#72323189

order	2048
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	C₂²	C₂²×#1333
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	C₂×#33607	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂×C₄×#131 ◁ C₂×#33326 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3430 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:

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

Polycyclic presentation:

2^11|a^2=e,e^2=h,g^2=i,h^2=k,i^2=j,j^2=k,c^b=ck,d^a=dg,d^c=df,e^d=ejk,g^a=gi,g^d=gijk,g^e=gk,h^d=hk,i^a=ij,i^d=ijk,j^a=jk,j^d=jk

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72323190

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