#72247271

order	4860
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, Hall	C₅×#6909 ⋊ C₄
derived	C₅×AΣL(1,𝔽₂₇) ⋊ C₃×C₄
Sylow	#6909 ⋊ AGL(1,𝔽₅)
Sylow	C₅ ⋊ #66330⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₅×AΣL(1,𝔽₂₇)	C₃×C₄
Frattini	C₃²	C₃×C₃⋊ΓL(1,𝔽₁₆)
Fitting	C₅×#6909	C₄
socle	C₃×C₅	C₃×#1084
prime radical	D₅×#6909	C₂
cosocle	D₅×AΣL(1,𝔽₂₇)	C₂×C₃
LCS	G ▷ C₅×AΣL(1,𝔽₂₇)
UCS	C₁

316 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 135

Polycyclic presentation:

2,3,2,3^2,5,3^2|a^2=c,e^3=h^2,d^a=d^2,d^b=dh,e^a=e^2h,e^b=eh^2,e^d=eg,f^a=f^3,f^c=f^4,g^a=gh^2,g^e=gh,h^a=h^2

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#4183⋊C₃, #72247336
maximal normal subgroup of	AGL(1,𝔽₅)×#4183⋊C₃

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