#72247256

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

Aliases

No aliases for this group.

Splittings

Fitting, Hall	C₅×C₃²⋊H(3,𝔽₃) ⋊ C₄
derived	C₃×C₅×H(3,𝔽₃) ⋊ C₃×C₄
Sylow	C₃²⋊H(3,𝔽₃) ⋊ AGL(1,𝔽₅)
Sylow	C₅ ⋊ #11757604

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₃⋊((C₃⋊ΓL(1,𝔽₁₆))⋊C₃)
commutator, hypocenter	C₃×C₅×H(3,𝔽₃)	C₃×C₄
Frattini	C₃²	C₃×C₃⋊ΓL(1,𝔽₁₆)
Fitting	C₅×C₃²⋊H(3,𝔽₃)	C₄
socle	C₃²×C₅	C₃×C₃⋊Dic₃
prime radical	D₅×C₃²⋊H(3,𝔽₃)	C₂
cosocle	C₃×D₅×H(3,𝔽₃)	C₂×C₃
LCS	G ▷ C₃×C₅×H(3,𝔽₃)
UCS	C₁ ◁ C₃

638 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 65 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:

2,3,2,3^3,5,3|a^2=c,d^a=d^2,d^b=df,e^a=e^2,e^d=eh,f^a=f^2,g^a=g^3,g^c=g^4

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#92300, #72247336, #72254514, …
maximal normal subgroup of	AGL(1,𝔽₅)×#92300, #72254514, #72254535

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