#72268181

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#10586518 ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂²	C₂³×C₂²⋊A₄
commutator, cosocle, hypocenter	Q₈²	C₂³×C₃
Fitting	#10586518	C₃
hypercenter	#120	C₂²⋊A₄
LCS	G ▷ Q₈²
UCS	C₁ ◁ C₂² ◁ #120

10926 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 10926 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 128

Polycyclic presentation:

2^3,3,2^6|e^2=j,f^2=j,g^2=i,h^2=i,b^a=bi,c^a=cij,e^d=f,f^d=efj,f^e=fj,g^d=gh,g^f=gj,h^d=g,h^e=hj,h^f=hj,h^g=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268182, #72268200, #72271367, …
maximal normal subgroup of	#72268200

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