#72253375

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#1783 ⋊ C₂
Sylow	C₃ ⋊ #418876

Subgroups

	subgroup	G/H
center	C₂²	#73857
commutator, cosocle	C₂³×C₃×C₄	C₂⁴
Frattini	C₂³×C₄	C₂³×D₃
Fitting	C₂×C₃×#1783	C₂
socle	C₂²×C₃	#2668
prime radical	C₂×#84038	C₂
hypercenter	C₂×#1783	D₃
hypocenter	C₃	#418876
LCS	G ▷ C₂³×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#1783

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

Polycyclic presentation:

2^9,3|a^2=h,b^2=ef,e^2=i,h^2=i,b^a=be,c^a=cfg,c^b=cg,d^a=dfgh,d^b=deghi,e^a=ei,e^b=ei,e^d=ei,f^a=fi,f^b=fi,g^a=gi,h^d=hi,j^a=j^2

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#418876, #72255104, #72271554
maximal normal subgroup of	D₃×#418876, #72255104, #72271554

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