#72270735

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12966 ⋊ C₂
Sylow	C₃ ⋊ #512119

Subgroups

	subgroup	G/H
center	C₂²	D₃×B(4,𝔽₂)
commutator, cosocle	C₂²×C₃×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#12966	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
prime radical	#10826821	C₂
hypercenter	#12966	D₃
hypocenter	C₃	#512119
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ #12966

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:

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

Polycyclic presentation:

2^9,3|a^2=g,b^2=gi,c^2=efi,f^2=i,b^a=be,c^a=cei,c^b=cefi,d^b=dgi,d^c=dghi,e^c=eh,f^a=fh,f^c=fh,f^d=fi,f^e=fi,h^b=hi,h^c=hi,j^d=j^2

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270741, #72270771, D₃×#512119
maximal normal subgroup of	#72270741, #72270771, D₃×#512119

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