#72224482

order	12096
automorphism	?
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, Sylow	C₂⁶ ⋊ #4428
Hall	#90631 ⋊ 3₋¹⁺²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	(C₂²⋊(C₂²⋊A₄))⋊C₇	C₃²
Fitting, socle	C₂⁶	#4428
prime radical	#72224477	C₃
hypocenter	#90631	3₋¹⁺²
LCS	G ▷ (C₂²⋊(C₂²⋊A₄))⋊C₇ ▷ #90631
UCS	C₁

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

3^3,7,2^6|b^3=c,b^a=bc,d^a=d^4,e^a=efghj,e^b=efg,e^c=egh,e^d=eh,f^a=j,f^b=fgij,f^c=fj,f^d=fg,g^a=fh,g^b=ef,g^c=ghij,g^d=efghi,h^a=eh,h^b=efgi,h^c=eghij,h^d=fgj,i^a=fghij,i^b=fhj,i^c=efhij,i^d=gij,j^a=fj,j^b=ei,j^c=f,j^d=hi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AΓL(1,𝔽₆₄)
maximal normal subgroup of	AΓL(1,𝔽₆₄)

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");;
gap> coll := SingleCollector(F, [3, 3, 3, 7, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.5*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 8, 1, F.5*F.8);;
gap> SetConjugate(coll, 8, 2, F.5*F.6*F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.6*F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.6*F.10);;
gap> SetConjugate(coll, 10, 2, F.5*F.9);;
gap> SetConjugate(coll, 10, 3, F.6);;
gap> SetConjugate(coll, 10, 4, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);