#72256263

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

#72219227 ⋊ D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#72209642
commutator, cosocle	C₂²×C₂²⋊(C₂²⋊A₄)	C₂³
Frattini, socle	C₂⁶	C₂²×S₄
Fitting	#72219227	D₃
hypocenter	C₂²⋊(C₂²⋊A₄)	#120
LCS	G ▷ C₂²×C₂²⋊(C₂²⋊A₄) ▷ C₂²⋊(C₂²⋊A₄)
UCS	C₁ ◁ C₂²

At least 22049 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has at least 22049 conjugacy classes of subgroups — too many to draw. Only its 73 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:

2^5,3,2^6|b^2=d,b^a=be,c^a=ce,c^b=cd,f^c=f^2,g^a=gi,g^b=gk,g^c=gh,g^f=h,h^a=hj,h^b=hl,h^f=gh,i^c=ij,i^f=j,j^f=ij,k^c=kl,k^f=l,l^f=kl

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256496
maximal normal subgroup of	#72256496

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.12);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> SetConjugate(coll, 11, 3, F.11*F.12);;
gap> SetConjugate(coll, 11, 6, F.12);;
gap> SetConjugate(coll, 12, 6, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);