#72270615

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13348 ⋊ C₂
Sylow	C₃ ⋊ #508087

Subgroups

	subgroup	G/H
center	C₂²	D₃×B(4,𝔽₂)
commutator, cosocle	C₂²×C₃×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#13348	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
prime radical	#10826765	C₂
hypercenter	#13348	D₃
hypocenter	C₃	#508087
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ #13348

3304 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)(5,9)(8,11)(13,18)(15,24)(17,25)(21,23);(1,2,3)(4,6,7,10)(5,8,9,11)(15,24)(17,25)(19,26)(20,27);(4,5)(6,8)(7,9)(10,11)(12,14)(13,17)(15,23)(16,22)(18,25)(19,27)(20,26)(21,24);(6,10)(8,11)(12,13)(14,20,22,27)(15,19)(16,18)(17,21,25,23)(24,26)

Polycyclic presentation:

2^9,3|a^2=hi,c^2=ghi,f^2=i,b^a=bf,c^a=cg,c^b=ce,d^a=dh,d^b=dghi,e^a=eh,f^b=fi,f^c=fh,f^d=fi,f^e=fi,g^b=gi,h^b=hi,j^d=j^2

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270629, #72270771, D₃×#508087
maximal normal subgroup of	#72270629, #72270771, D₃×#508087

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