#72253365

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#1903 ⋊ C₂
Sylow	C₃ ⋊ #418876

Subgroups

	subgroup	G/H
center	C₂²	#74910
commutator, cosocle	C₂³×C₃×C₄	C₂⁴
Frattini	C₂³×C₄	C₂³×D₃
Fitting	C₂×C₃×#1903	C₂
socle	C₂²×C₃	#2668
prime radical	C₂×#83944	C₂
hypercenter	C₂×#1903	D₃
hypocenter	C₃	#418876
LCS	G ▷ C₂³×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#1903

4956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9,3|a^2=f,b^2=eg,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cfhi,c^b=cg,d^a=df,d^b=defi,e^d=ei,f^b=fi,f^d=fi,g^a=gi,h^d=hi,j^c=j^2

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#418876, #72255039, #72270021
maximal normal subgroup of	D₃×#418876, #72255039, #72270021

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