#72226981

order	4096
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₈²
commutator	C₈²	C₂⁴×C₄
Frattini, cosocle	C₂×C₈²	C₂⁵
prime radical	C₂×D₁₆²	C₂
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

9477 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 5 generators on 36 points:

(22,26)(23,29)(25,31)(27,34)(28,33)(30,35)(32,36);(6,10)(7,13)(9,15)(11,18)(12,17)(14,19)(16,20)(22,26)(23,29)(25,31)(27,34)(28,33)(30,35)(32,36);(1,2,3,4)(5,6,8,10)(7,14,16,11)(9,12,17,15)(13,18,20,19)(21,22,24,26)(23,30,32,27)(25,28,33,31)(29,34,36,35);(5,7,15,18,10,14,17,20,8,16,12,19,6,11,9,13)(21,23,31,34,26,30,33,36,24,32,28,35,22,27,25,29);(21,23,31,34,26,30,33,36,24,32,28,35,22,27,25,29)

Polycyclic presentation:

2^12|c^2=f,d^2=gi,e^2=hj,g^2=ik,h^2=jl,i^2=k,j^2=l,c^a=cl,c^b=ck,d^a=dh,d^b=dg,d^c=dk,e^a=eh,e^b=eh,e^c=el,g^a=gj,g^b=gi,h^a=hj,h^b=hj,i^a=il,i^b=ik,j^a=jl,j^b=jl

Conjugacy classes & character table

268 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227169
maximal normal subgroup of	#72227169
maximal quotient of	#72226982

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