#72225236

order	4096
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#212918
commutator	C₈²	C₂³×C₈
Frattini, cosocle	C₄×C₈²	C₂⁴
socle	C₂²	#72216170
prime radical	#72216236	C₄
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #126522 ◁ G
LECS	G ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

2917 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:

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

Polycyclic presentation:

2^12|a^2=e,c^2=fi,d^2=gj,e^2=h,f^2=ik,g^2=jl,h^2=k,i^2=k,j^2=l,c^a=cik,c^b=cf,d^a=djl,d^b=dg,e^c=ek,e^d=el,f^a=fk,f^b=fi,g^a=gl,g^b=gj,i^b=ik,j^b=jl

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225268, #72225288
maximal normal subgroup of	#72225268, #72225288
maximal quotient of	#72225268

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