#72254155

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12581 ⋊ C₂
Sylow	C₃ ⋊ #147577

Subgroups

	subgroup	G/H
center	C₂²	#70035
commutator	C₂³×C₃×C₄	C₂²×C₄
Frattini	C₂²×#41	C₂²×D₃
Fitting	C₃×#12581	C₂
socle	C₂²×C₃	#1781
prime radical	#10978440	C₂
cosocle	C₂²×C₃×#41	C₂³
hypercenter	#12581	D₃
hypocenter	C₃	#147577
LCS	G ▷ C₂³×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ #12581

3924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 23 points:

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

Polycyclic presentation:

2^9,3|a^2=d,b^2=e,f^2=i,g^2=i,b^a=bf,c^a=cd,c^b=cegi,d^b=di,e^a=eh,e^c=ehi,f^b=fhi,f^c=fi,f^e=fi,g^a=gh,g^b=ghi,g^c=ghi,g^e=gi,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	#72254161, #72254274, D₃×#147577
maximal normal subgroup of	#72254161, #72254274, D₃×#147577

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