#72270474

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12988 ⋊ C₂
Sylow	C₃ ⋊ #500168

Subgroups

	subgroup	G/H
center	C₂	#10828142
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#12988	C₂
socle	C₂×C₃	#24033
prime radical	#11671078	C₂
hypercenter	#12988	D₃
hypocenter	C₃	#500168
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2932 ◁ #12988

1956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 35 points:

(2,3)(6,16)(7,22)(8,23)(10,25)(11,29)(12,14)(15,32)(17,18)(19,33)(20,30)(21,31)(24,34)(26,27)(28,35);(1,2,3)(6,15,19,18)(7,21)(10,24,28,27)(11,20)(16,32,33,17)(22,31)(25,34,35,26)(29,30);(4,6,7,17,23,15,31,33,9,19,21,32,8,18,22,16)(5,10,11,26,12,27,30,25,13,28,20,34,14,24,29,35);(4,5)(6,10)(7,20)(8,14)(9,13)(11,21)(12,23)(15,27)(16,25)(17,34)(18,24)(19,28)(22,30)(26,32)(29,31)(33,35)

Polycyclic presentation:

2^9,3|b^2=e,c^2=gi,e^2=hi,f^2=i,h^2=i,b^a=bg,c^a=cgi,c^b=cf,d^b=defhi,e^a=ei,e^c=ei,e^d=eh,f^a=fi,f^d=fi,g^b=gi,h^d=hi,j^d=j^2

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#500168, #72271460
maximal normal subgroup of	D₃×#500168, #72271460

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