#72239526

order	4096
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	#2679
commutator	C₄³	C₂²×C₄²
Frattini, cosocle	C₂²×C₄³	C₂⁴
socle	C₂³	#153953
prime radical	#72239493	C₂²
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄³ ◁ G
LECS	G ▷ C₂²×C₄³ ▷ C₂³ ▷ C₁

34109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2)(3,8)(4,6)(5,7)(9,10,12,13)(11,15,16,14)(17,18)(19,21)(20,23)(22,24);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,20)(21,24)(22,23);(2,6,7,8)(11,16)(14,15)(18,23,24,19);(11,15,16,14)(18,19,24,23)

Polycyclic presentation:

2^12|a^2=jkl,b^2=k,c^2=e,d^2=f,g^2=j,h^2=k,i^2=l,b^a=bkl,c^a=cg,c^b=cghk,d^a=dghkl,d^b=di,e^a=ej,e^b=ejk,f^a=fjk,f^b=fl,g^a=gj,g^b=gjk,h^a=hk,i^a=ijk,i^b=il

Conjugacy classes & character table

424 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72239537
Fitting subgroup of	#72239537
maximal subgroup of	#72239537
maximal normal subgroup of	#72239537

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