#72225374

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#72214602
commutator	C₄×C₁₆	C₂³×C₈
Frattini, cosocle	C₄×#346	C₂⁴
socle	C₂	#72216371
prime radical	#72225339	C₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #72216161 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,12)(3,13,17,42)(4,20,22,55)(5,24,25,56)(6,27,26,10)(8,28,31,18)(9,32,33,23)(11,35,38,14)(15,44,19,40)(16,39,52,61)(21,54,47,36)(29,59,60,64)(30,53,57,50)(34,48,37,46)(41,58,63,45)(43,49,62,51);(1,3,14,45,5,16,47,50,26,15,46,59,32,49,22,28,7,17,35,58,25,52,21,53,6,19,48,64,23,51,4,18)(2,8,20,43,9,29,37,40,27,57,54,61,56,63,38,42,12,31,55,62,33,60,34,44,10,30,36,39,24,41,11,13);(2,9,10,24,12,33,27,56)(3,15,17,19)(4,21,22,47)(5,25)(8,29,30,41,31,60,57,63)(11,36,37,20,38,54,34,55)(13,39,40,43,42,61,44,62)(14,46,35,48)(16,51,52,49)(23,32)(45,58)(59,64);(2,9,27,56,12,33,10,24)(3,16,15,49,17,52,19,51)(11,20,37,54,38,55,34,36)(18,45,50,59,28,58,53,64)

Polycyclic presentation:

2^12|a^2=l,b^2=fi,c^2=e,d^2=gj,e^2=h,f^2=ik,g^2=j,i^2=kl,k^2=l,b^a=bf,c^a=cik,c^b=cg,d^a=dik,d^b=dgi,d^c=dj,e^a=ekl,e^b=ej,f^a=fi,f^c=fk,f^e=fl,g^a=gkl,g^b=gjk,h^a=hl,i^a=ik,i^c=il,j^a=jl,j^b=jl,k^a=kl

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227231, #72227291
maximal normal subgroup of	#72227231, #72227291

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