#72323108

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#12067
commutator	C₂×C₁₆	C₂⁴×C₈
Frattini, cosocle	#1311	C₂⁵
socle	C₂²	C₂²×#12067
prime radical	#72234362	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₈² ◁ C₂²×#7053 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4317 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 5 generators on 96 points:

(1,2,5,10,7,11,17,13,6,9,20,25,21,26,31,29)(3,12,15,19,16,27,28,18,14,8,22,23,30,24,32,4)(33,34)(35,45,48,73,47,74,60,76)(36,50,53,86)(37,55)(38,59)(39,41)(40,49,63,85,64,81,70,82)(42,54,65,91)(43,58)(44,61)(46,72,79,75,78,67,56,94)(51,69,88,62,84,96,80,92)(52,71,89,87)(57,93)(66,77)(68,83,95,90);(1,3)(2,8)(4,13)(5,15)(6,14)(7,16)(9,12)(10,23)(11,24)(17,28)(18,29)(19,25)(20,22)(21,30)(26,27)(31,32)(33,35)(34,40)(36,51)(37,56)(38,60)(39,47)(41,64)(42,67)(43,69)(44,70)(45,71)(46,77)(48,61)(49,83)(50,73)(52,88)(53,84)(54,85)(55,92)(57,79)(58,78)(59,63)(62,93)(65,72)(66,96)(68,94)(74,87)(75,95)(76,86)(80,89)(81,90)(82,91);(1,4,17,27,21,23,5,12,6,18,31,24,7,19,20,8)(2,3,13,28,26,30,10,15,9,14,29,32,11,16,25,22)(35,46,60,56,47,78,48,79)(36,52,53,89)(37,57)(40,62,70,69,64,92,63,96)(42,68,65,95)(43,66)(45,72,76,94,74,67,73,75)(49,84,82,88,81,51,85,80)(50,71,86,87)(54,83,91,90)(55,93)(58,77);(2,9)(3,14)(10,25)(11,26)(13,29)(15,22)(16,30)(28,32);(34,41)(35,47)(42,65)(43,66)(44,59)(46,78)(48,60)(49,81)(50,86)(51,84)(55,93)(56,79)(68,95)(71,87)(80,88)(82,85)

Polycyclic presentation:

2^12|a^2=f,c^2=fh,f^2=h,g^2=i,h^2=j,i^2=k,k^2=l,b^a=bg,c^b=cikl,d^a=dj,d^b=dj,e^a=el,e^b=el,f^b=fkl,g^a=gi,g^b=gikl,g^c=gk,g^f=gl,h^b=hl,i^a=ik,i^b=ikl,i^c=il,k^a=kl,k^b=kl

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

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