#72225554

order	4096
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#72216237
commutator	C₄×C₈	C₂²×C₄×C₈
Frattini, cosocle	C₄×#346	C₂⁴
socle	C₂	#72225553
prime radical	C₂×#508	#97
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#346 ◁ C₄×#7333 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^12|a^2=l,b^2=f,c^2=e,d^2=gij,e^2=h,f^2=i,g^2=jl,i^2=k,j^2=l,k^2=l,c^a=cg,c^b=cgikl,d^a=dg,d^b=dg,d^c=djk,e^a=ej,e^b=ejk,f^c=fk,f^e=fl,g^a=gj,g^b=gj,g^c=gl,h^a=hl,i^c=il,j^a=jl,j^b=jl

Conjugacy classes & character table

268 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227231, #72227291, #72258194
maximal normal subgroup of	#72227231, #72227291, #72258194

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