#72225288

order	8192
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72216176
commutator	C₈×C₁₆	C₂³×C₈
Frattini, cosocle	C₈×#346	C₂⁴
prime radical	#72216368	C₄
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₈² ◁ #72216186 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(2,6,7,15)(3,9)(4,12)(8,10,13,16)(11,14)(18,23,25,37,26,38,43,40)(19,28,30,47)(20,32)(21,35)(24,36,41,27,42,44,39,45)(29,46,48,31)(33,34);(1,2)(3,10)(4,13)(5,7)(6,14)(8,12)(9,15)(11,16)(18,24)(19,29)(20,33)(21,35)(23,36)(25,41)(26,42)(27,37)(28,46)(30,48)(31,47)(32,34)(38,44)(39,43)(40,45);(17,18,19,23,20,27,28,39,21,25,31,40,32,44,48,42,22,26,30,38,34,45,47,41,35,43,46,37,33,36,29,24);(1,2,3,6,4,8,9,16,5,7,11,15,12,13,14,10)

Polycyclic presentation:

2^13|a^2=e,c^2=f,d^2=g,e^2=h,f^2=i,g^2=j,i^2=k,j^2=l,k^2=m,b^a=bg,c^a=cf,c^b=cfikm,d^a=dg,d^b=dgjl,e^b=el,e^c=ekm,e^d=el,f^a=fi,f^b=fikm,f^e=fm,g^a=gj,g^b=gjl,h^c=hm,i^a=ik,i^b=ikm,j^a=jl,j^b=jl,k^a=km,k^b=km

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225329
maximal normal subgroup of	#72225329
maximal quotient of	#72225329

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