#72225150

order	4096
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₄×#322
commutator	C₄×C₈	C₄²×C₈
Frattini, cosocle	C₂×#7052	C₂³
socle	C₂²	#72216178
prime radical	C₂³	#122220
LCS	G ▷ C₄×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ #72216182 ◁ G
LECS	G ▷ C₂×#7052 ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^12|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,g^2=i,h^2=j,i^2=k,j^2=l,b^a=bhjl,c^a=ci,c^b=cl,d^b=djl,d^c=dk,e^a=ejl,e^d=el,f^a=fk,g^b=gl,h^a=hl

Conjugacy classes & character table

352 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225192, #72225300, #72225307
maximal normal subgroup of	#72225192, #72225300, #72225307

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