#72323158

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#72225169
commutator	C₂²×C₃₂	C₂²×C₈
Frattini, cosocle	C₂×#7330	C₂³
socle	C₂	#72323154
prime radical	#512934	C₈
LCS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ C₂²×#1311 ◁ C₂²×#7330 ◁ G
LECS	G ▷ C₂×#7330 ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1029 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 128 points:

(1,2)(3,15,16,55,18,56,62,58,20,53,64,76,37,59,33,107)(4,21,23,77,24,78,85,80)(5,27,29,95)(6,32)(7,36)(8,10)(9,17,40,66,41,67,111,61,39,69,104,90,52,70,51,120)(11,25,47,91,43,88,108,86)(12,30,44,99)(13,34)(14,38)(19,72,74,57,71,46,28,97,75,118,63,121,101,115,98,125)(22,49,81,110,82,112,127,94,83,117,128,42,84,114,68,113)(26,93,87,79,89,54,122,123)(31,103,100,96)(35,105)(45,60)(48,119,106,92,116,65,124,126)(50,73,109,102);(9,39)(15,53)(40,104)(41,52)(42,110)(46,115)(49,117)(51,111)(55,76)(56,59)(57,121)(58,107)(72,118)(94,113)(97,125)(112,114);(1,3)(2,9)(4,22)(5,28)(6,33)(7,37)(8,20)(10,39)(11,46)(12,49)(13,51)(14,52)(15,54)(16,60)(17,65)(18,38)(19,73)(21,76)(23,82)(24,83)(25,90)(26,81)(27,94)(29,98)(30,101)(31,74)(32,104)(34,64)(35,62)(36,41)(40,105)(42,103)(43,115)(44,117)(45,111)(47,118)(48,97)(50,114)(53,93)(55,78)(56,79)(57,124)(58,77)(59,123)(61,91)(63,100)(66,88)(67,92)(68,122)(69,119)(70,126)(71,99)(72,108)(75,102)(80,107)(84,85)(86,120)(87,127)(89,128)(95,113)(96,110)(106,121)(109,112)(116,125)

Polycyclic presentation:

2^12|a^2=d,d^2=g,f^2=h,g^2=i,h^2=j,j^2=k,k^2=l,b^a=bi,c^a=cf,c^b=ce,d^c=djkl,e^a=el,f^a=fh,f^c=fhjkl,f^d=fk,g^c=gkl,g^f=gl,h^a=hj,h^c=hjkl,h^d=hl,i^c=il,j^a=jk,j^c=jkl,k^a=kl,k^c=kl

Conjugacy classes & character table

184 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.4);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.10*F.11*F.12);;
gap> SetConjugate(coll, 5, 1, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11*F.12);;
gap> SetConjugate(coll, 7, 6, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11*F.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> SetConjugate(coll, 11, 3, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);