#72323190

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×#1333
commutator	C₂²×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×#1311	C₂⁴
prime radical	#72320231	C₄
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ C₂×C₈×#131 ◁ C₂×#479943 ◁ G
LECS	G ▷ C₂×#1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4607 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 4 generators on 68 points:

(5,6,8,12,9,13,18,24,10,14,19,25,20,26,30,34)(7,11,16,22,17,23,29,33,15,21,27,31,28,32,35,36)(38,43,44,57,45,58,55,60)(39,47,49,65)(40,51)(41,54)(46,63,61,59,62,48,52,67)(50,56,68,66)(53,64);(5,7)(6,11)(8,16)(9,17)(10,15)(12,22)(13,23)(14,21)(18,29)(19,27)(20,28)(24,33)(25,31)(26,32)(30,35)(34,36);(1,2)(3,4)(7,15)(11,21)(16,27)(17,28)(22,31)(23,32)(29,35)(33,36);(2,4)(37,38)(39,48)(40,52)(41,55)(42,45)(43,56)(44,54)(46,64)(47,57)(49,63)(50,67)(51,62)(53,61)(58,66)(59,68)(60,65)

Polycyclic presentation:

2^12|a^2=e,e^2=h,g^2=i,h^2=j,i^2=k,k^2=l,c^b=cj,d^a=dg,d^c=df,e^d=ekl,g^a=gi,g^d=gikl,g^e=gl,h^d=hl,i^a=ik,i^d=ikl,k^a=kl,k^d=kl

Conjugacy classes & character table

268 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72323191

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