#72225844

order	1536
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #126544

Subgroups

	subgroup	G/H
center	C₄²	C₄²×D₃
commutator	C₂×C₃×C₄	C₄³
Frattini	C₂×C₄×C₈	C₂²×D₃
Fitting	C₃×C₄×#347	C₂
socle	C₂²×C₃	C₄×#97
prime radical	C₂³×C₃	C₄×M₁₆
cosocle	C₂×C₃×C₄×C₈	C₂³
hypercenter	C₄×#347	D₃
hypocenter	C₃	#126544
LCS	G ▷ C₂×C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₄² ◁ C₄×#347

708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 384

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

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

Polycyclic presentation:

2^9,3|a^2=d,b^2=f,c^2=e,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bhi,c^a=cgi,d^b=di,d^c=di,e^a=ei,f^a=fi,j^b=j^2

Conjugacy classes & character table

216 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#126544
maximal normal subgroup of	D₃×#126544
maximal quotient of	#72226214

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