#72271194

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×H(7,𝔽₂) ⋊ C₂
Sylow	C₃ ⋊ #6628896

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#953
commutator, cosocle	C₂⁴×C₃	C₂⁵
Frattini	C₂⁴	C₂⁴×D₃
Fitting	C₂×C₃×H(7,𝔽₂)	C₂
socle	C₂²×C₃	C₂²×#120
hypercenter	C₂×H(7,𝔽₂)	D₃
hypocenter	C₃	#6628896
LCS	G ▷ C₂⁴×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³×D₄ ◁ C₂×H(7,𝔽₂)

35824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^9,3|e^2=i,c^b=cf,d^a=di,d^b=dh,e^b=eg,e^c=ei,f^e=fi,g^c=gi,j^b=j^2

Conjugacy classes & character table

222 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#6628896
maximal normal subgroup of	D₃×#6628896

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