#72231972

order	4096
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×#322
commutator	C₂×C₈	C₂³×C₄×C₈
Frattini, cosocle	#1311	C₂⁵
socle	C₂²	C₂³×#1313
prime radical	#3444	#97
LCS	G ▷ C₂×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #72231856 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5327 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)(35,45,47,68,46,67,49,70)(36,50,51,72)(37,53)(41,60,61,85,56,80,63,87)(42,64,57,82)(43,58)(48,74,75,93,71,94,73,69)(52,78,77,76)(54,79)(62,89,90,95,81,96,88,86)(65,92,83,91)(66,84);(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)(45,60)(46,56)(47,61)(48,62)(49,63)(50,64)(51,57)(52,65)(53,58)(54,66)(55,59)(67,80)(68,85)(69,86)(70,87)(71,81)(72,82)(73,88)(74,89)(75,90)(76,91)(77,83)(78,92)(79,84)(93,95)(94,96);(3,11)(7,17)(12,23)(13,24)(18,27)(19,28)(25,31)(29,32)(35,46)(41,56)(45,67)(47,49)(48,71)(60,80)(61,63)(62,81)(68,70)(69,93)(73,75)(74,94)(85,87)(86,95)(88,90)(89,96);(34,40)(41,56)(42,57)(43,58)(44,59)(60,80)(61,63)(62,81)(64,82)(65,83)(66,84)(85,87)(86,95)(88,90)(89,96)(91,92);(33,35,36,48,37,45,52,69,38,49,50,75,54,70,76,74,39,46,51,71,53,67,77,93,55,47,72,73,79,68,78,94)(34,41,42,62,43,60,65,86,44,63,64,90,66,87,91,89,40,56,57,81,58,80,83,95,59,61,82,88,84,85,92,96)

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=ei,e^c=el,f^e=fk,g^a=gk,g^f=gl,h^e=hl,i^a=il

Conjugacy classes & character table

448 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, 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.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.12);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
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.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);