#72271168

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Splittings

Fitting	C₃×D₄×#95 ⋊ C₂
Sylow	C₃ ⋊ #518019

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂⁴×D₃
commutator	C₂³×C₃	C₂⁴×C₄
Fitting	C₃×D₄×#95	C₂
socle, cosocle	C₂⁴×C₃	C₂⁵
prime radical	C₂²×D₄×D₁₂	C₂
hypercenter	D₄×#95	D₃
hypocenter	C₃	#518019
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ D₄×#95

19300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 384

Polycyclic presentation:

2^9,3|a^2=i,b^2=g,b^a=bf,d^a=di,d^b=dfh,e^c=efh,j^d=j^2

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#518019, #72271229, #72271317
maximal normal subgroup of	D₃×#518019, #72271229, #72271317

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