#72270634

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#1393 ⋊ C₂
Sylow	C₃ ⋊ #147779

Subgroups

	subgroup	G/H
center	C₂³	#4741
commutator	C₂²×C₃×D₄	C₂²×C₄
Frattini	C₂³×D₄	C₂²×D₃
Fitting	C₂×C₃×#1393	C₂
socle	C₂³×C₃	B(4,𝔽₂)
prime radical	C₂×#76464	C₂
cosocle	C₂³×C₃×D₄	C₂³
hypercenter	C₂×#1393	D₃
hypocenter	C₃	#147779
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×D₄ ◁ C₂×#1393

4592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 19 points:

(2,3)(9,11)(16,18);(1,2,3)(4,5,6,7)(12,13)(14,16,19,18)(15,17);(8,9,10,11)(13,16,17,18)

Polycyclic presentation:

2^9,3|a^2=e,b^2=def,d^2=i,b^a=bd,c^a=ce,c^b=ceg,d^b=dh,d^c=dhi,e^b=ei,f^a=fhi,f^b=fhi,f^c=fhi,f^d=fi,g^a=ghi,g^b=gi,g^d=gi,g^f=gi,h^b=hi,j^c=j^2

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#147779
maximal normal subgroup of	D₃×#147779
maximal quotient of	#72270635

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