#72224504

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, Sylow

C₃⁴ ⋊ #365

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator	#10620192	C₂×C₄
Fitting, socle, hypocenter	C₃⁴	#365
prime radical	C₃⋊D₃²	SD₁₆
cosocle	#13049788	C₂²
LCS	G ▷ #10620192 ▷ #66472 ▷ C₃⋊(C₃⋊(C₃⋊D₃)) ▷ C₃⁴
UCS	C₁

99 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 2 generators on 18 points:

(1,9,8,10)(2,17,7,11,18,6)(4,14,12,5)(13,16,15);(1,2)(3,6,5,13,9,15,14,18,8,16,4,11,12,17,10,7)

Polycyclic presentation:

2^6,3^4|a^2=d,b^2=c,d^2=e,e^2=f,b^a=bdef,c^a=cf,d^b=de,e^b=ef,g^a=i,g^b=h,g^c=g^2hj^2,g^d=ij^2,g^e=gi^2j,g^f=g^2,h^a=gj,h^b=g^2hj^2,h^c=h^2i^2j^2,h^d=gh^2i^2j^2,h^e=g^2h^2ij,h^f=h^2,i^a=ij^2,i^b=ghi^2j,i^c=h^2i^2j,i^d=g^2h,i^e=gh^2i,i^f=i^2,j^a=gh^2ij^2,j^b=g^2hi,j^c=hi^2,j^d=ghij^2,j^e=g^2h^2i^2j^2,j^f=j^2

Conjugacy classes & character table

21 conjugacy classes — view the full character table.

Relationships

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