#72271050

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#1304 ⋊ C₂
Sylow	C₃ ⋊ C₂²⋊#1304

Subgroups

	subgroup	G/H
center	C₂³	D₃×#95
commutator	C₂⁴×C₃	C₂×C₄²
Frattini	C₂⁶	C₂²×D₃
Fitting	C₂×C₃×#1304	C₂
socle	C₂³×C₃	C₂×#95
prime radical	C₂⁶×D₃	C₂²
cosocle	C₂⁶×C₃	C₂³
hypercenter	C₂×#1304	D₃
hypocenter	C₃	C₂²⋊#1304
LCS	G ▷ C₂⁴×C₃ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#1304

18364 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^9,3|a^2=d,b^2=ef,b^a=bde,c^a=cdg,c^b=cdg,d^b=di,e^a=ei,e^b=eh,f^a=fh,f^b=fh,g^b=gi,j^c=j^2

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271058, #72271081, D₃×C₂²⋊#1304
maximal normal subgroup of	#72271058, #72271081, D₃×C₂²⋊#1304

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