#72255403

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#772778 ⋊ C₂
Sylow	C₃ ⋊ #72210469

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂⁵×D₃
commutator, socle, cosocle	C₂⁴×C₃	C₂⁶
Fitting	C₃×#772778	C₂
hypercenter	#772778	D₃
hypocenter	C₃	#72210469
LCS	G ▷ C₂⁴×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ #772778

At least 25012 conjugacy classes of subgroups; only the 4799 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10,3|a^2=j,e^2=gh,b^a=bj,c^a=cj,d^a=dg,d^b=di,e^b=ej,e^c=egh,e^d=egi,f^a=fgh,f^b=fi,f^e=fgh,k^f=k^2

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72210469
maximal normal subgroup of	D₃×#72210469

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