#72272620

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72272580 ⋊ C₃
derived	H(3,𝔽₄) ⋊ C₂⁴×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	H(3,𝔽₄)	C₂⁴×C₃
Frattini, socle	C₂²	C₂⁴×C₂²⋊A₄
Fitting	#72272580	C₃
LCS	G ▷ H(3,𝔽₄)
UCS	C₁

At least 13127 conjugacy classes of subgroups; only the 537 normal subgroups are shown.

normal subgroups only

This group has at least 13127 conjugacy classes of subgroups — too many to draw. Only its 537 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:

2^4,3,2^6|h^2=j,i^2=j,e^a=ek,e^b=ej,f^c=fjk,f^d=fj,f^e=fg,g^c=gj,g^d=gk,g^e=f,h^a=hk,h^b=hjk,h^c=hj,h^e=fij,h^f=hj,h^g=hjk,i^a=ij,i^b=ik,i^d=ij,i^e=ghij,i^f=ik,i^g=ij,i^h=ij,j^e=jk,k^e=j

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72272663
maximal normal subgroup of	#72272663

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