#72248484

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#10745 ⋊ C₂
Sylow	C₃ ⋊ #147819

Subgroups

	subgroup	G/H
center	C₂²	#69598
commutator	C₂²×C₃×D₄	C₂²×C₄
Frattini	C₂³×D₄	C₂²×D₃
Fitting	C₃×#10745	C₂
socle	C₂²×C₃	#1790
prime radical	#11002551	C₂
cosocle	C₂³×C₃×D₄	C₂³
hypercenter	#10745	D₃
hypocenter	C₃	#147819
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #10745

2582 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:

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

Polycyclic presentation:

2^9,3|a^2=h,b^2=ef,c^2=d,g^2=i,b^a=be,c^a=cd,c^b=cfgi,e^a=ei,e^b=ei,e^c=eh,f^b=fi,f^c=fh,g^a=ghi,g^b=gi,g^c=gh,g^e=gi,g^f=gi,h^b=hi,j^a=j^2

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#147819, #72248514, #72248542
maximal normal subgroup of	D₃×#147819, #72248514, #72248542
maximal quotient of	#72269990

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