#72286899

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#7385
commutator	C₂×C₃₂	C₂²×C₈
Frattini, cosocle	C₂×#1397	C₂³
prime radical	C₂³×D₃₂	C₄
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ C₂²×C₄×C₁₆ ◁ C₂²×#1397 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1679 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^11|b^2=hi,c^2=dfhjk,d^2=gik,f^2=h,g^2=ij,h^2=j,i^2=jk,j^2=k,c^a=ce,c^b=cd,d^b=dg,d^c=djk,f^b=fi,f^c=fjk,f^d=fk,g^b=gi,g^c=gk,h^b=hjk,h^c=hk,i^b=ij,j^b=jk

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286964, #72323034, #72323081
maximal normal subgroup of	#72286964, #72323034, #72323081

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