#72270889

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #147813

Subgroups

	subgroup	G/H
center	C₂²	#72142
commutator	C₂²×C₃×D₄	C₂²×C₄
Frattini	C₂²×#41	C₂²×D₃
Fitting, prime radical	C₃×#37269	C₂
socle	C₂²×C₃	#1790
cosocle	C₂²×C₃×#41	C₂³
hypercenter	#37269	D₃
hypocenter	C₃	#147813
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ #37269

2412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3)(5,7)(9,14)(10,17)(11,20)(13,22)(15,16)(19,23);(2,3)(5,7)(8,9)(10,14)(11,19,20,13)(12,16)(15,17)(18,23,21,22);(4,5)(6,7)(9,13)(14,22)(15,23)(16,19)

Polycyclic presentation:

2^9,3|b^2=e,d^2=i,b^a=bf,c^a=cfg,c^b=cd,d^a=dh,d^b=di,d^c=di,e^a=eh,f^b=fh,f^c=fhi,f^d=fi,f^e=fi,g^b=gh,g^c=ghi,g^d=gi,g^e=gi,h^b=hi,j^b=j^2

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270902, D₃×#147813, #72270949
maximal normal subgroup of	#72270902, D₃×#147813, #72270949

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