#72270553

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#7307 ⋊ C₂
Sylow	C₃ ⋊ #128057

Subgroups

	subgroup	G/H
center	C₂²	D₃×#342
commutator	C₂²×C₃×C₄	C₂×C₄²
Frattini	C₂³×D₄	C₂²×D₃
Fitting	C₃×#7307	C₂
socle	C₂²×C₃	C₂×#342
prime radical	C₂³×#237	C₂²
cosocle	C₂³×C₃×D₄	C₂³
hypercenter	#7307	D₃
hypocenter	C₃	#128057
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #7307

3132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 27 points:

(2,3)(6,11)(8,10)(12,14,16,22)(13,18,20,27)(15,24,23,21)(17,25,19,26);(1,2,3)(4,6)(5,8,9,10)(7,11)(13,17)(15,23)(18,26)(19,20)(21,24)(25,27);(4,5)(6,10)(7,9)(8,11)(12,13)(14,18)(15,19,23,17)(16,20)(21,25,24,26)(22,27)

Polycyclic presentation:

2^9,3|a^2=e,b^2=d,c^2=i,f^2=i,b^a=bf,c^b=cgh,d^a=dg,e^b=eh,f^a=fhi,f^b=fgi,f^c=fi,f^e=fi,h^a=hi,j^c=j^2

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#128057
maximal normal subgroup of	D₃×#128057

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