#72286964

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#151085
commutator	C₂×C₃₂	C₂³×C₈
Frattini, cosocle	C₂×#1397	C₂⁴
prime radical	#72286862	C₄
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ C₂²×#1311 ◁ C₂²×#7330 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

2972 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);(6,12,17,35)(7,19)(8,25)(9,27)(10,29)(13,24)(14,31,44,62)(15,33,46,36)(16,47,49,64)(18,41,42,61)(20,38)(21,26)(22,51)(23,58)(28,56)(30,60)(32,63,50,34)(37,48,65,43)(39,57)(40,52)(45,67,68,66)(53,59)(54,55);(2,4)(5,6,13,37,8,16,23,31,27,14,40,47,53,43,7,12,10,15,19,34,28,45,52,41,30,42,58,66,21,32,24,36,11,17,38,65,26,49,57,62,60,44,55,64,56,48,22,35,29,46,51,63,59,68,54,61,9,18,39,67,25,50,20,33);(6,14,15,42,17,44,46,18)(7,20,22,24)(8,26)(12,31,33,61,35,62,36,41)(13,19,38,51)(16,48,45,50,49,43,68,32)(21,25)(23,54,57,52)(28,59)(34,64,37,66,63,47,65,67)(39,40,58,55)(53,56)

Polycyclic presentation:

2^12|b^2=i,c^2=ehkl,d^2=g,e^2=hjl,g^2=i,h^2=jk,j^2=kl,k^2=l,c^a=cf,c^b=ce,d^c=dj,e^b=eh,e^d=ekl,g^c=gkl,g^e=gl,h^b=hj,h^d=hl,i^c=il,j^b=jk,k^b=kl

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287333, #72323149
maximal normal subgroup of	#72287333, #72323149

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