#72234318

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	C₂³×#1368
commutator	C₂×C₁₆	C₂⁴×C₈
Frattini, cosocle	#1311	C₂⁵
prime radical	#3444	#106
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #6342481 ◁ #72231856 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4385 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:

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

Polycyclic presentation:

2^12|a^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=k,k^2=l,c^b=cj,d^b=dl,e^a=eg,f^e=fkl,g^a=gi,g^f=gl,h^e=hl,i^a=ik,k^a=kl

Conjugacy classes & character table

328 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323191
maximal normal subgroup of	#72323191

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