#72248533

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#2403 ⋊ C₂
Sylow	C₃ ⋊ #7792761

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁵×D₃
commutator, socle, cosocle	C₂³×C₃	C₂⁶
Fitting	C₂×C₃×#2403	C₂
hypercenter	C₂×#2403	D₃
hypocenter	C₃	#7792761
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×#2403

At least 6074 conjugacy classes of subgroups; only the 3673 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^9,3|a^2=i,b^2=h,e^2=g,c^a=ci,d^a=dh,e^b=ei,e^c=eg,f^b=fh,f^e=fg,j^a=j^2

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72248542, D₃×#7792761, #72248599
maximal normal subgroup of	#72248542, D₃×#7792761, #72248599
maximal quotient of	#72270923

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