#72272023

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#21776 ⋊ C₂
Sylow	C₃ ⋊ #493480

Subgroups

	subgroup	G/H
center	C₂	#10794323
commutator	C₂²×C₃×C₄	C₂³×C₄
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#21776	C₂
socle	C₂×C₃	#20585
prime radical	#11704065	C₂
cosocle	C₂²×C₃×D₄	C₂⁴
hypercenter	#21776	D₃
hypocenter	C₃	#493480
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #3484 ◁ #21776

5234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 4 generators on 19 points:

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

Polycyclic presentation:

2^9,3|b^2=e,d^2=h,f^2=i,c^a=ch,c^b=cg,d^b=df,e^d=eh,f^b=fhi,f^e=fi,g^a=gi,h^b=hi,j^d=j^2

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72272028, #72272110, D₃×#493480
maximal normal subgroup of	#72272028, #72272110, D₃×#493480

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, 2, F.5);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 4, 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, 7, 1, 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);