#72248029

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#1781 ⋊ C₂
Sylow	C₃ ⋊ #147555

Subgroups

	subgroup	G/H
center	C₂³	C₂×#806
commutator	C₂⁵×C₃	C₂²×C₄
Frattini	C₂³×D₄	C₂²×D₃
Fitting	C₂×C₃×#1781	C₂
socle	C₂³×C₃	C₂×#99
prime radical	C₂×#84052	C₂
cosocle	C₂³×C₃×D₄	C₂³
hypercenter	C₂×#1781	D₃
hypocenter	C₃	#147555
LCS	G ▷ C₂⁵×C₃ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂⁴×D₄ ◁ C₂×#1781

6768 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Polycyclic presentation:

2^9,3|a^2=d,b^a=be,c^a=cf,c^b=cefg,d^c=dh,f^a=fh,f^d=fi,g^a=gh,g^d=gi,h^a=hi,j^c=j^2

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#147555, #72248482, #72248495
maximal normal subgroup of	D₃×#147555, #72248482, #72248495
maximal quotient of	#72255199

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