#72248696

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#115204 ⋊ C₂
Sylow	C₃ ⋊ #72210600

Subgroups

	subgroup	G/H
center	C₂²	#10726564
commutator, cosocle	C₂⁶×C₃	C₂⁴
Frattini	C₂⁶	C₂³×D₃
Fitting	C₃×#115204	C₂
socle	C₂²×C₃	#14715
hypercenter	#115204	D₃
hypocenter	C₃	#72210600
LCS	G ▷ C₂⁶×C₃ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ #115204

At least 9338 conjugacy classes of subgroups; only the 489 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10,3|a^2=f,d^2=j,b^a=bf,c^a=cfg,c^b=ce,d^a=dh,e^a=ei,f^c=fi,g^a=gi,g^c=gi,g^d=gj,h^c=hj,k^b=k^2,k^d=k^2

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72210600
maximal normal subgroup of	D₃×#72210600
maximal quotient of	#72255255

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 11, 2, F.11^2);;
gap> SetConjugate(coll, 11, 4, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);