#72340460

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

Polycyclic presentation:

2^11|a^2=g,c^2=k,e^2=j,g^2=k,b^a=bf,c^a=ce,c^b=cjk,d^c=dfk,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	#72340463
maximal normal subgroup of	#72340463

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.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, 3, F.4*F.6*F.11);;
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);