#72322015

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#72225501
commutator	C₂×C₃₂	C₂⁴×C₄
Frattini, cosocle	#1397	C₂⁵
prime radical	#72321842	C₂
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ #479899 ◁ #72231854 ◁ G
LECS	G ▷ #1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

3896 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 128 points:

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

Polycyclic presentation:

2^12|d^2=g,e^2=hj,f^2=hj,g^2=i,h^2=jk,j^2=kl,k^2=l,b^a=bi,c^a=cf,c^b=cl,d^a=dk,d^c=djk,e^a=eh,e^c=eh,e^d=ekl,f^a=fh,f^b=fl,f^c=fh,f^d=fkl,g^a=gl,g^c=gk,g^e=gl,g^f=gl,h^a=hj,h^c=hj,h^d=hl,i^c=il,j^a=jk,j^c=jk,k^a=kl,k^c=kl

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323149
maximal normal subgroup of	#72323149

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