#72225136

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225133
commutator	C₂²×C₁₆	C₂⁴×C₄
Frattini, cosocle	C₂×#346	C₂⁵
prime radical	#72225019	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #60204 ◁ #72216670 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

At least 6726 conjugacy classes of subgroups; only the 1185 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^12|a^2=h,b^2=f,c^2=fi,f^2=i,h^2=j,j^2=k,k^2=l,b^a=bhjkl,c^a=cjkl,e^a=eg,e^b=ei,e^d=ei,f^a=fkl,g^a=gl,g^d=gl,h^b=hj,h^c=hk,h^e=hl,h^f=hl,i^a=il,j^b=jk,j^c=jl,k^b=kl

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

holomorph of

M₆₄

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