊#1043
hypocenter	C₃×C₅²	#390
LCS	G ▷ C₂²×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂² ◁ C₂×D₄

2064 conjugacy classes of subgroups; only the 245 normal subgroups are shown.

normal subgroups only

This group has 2064 conjugacy classes of subgroups — too many to draw. Only its 245 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) = 1200

Polycyclic presentation:

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

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#940×#1043, #127×C₃⋊#1043, D₃×#13104792
maximal normal subgroup of	#940×#1043, #127×C₃⋊#1043, D₃×#13104792

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