#72323147

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	#72232017	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#131 ◁ #72231879 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4173 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:

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

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,d^c=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

250 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, 4, 3, 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);