#72340520

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#213828
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂²×#127	C₂⁴
prime radical	#72337659	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#127 ◁ G
LECS	G ▷ C₂²×#127 ▷ C₂⁴ ▷ C₂² ▷ C₁

4921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^11|a^2=g,c^2=jk,e^2=j,g^2=k,b^a=bf,c^a=ce,c^b=cjk,d^a=dk,d^c=df,e^a=ei,e^b=ej,e^c=ej,e^d=eh,f^a=fhj,f^e=fj,g^b=ghj,g^c=gij,g^f=gk,h^a=hk,h^c=hj,i^b=ij,i^d=ik

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340523, #72340564
maximal normal subgroup of	#72340523, #72340564

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