#72248282

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12707 ⋊ C₂
Sylow	C₃ ⋊ #511793

Subgroups

	subgroup	G/H
center	C₂	#10943960
commutator, cosocle	C₂²×C₃×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#12707	C₂
socle	C₂×C₃	#32898
hypercenter	#12707	D₃
hypocenter	C₃	#511793
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ #12707

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#511793
maximal normal subgroup of	D₃×#511793
maximal quotient of	#72248495

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