#72234414

order	4096
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₁₆×C₃₂ · Schur multiplier (H₂) = C₂×C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄×C₈	C₈²
commutator	C₈	C₁₆×C₃₂
Frattini, cosocle	C₄×#7052	C₂²
socle, prime radical	C₂³	#92896
LCS	G ▷ C₈ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×#7052 ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

849 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 112
transitive degree	μ_t(G) = 4096

A minimal faithful permutation representation — 2 generators on 112 points:

(1,2)(3,9,10,15)(4,8)(5,7)(6,11,13,16)(12,14)(17,18,19,23,20,24,27,33,21,25,28,34,30,36,39,43,22,26,29,35,31,37,40,44,32,38,41,45,42,46,47,48)(49,50)(51,63,64,82,67,92,69,93)(52,70,71,88)(53,59)(54,77)(55,61)(56,62)(57,66,79,95,81,101,83,68)(58,84,85,74)(60,78)(65,96,97,91,99,110,94,111)(72,87,104,106)(73,86,105,103)(75,89)(76,90)(80,107,108,98,109,100,102,112);(1,3)(2,6,7,13)(4,11)(5,10)(8,9,14,15)(12,16)(49,51,52,65,54,68,73,100,55,69,74,97,76,101,104,112,56,67,71,99,78,95,105,107,75,64,84,94,53,66,72,98)(50,57,58,80,60,82,87,96,61,83,88,108,59,63,86,91,62,81,85,109,77,93,106,110,89,79,70,102,90,92,103,111)

Polycyclic presentation:

2^12|a^2=c,b^2=d,c^2=f,d^2=g,e^2=hl,f^2=i,g^2=j,h^2=l,i^2=k,j^2=l,b^a=be,c^b=ch,d^a=dhl,d^c=dl,f^b=fl,g^a=gl

Conjugacy classes & character table

736 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72234417

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