#72248285

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#62863 ⋊ C₂
Sylow	C₃ ⋊ #511793

Subgroups

	subgroup	G/H
center	C₂	#10943958
commutator, cosocle	C₂²×C₃×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#62863	C₂
socle	C₂×C₃	#32898
hypercenter	#62863	D₃
hypocenter	C₃	#511793
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ #62863

4952 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:

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

Polycyclic presentation:

2^9,3|a^2=hi,b^2=e,c^2=i,g^2=i,b^a=bg,c^b=cf,d^b=def,d^c=di,e^a=ei,f^a=fh,g^c=ghi,g^d=gh,g^f=gi,h^b=hi,j^b=j^2

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#511793
maximal normal subgroup of	D₃×#511793
maximal quotient of	#72248578

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