#72220575

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#612424 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#120⋊C₃
commutator, cosocle	C₂⁶	C₂³×C₃
Frattini, socle	C₂⁴	C₂³×A₄
Fitting	#612424	C₃
hypercenter	C₂×D₄	#120⋊C₃
hypocenter	C₂⁴	C₃×#120
LCS	G ▷ C₂⁶ ▷ C₂⁴
UCS	C₁ ◁ C₂² ◁ C₂×D₄

11757 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 11757 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2^6|b^2=f,c^2=ef,b^a=bf,c^a=cf,c^b=ce,g^a=gi,g^b=gi,g^d=gh,h^a=hj,h^b=hj,h^d=g,i^d=ij,j^d=i

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267166, C₂×#72220575
maximal normal subgroup of	#72222918, C₂×#72220575
maximal quotient of	#72220576, #72256423, #72341393, …

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