#72270598

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#2684 ⋊ C₂
Sylow	C₃ ⋊ #508087

Subgroups

	subgroup	G/H
center	C₂²	C₂×#5244
commutator, cosocle	C₂²×C₃×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₂×C₃×#2684	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
prime radical	#10826764	C₂
hypercenter	C₂×#2684	D₃
hypocenter	C₃	#508087
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ C₂×#2684

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:

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

Polycyclic presentation:

2^9,3|a^2=h,b^2=ghi,f^2=i,b^a=bg,c^a=ce,c^b=cefi,d^c=dghi,e^a=ei,e^b=eh,f^a=fhi,f^b=fh,f^e=fi,g^c=gi,h^c=hi,j^c=j^2

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270609, #72270770, D₃×#508087
maximal normal subgroup of	#72270609, #72270770, 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);;
gap> SetPower(coll, 2, F.7*F.8*F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);