#72271900

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12753 ⋊ C₂
Sylow	C₃ ⋊ #493635

Subgroups

	subgroup	G/H
center	C₂	#10794452
commutator	C₂×C₃×D₄	C₂³×C₄
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#12753	C₂
socle	C₂×C₃	#20609
prime radical	#11707698	C₂
cosocle	C₂²×C₃×D₄	C₂⁴
hypercenter	#12753	D₃
hypocenter	C₃	#493635
LCS	G ▷ C₂×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×D₄ ◁ #12753

5326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 4 generators on 19 points:

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

Polycyclic presentation:

2^9,3|b^2=eg,d^2=h,f^2=i,b^a=be,c^b=cefi,d^b=df,d^c=dh,e^b=ei,e^c=eh,e^d=eh,f^a=fhi,f^b=fhi,f^c=fh,f^e=fi,g^a=gi,g^b=gi,h^b=hi,j^d=j^2

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#493635, #72271950, #72272102
maximal normal subgroup of	D₃×#493635, #72271950, #72272102

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