#72248200

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#1783 ⋊ C₂
Sylow	C₃ ⋊ #147567

Subgroups

	subgroup	G/H
center	C₂²	#69498
commutator	C₂⁵×C₃	C₂²×C₄
Frattini	C₂³×D₄	C₂²×D₃
Fitting	C₂×C₃×#1783	C₂
socle	C₂²×C₃	#1781
prime radical	#10967723	C₂
cosocle	C₂³×C₃×D₄	C₂³
hypercenter	C₂×#1783	D₃
hypocenter	C₃	#147567
LCS	G ▷ C₂⁵×C₃ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ C₂×#1783

4902 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 23 points:

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

Polycyclic presentation:

2^9,3|a^2=d,b^2=df,d^2=i,b^a=bg,c^a=ce,c^b=cefgh,d^b=dh,d^c=di,e^a=ei,f^a=fhi,f^b=fh,f^c=fi,f^d=fi,g^a=gh,g^b=ghi,g^c=gi,g^d=gi,h^a=hi,h^b=hi,j^b=j^2

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#147567, #72248347, #72248599
maximal normal subgroup of	D₃×#147567, #72248347, #72248599
maximal quotient of	#72269885

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.4);;
gap> SetPower(coll, 2, F.4*F.6);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*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);