#72233347

order	4096
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72233105
commutator	C₂⁴×D₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄²	C₂⁴
prime radical	#72232676	C₂
LCS	G ▷ C₂⁴×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×D₄² ◁ G
LECS	G ▷ C₂²×D₄² ▷ C₂⁴ ▷ C₂² ▷ C₁

At least 38680 conjugacy classes of subgroups; only the 735 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^12|d^2=e,e^2=k,c^b=cjl,d^a=dh,d^b=dg,d^c=df,e^a=ei,e^c=eij,f^a=fl,f^b=fijkl,f^d=fij,f^e=fl,g^c=gijl,g^f=gl,h^c=hl,h^d=hi,h^e=hk,i^d=ik,j^d=jkl

Conjugacy classes & character table

139 conjugacy classes — view the full character table.

Relationships

automorphism group of

#1534

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