#72220386

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃³×D₄ ⋊ C₂³
Sylow	C₃³ ⋊ C₂×#453

Subgroups

	subgroup	G/H
center	C₂	#13172938
commutator, cosocle	C₂×C₃³×C₄	C₂⁴
Frattini	C₂×C₄	C₂×D₃³
Fitting	C₂×C₃³×D₄	C₂³
socle	C₂×C₃³	C₂×#120
hypercenter	C₂×D₄	D₃³
hypocenter	C₃³	C₂×#453
LCS	G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

4706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,3^3|a^2=g,b^2=e,f^2=g,b^a=bf,c^a=cfg,c^b=ce,e^a=eg,f^a=fg,f^c=fg,h^a=h^2,h^b=h^2,h^d=h^2,i^a=h^2i,i^b=i^2,i^d=i^2,j^d=j^2

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of

D₃×#11864777

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