#72258706

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×AGL(1,ℤ/16) ⋊ C₂
Sylow	C₃ ⋊ #495850

Subgroups

	subgroup	G/H
center	C₂²	#79869
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₂×C₃×AGL(1,ℤ/16)	C₂
socle	C₂²×C₃	#3109
prime radical	C₂×#71514	C₂
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	C₂×AGL(1,ℤ/16)	D₃
hypocenter	C₃	#495850
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×AGL(1,ℤ/16)

1518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 23 points:

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

Polycyclic presentation:

2^9,3|c^2=e,d^2=g,f^2=hi,h^2=i,b^a=bf,c^a=cfh,d^a=dhi,d^b=dg,d^c=dg,e^a=ei,f^a=fh,f^b=fh,f^c=fhi,g^a=gi,h^a=hi,h^b=hi,h^c=hi,j^d=j^2

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#495850
maximal normal subgroup of	D₃×#495850

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