#72258924

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#5021361 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂²	C₂×#560⋊C₃
commutator, cosocle	C₂²×C₄²	C₂³×C₃
Frattini, socle	C₂⁴	C₂³×A₄
Fitting	#5021361	C₃
hypercenter	C₂³	#560⋊C₃
hypocenter	C₄²	C₃×#120
LCS	G ▷ C₂²×C₄² ▷ C₄²
UCS	C₁ ◁ C₂² ◁ C₂³

1943 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 1943 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^3,3,2^6|a^2=e,g^2=i,h^2=j,b^a=bf,c^a=ce,g^a=gi,g^b=gij,g^c=gi,g^d=gh,h^a=hj,h^b=hi,h^c=hj,h^d=gj,i^d=ij,j^d=i

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258930, #72258945
maximal normal subgroup of	#72258930, #72258945

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