#72270056

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#34171 ⋊ C₂
Sylow	C₃ ⋊ #359501

Subgroups

	subgroup	G/H
center	C₂³	C₂×#953
commutator	C₂⁴×C₃	C₂³×C₄
Frattini	C₂⁵	C₂³×D₃
Fitting	C₃×#34171	C₂
socle	C₂³×C₃	C₂×#120
prime radical	C₂³×#937	C₂
cosocle	C₂⁵×C₃	C₂⁴
hypercenter	#34171	D₃
hypocenter	C₃	#359501
LCS	G ▷ C₂⁴×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ #34171

24472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^9,3|b^2=f,b^a=be,c^b=ce,d^b=defh,d^c=deg,g^b=gi,h^c=hi,j^d=j^2

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270082, #72270221, #72270302, …
maximal normal subgroup of	#72270082, #72270221, #72270302, …

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