#72239401

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, socle	C₂³	#153953
commutator	C₄³	C₂²×C₄²
Frattini, cosocle	C₂²×C₄³	C₂⁴
prime radical	#72239255	C₂²
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄³ ◁ G
LECS	G ▷ C₂²×C₄³ ▷ C₂³ ▷ C₁

19659 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(3,10,11,12)(8,15,16,14)(17,18)(19,22)(20,25)(21,23)(24,29)(26,31)(27,30)(28,32)(34,39,41,42)(35,43)(40,47,48,46)(44,45);(2,6,9,13)(3,10,11,12)(8,16)(14,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,32)(25,31)(28,29)(35,44,43,45)(40,46,48,47);(1,2,7,6,5,9,4,13)(3,8,12,14,11,16,10,15)(18,23,25,24)(19,26)(22,30,31,32)(27,28)(33,34)(35,40)(36,42)(37,39)(38,41)(43,48)(44,47)(45,46);(1,3)(2,8)(4,10)(5,11)(6,14)(7,12)(9,16)(13,15)(18,24,25,23)(19,27,26,28)(33,35,37,45,38,43,36,44)(34,40,39,46,41,48,42,47)

Polycyclic presentation:

2^12|a^2=e,b^2=f,c^2=fgjk,d^2=efijk,g^2=j,h^2=kl,i^2=l,c^a=ch,c^b=cg,d^a=dghikl,d^b=dghjkl,e^c=el,e^d=ej,f^c=fj,f^d=fjl,g^a=gk,g^c=gj,g^d=gjl,h^a=hk,h^b=hk,h^c=hl,i^a=ik,i^b=ik,i^c=ijl,i^d=il

Conjugacy classes & character table

262 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72239408
Fitting subgroup of	#72239408
maximal subgroup of	#72239408
maximal normal subgroup of	#72239408

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