#72225876

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#8822 ⋊ C₂
Sylow	C₃ ⋊ #213274

Subgroups

	subgroup	G/H
center	C₂×C₄	D₃×#127
commutator	C₃×C₄²	C₂³×C₄
Frattini	C₂×C₄²	C₂³×D₃
Fitting	C₃×#8822	C₂
socle	C₂²×C₃	C₄×#127
prime radical	#10782888	C₂
cosocle	C₂×C₃×C₄²	C₂⁴
hypercenter	#8822	D₃
hypocenter	C₃	#213274
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #8822

1612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 27 points:

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

Polycyclic presentation:

2^9,3|a^2=e,b^2=egi,c^2=f,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bg,c^a=cf,c^b=ch,d^a=dgi,d^b=di,f^a=fh,g^a=gi,j^b=j^2

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225880, #72226144
maximal normal subgroup of	#72225880, #72226144
maximal quotient of	#72226144

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