#72247405

order	4860
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, Hall	C₅×#6891 ⋊ C₄
Sylow	#6891 ⋊ AGL(1,𝔽₅)
Sylow	C₅ ⋊ #11757563

Subgroups

	subgroup	G/H
center	C₃	C₃×(C₃⋊ΓL(1,𝔽₁₆))⋊C₃
commutator	C₃³×C₅	C₃²×C₄
Frattini	C₃²	C₃²×ΓL(1,𝔽₁₆)
Fitting	C₅×#6891	C₄
socle	C₃²×C₅	C₃²×Dic₃
prime radical	C₃⁴×D₅	C₂×C₃
cosocle	C₃³×D₅	C₂×C₃²
hypercenter	C₃²	(C₃⋊ΓL(1,𝔽₁₆))⋊C₃
hypocenter	C₃²×C₅	C₄×3₋¹⁺²
LCS	G ▷ C₃³×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₃ ◁ C₃²

372 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 90

Polycyclic presentation:

2,3^2,2,3^3,5|a^2=d,c^3=e,c^b=ce,f^a=f^2,f^c=fg,g^a=g^2,h^a=h^3,h^d=h^4

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#92259, #72247464, #72247468
maximal normal subgroup of	AGL(1,𝔽₅)×#92259, #72247464, #72247468

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