#72248281

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#23895 ⋊ C₂
Sylow	C₃ ⋊ #511793

Subgroups

	subgroup	G/H
center	C₂	#10943957
commutator, cosocle	C₂²×C₃×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#23895	C₂
socle	C₂×C₃	#32898
hypercenter	#23895	D₃
hypocenter	C₃	#511793
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ #23895

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:

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

Polycyclic presentation:

2^9,3|a^2=hi,b^2=ef,g^2=i,b^a=be,c^a=ci,c^b=cef,d^b=degi,d^c=di,e^a=ei,e^b=ei,e^c=ei,e^d=eh,f^b=fi,f^c=fi,f^d=fh,g^a=gh,g^b=gi,g^c=gi,g^d=gh,g^e=gi,g^f=gi,h^b=hi,j^d=j^2

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#511793
maximal normal subgroup of	D₃×#511793
maximal quotient of	#72248482

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