#72258755

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72228377 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#560⋊C₃
commutator, cosocle	C₂²×C₄²	C₂⁴×C₃
Frattini, socle	C₂⁴	C₂⁴×A₄
Fitting	#72228377	C₃
hypercenter	C₄²	#560⋊C₃
hypocenter	C₄²	C₂×C₃×#127
LCS	G ▷ C₂²×C₄² ▷ C₄²
UCS	C₁ ◁ C₂² ◁ C₄²

10788 conjugacy classes of subgroups; only the 330 normal subgroups are shown.

normal subgroups only

This group has 10788 conjugacy classes of subgroups — too many to draw. Only its 330 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:

2^4,3,2^6|c^2=f,d^2=g,h^2=j,i^2=k,c^b=cf,d^b=dg,e^c=ejk,h^a=hk,h^b=hj,h^e=hi,i^a=ijk,i^b=ik,i^e=hk,j^e=jk,k^e=j

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258804
maximal normal subgroup of	#72258804

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 5, F.8*F.11);;
gap> SetConjugate(coll, 10, 5, F.10*F.11);;
gap> SetConjugate(coll, 11, 5, F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);