#72226710

order	4096
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72226708
commutator, Frattini, cosocle	C₂×C₄×M₁₆	C₂⁵
LCS	G ▷ C₂×C₄×M₁₆ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×M₁₆ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

At least 12483 conjugacy classes of subgroups; only the 1177 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^12|a^2=il,e^2=f,f^2=j,h^2=k,k^2=l,b^a=bil,e^a=eh,e^b=ej,e^c=eg,e^d=ef,f^a=fk,f^d=fj,g^a=gi,h^a=hk,h^c=hil,h^d=hk,h^g=hl,i^e=il,j^a=jl,k^a=kl,k^d=kl

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

holomorph of

#106

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