#72255505

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#3021 ⋊ C₂
Sylow	C₃ ⋊ #498909

Subgroups

	subgroup	G/H
center	C₂²	C₂×#5032
commutator, cosocle	C₂²×C₃×Q₈	C₂⁴
Frattini	C₂²×Q₈	C₂³×D₃
Fitting	C₂×C₃×#3021	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
hypercenter	C₂×#3021	D₃
hypocenter	C₃	#498909
LCS	G ▷ C₂²×C₃×Q₈ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#3021

4618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

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

Polycyclic presentation:

2^9,3|a^2=g,b^2=e,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=cei,d^a=dhi,d^b=dhi,d^c=dhi,e^a=eh,e^c=ei,e^d=ei,f^a=fi,f^b=fhi,f^c=fhi,f^d=fi,f^e=fi,h^b=hi,j^a=j^2

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#498909, #72270902, #72270922
maximal normal subgroup of	D₃×#498909, #72270902, #72270922

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