#72323144

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, socle	C₂²	C₂²×#12067
commutator	C₂×C₁₆	C₂⁴×C₈
Frattini, cosocle	#1311	C₂⁵
prime radical	#72260454	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#131 ◁ #72231875 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4442 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:

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

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^b=dj,e^d=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

274 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, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 4, 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);