#72253909

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#2134 ⋊ C₂
Sylow	C₃ ⋊ #498909

Subgroups

	subgroup	G/H
center	C₂²	C₂×#4741
commutator, cosocle	C₂²×C₃×Q₈	C₂⁴
Frattini	C₂²×Q₈	C₂³×D₃
Fitting	C₂×C₃×#2134	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
hypercenter	C₂×#2134	D₃
hypocenter	C₃	#498909
LCS	G ▷ C₂²×C₃×Q₈ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#2134

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:

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

Polycyclic presentation:

2^9,3|a^2=e,b^2=i,e^2=i,f^2=i,b^a=bei,c^a=cefi,c^b=cefgi,d^c=dhi,e^b=ei,e^c=eh,f^a=fhi,f^c=fhi,f^d=fi,f^e=fi,h^a=hi,h^b=hi,j^b=j^2

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72253930, #72253950, D₃×#498909
maximal normal subgroup of	#72253930, #72253950, D₃×#498909

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