UT(4,𝔽₄) #72228247

order	4096
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂⁷×C₄⁴

Aliases

unitriangular

UT(4,𝔽₄)

Subgroups

	subgroup	G/H
center, socle	C₂²	#72209699
commutator, Frattini, cosocle	C₂⁶	C₂⁶
LCS	G ▷ C₂⁶ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

159694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 64 points:

(1,2)(3,9)(4,10)(5,11)(6,13)(7,14)(8,12)(15,27)(16,28)(17,30)(18,29)(19,33)(20,32)(21,31)(22,34)(23,35)(24,36)(25,37)(26,38)(39,52)(40,49)(41,51)(42,50)(43,55)(44,53)(45,54)(46,57)(47,58)(48,56)(59,64)(60,62)(61,63);(1,3)(2,9)(4,15)(5,16)(6,26)(7,18)(8,17)(10,27)(11,28)(12,30)(13,38)(14,29)(19,41)(20,39)(21,42)(22,40)(23,45)(24,43)(25,44)(31,50)(32,52)(33,51)(34,49)(35,54)(36,55)(37,53)(46,59)(47,60)(48,61)(56,63)(57,64)(58,62);(10,31)(11,35)(15,39)(16,43)(27,49)(28,53)(32,34)(33,58)(36,37)(40,42)(41,61)(44,45)(50,52)(51,64)(54,55)(56,57)(59,60)(62,63);(10,32)(11,36)(15,40)(16,44)(27,50)(28,54)(31,34)(33,56)(35,37)(39,42)(41,59)(43,45)(49,52)(51,62)(53,55)(57,58)(60,61)(63,64);(1,4)(2,10)(3,15)(5,19)(6,20)(7,21)(8,22)(9,27)(11,33)(12,34)(13,32)(14,31)(16,41)(17,40)(18,42)(23,46)(24,47)(25,48)(26,39)(28,51)(29,50)(30,49)(35,57)(36,58)(37,56)(38,52)(43,60)(44,61)(45,59)(53,63)(54,64)(55,62);(1,5)(2,11)(3,16)(4,19)(6,23)(7,25)(8,24)(9,28)(10,33)(12,36)(13,35)(14,37)(15,41)(17,43)(18,44)(20,46)(21,48)(22,47)(26,45)(27,51)(29,53)(30,55)(31,56)(32,57)(34,58)(38,54)(39,59)(40,60)(42,61)(49,62)(50,63)(52,64)

Polycyclic presentation:

2^12|c^a=ci,c^b=cj,d^a=dj,d^b=dij,e^c=eg,e^d=eh,f^c=fh,f^d=fgh,g^a=gk,g^b=gkl,h^a=hkl,h^b=hl,i^e=ik,i^f=ikl,j^e=jkl,j^f=jl

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72241454
Fitting subgroup of	#72241454
maximal subgroup of	#72241454
maximal normal subgroup of	#72241454

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

UT(4,𝔽4) #72228247