#72224495

order	12960
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, Sylow	C₃⁴ ⋊ #3930
Hall	C₃⁴⋊C₅ ⋊ M₃₂

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator	(C₃⋊(C₃⋊(C₃⋊D₃)))⋊C₅	C₂×C₈
Fitting, socle	C₃⁴	#3930
prime radical	#11717694⋊C₂	C₈
cosocle	#72210626	C₂²
hypocenter	C₃⁴⋊C₅	M₃₂
LCS	G ▷ (C₃⋊(C₃⋊(C₃⋊D₃)))⋊C₅ ▷ C₃⁴⋊C₅
UCS	C₁

86 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 81

A minimal faithful permutation representation — 2 generators on 81 points:

(1,57,62,66,64,25)(2,22,75,19,37,72)(3,51,65,36,4,69)(5,76,77,12,52,11)(6,40,55,28,39,68)(7,32,78)(8,74,30,17,45,9)(10,43,29)(13,14,38,80,71,27)(15,81,18,58,16,60)(20,35,61,34,56,21)(23,53,44,63,54,24)(26,41,50,73,48,42)(31,47,79,59,70,67)(33,46,49);(2,23,68,77,38,31,17,46,56,29,11,53,62,43,4,52,5,67,33,50,9,55,40,47,39,32,49,69,45,36,16,81,28,71,21,30,26,65,13,63,10,74,72,80,42,51,54,66,24,37,8,61,64,75,15,14,27,60,57,35,44,78,19,58,7,6,25,79,76,70,3,59,22,73,48,12,34,18,41,20)

Polycyclic presentation:

2^5,5,3^4|a^2=c,c^2=d,d^2=e,b^a=be,f^b=f^4,g^a=g^2i,g^b=g^2hj,g^c=gj^2,g^d=j,g^e=g^2,g^f=g^2h^2,h^a=hi^2j^2,h^b=i^2,h^c=g^2i^2j,h^d=ghij^2,h^e=h^2,h^f=gi,i^a=ij^2,i^b=h^2,i^c=g^2h^2i^2,i^d=ghi^2,i^e=i^2,i^f=hj,j^a=ghi^2j^2,j^b=hij,j^c=gj,j^d=g^2,j^e=j^2,j^f=g^2h^2i^2

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AΓL(1,𝔽₈₁)
maximal normal subgroup of	AΓL(1,𝔽₈₁)

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