#72270591

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#2380 ⋊ C₂
Sylow	C₃ ⋊ #512119

Subgroups

	subgroup	G/H
center	C₂²	C₂×#5244
commutator, cosocle	C₂²×C₃×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₂×C₃×#2380	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
prime radical	#10826818	C₂
hypercenter	C₂×#2380	D₃
hypocenter	C₃	#512119
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ C₂×#2380

2960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270609, D₃×#512119, #72270790
maximal normal subgroup of	#72270609, D₃×#512119, #72270790

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