#72271651

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×AGL(1,ℤ/16) ⋊ C₂
Sylow	C₃ ⋊ #499269

Subgroups

	subgroup	G/H
center	C₂	#10823408
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₂×C₃×AGL(1,ℤ/16)	C₂
socle	C₂×C₃	#23573
hypercenter	C₂×AGL(1,ℤ/16)	D₃
hypocenter	C₃	#499269
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×AGL(1,ℤ/16)

1638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 35 points:

(1,2,3)(5,12,14,15)(6,18,20,21)(7,24)(10,16)(11,29,30,13)(17,22,34,19)(23,31)(26,32);(5,11)(6,17)(7,23)(8,25)(10,26)(12,29)(13,15)(14,30)(16,32)(18,22)(19,21)(20,34)(24,31)(27,33);(2,3)(4,6,7,22,25,18,31,17,9,20,24,19,8,21,23,34)(5,10,13,33,12,32,30,35,14,16,29,27,15,26,11,28);(4,5,7,13,25,12,31,30,9,14,24,29,8,15,23,11)(6,16,19,33,18,26,34,35,20,10,22,27,21,32,17,28)

Polycyclic presentation:

2^9,3|a^2=e,b^2=efg,d^2=g,e^2=hi,h^2=i,b^a=bf,c^a=ce,c^b=cegi,d^a=dh,d^b=dh,e^b=ei,e^c=eh,e^d=ei,f^a=fi,g^a=gi,g^b=gi,h^c=hi,j^b=j^2

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#499269
maximal normal subgroup of	D₃×#499269

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