#72248479

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#1913 ⋊ C₂
Sylow	C₃ ⋊ #147819

Subgroups

	subgroup	G/H
center	C₂²	#69598
commutator	C₂²×C₃×D₄	C₂²×C₄
Frattini	C₂³×D₄	C₂²×D₃
Fitting	C₂×C₃×#1913	C₂
socle	C₂²×C₃	#1790
prime radical	#11002553	C₂
cosocle	C₂³×C₃×D₄	C₂³
hypercenter	C₂×#1913	D₃
hypocenter	C₃	#147819
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ C₂×#1913

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

Polycyclic presentation:

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

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72248482, D₃×#147819, #72248538
maximal normal subgroup of	#72248482, D₃×#147819, #72248538
maximal quotient of	#72270467

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