#72254117

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#2120 ⋊ C₂
Sylow	C₃ ⋊ #147573

Subgroups

	subgroup	G/H
center	C₂²	#70035
commutator	C₂³×C₃×C₄	C₂²×C₄
Frattini	C₂²×#41	C₂²×D₃
Fitting	C₂×C₃×#2120	C₂
socle	C₂²×C₃	#1781
prime radical	#10978163	C₂
cosocle	C₂²×C₃×#41	C₂³
hypercenter	C₂×#2120	D₃
hypocenter	C₃	#147573
LCS	G ▷ C₂³×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#2120

3750 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:

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

Polycyclic presentation:

2^9,3|a^2=h,b^2=dfi,c^2=deh,f^2=i,g^2=i,b^a=bd,c^a=cde,c^b=cg,d^a=di,d^b=dh,d^c=di,e^a=ei,e^b=eh,e^c=ei,f^a=fhi,f^b=fh,f^c=fhi,f^d=fi,f^e=fi,g^a=gi,g^b=ghi,g^f=gi,h^b=hi,j^c=j^2

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72254126, #72254161, D₃×#147573, …
maximal normal subgroup of	#72254126, #72254161, D₃×#147573, …

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.4*F.6*F.9);;
gap> SetPower(coll, 3, F.4*F.5*F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
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.8*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);