#72323182

order	4096
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#181886
commutator	C₂²×C₈	C₂²×C₄×C₈
Frattini, cosocle	C₂×#1397	C₂⁴
socle	C₂	#72323179
prime radical	#3444	M₃₂
LCS	G ▷ C₂²×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#1397 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

2597 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 128 points:

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

Polycyclic presentation:

2^12|a^2=e,d^2=g,e^2=h,g^2=i,h^2=j,i^2=k,j^2=k,k^2=l,b^a=bl,c^a=cf,c^b=cijk,d^a=di,e^d=ek,f^b=fl,g^a=gk,g^e=gl,h^d=hl,i^a=il

Conjugacy classes & character table

352 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.5);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.12);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.12);;
gap> SetConjugate(coll, 9, 1, F.9*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);