#72253846

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂³×C₃×H(5,𝔽₂) ⋊ C₂
Sylow	C₃ ⋊ #7631286

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁵×D₃
commutator, socle, cosocle	C₂³×C₃	C₂⁶
Fitting	C₂³×C₃×H(5,𝔽₂)	C₂
hypercenter	C₂³×H(5,𝔽₂)	D₃
hypocenter	C₃	#7631286
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂³×H(5,𝔽₂)

52444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Polycyclic presentation:

2^9,3|d^a=di,e^a=eh,e^b=eg,f^a=fh,f^b=fg,f^c=fg,f^d=fg,j^a=j^2

Conjugacy classes & character table

258 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#7631286
maximal normal subgroup of	D₃×#7631286
maximal quotient of	#72271201, #72271583

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 3]);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);