#72234105

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	C₂²	#72233203
commutator, Frattini, cosocle	C₂²×C₄×D₄	C₂⁵
LCS	G ▷ C₂²×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#37674 ◁ G
LECS	G ▷ C₂²×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

At least 40144 conjugacy classes of subgroups; only the 1047 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:

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

Polycyclic presentation:

2^12|a^2=i,d^2=l,h^2=k,c^b=cij,d^a=dh,d^b=dg,d^c=df,e^a=ei,e^c=el,f^a=fj,f^b=fjk,g^c=gj,g^f=gl,h^c=hjl,h^d=hk,h^e=hk,h^f=hl,i^d=ik,j^d=jl

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

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