#72255121

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12852 ⋊ C₂
Sylow	C₃ ⋊ #508087

Subgroups

	subgroup	G/H
center	C₂²	C₂×#4741
commutator, cosocle	C₂²×C₃×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#12852	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
prime radical	#10826769	C₂
hypercenter	#12852	D₃
hypocenter	C₃	#508087
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #12852

3304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 27 points:

(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,20)(13,14)(15,19)(17,18)(21,22)(23,24)(25,26,27);(2,5)(3,7)(9,11)(10,15,18,23)(12,21,13,20)(14,22)(16,19)(17,24)(26,27);(1,2,4,5)(3,6,7,8)(15,23)(16,17)(19,24)(20,21);(3,7)(6,8)(9,10,12,17)(11,19,21,15)(13,16,14,18)(20,23,22,24)

Polycyclic presentation:

2^9,3|a^2=e,b^2=g,c^2=ef,f^2=i,b^a=befi,c^a=cf,c^b=cefg,d^a=dgi,d^b=dh,d^c=dgh,e^b=eh,e^c=ehi,f^a=fh,f^b=fh,f^c=fhi,f^d=fi,f^e=fi,h^a=hi,h^c=hi,j^c=j^2

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255171, D₃×#508087, #72270790
maximal normal subgroup of	#72255171, D₃×#508087, #72270790

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