#72234842

order	4096
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#23900
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁶
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₂²⋊C₄ ◁ C₂²×#3484 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

At least 36529 conjugacy classes of subgroups; only the 5183 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^12|b^2=jl,e^2=jl,f^2=l,g^2=kl,b^a=bjl,e^a=eh,e^b=eh,e^c=eh,e^d=ejl,f^a=fi,f^b=fil,f^e=fg,g^a=gj,g^b=gj,g^c=gk,g^d=gl,g^e=gk,g^f=gkl,h^f=hk,i^b=il,i^e=ijkl,i^g=il,j^f=jl

Conjugacy classes & character table

214 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238442, #72262341
maximal normal subgroup of	#72238442, #72262341
automorphism group of	#1466, #1467

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