#72287333

order	8192
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#72224639
commutator	C₂×C₃₂	C₂³×C₁₆
Frattini, cosocle	C₂×#7330	C₂⁴
prime radical	#72286862	C₈
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ C₂²×#1311 ◁ C₂²×#93668 ◁ G
LECS	G ▷ C₂×#7330 ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3895 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^13|b^2=e,c^2=eh,d^2=f,e^2=h,f^2=i,h^2=j,i^2=k,k^2=l,l^2=m,d^a=dg,d^b=df,d^c=di,e^d=eklm,f^b=fi,f^c=fk,f^e=fl,h^d=hlm,h^f=hm,i^b=ik,i^c=il,i^e=im,j^d=jm,k^b=kl,k^c=km,l^b=lm

Conjugacy classes & character table

284 conjugacy classes — view the full character table.

Relationships

automorphism group of

#7545, #7546

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", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.5*F.8);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 11, F.12);;
gap> SetPower(coll, 12, F.13);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.11*F.12*F.13);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.12*F.13);;
gap> SetConjugate(coll, 8, 6, F.8*F.13);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.12);;
gap> SetConjugate(coll, 9, 5, F.9*F.13);;
gap> SetConjugate(coll, 10, 4, F.10*F.13);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> SetConjugate(coll, 11, 3, F.11*F.13);;
gap> SetConjugate(coll, 12, 2, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);