#72248833

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#8101 ⋊ C₂
Sylow	C₃ ⋊ #106766

Subgroups

	subgroup	G/H
center	C₂²	(C₂×D₃)⋊#95
commutator	C₂⁴×C₃	C₂×C₄²
Frattini	C₂⁶	C₂²×D₃
Fitting	C₃×#8101	C₂
socle	C₂²×C₃	C₂²⋊#95
prime radical	C₂⁴×#67	C₂²
cosocle	C₂⁶×C₃	C₂³
hypercenter	#8101	D₃
hypocenter	C₃	#106766
LCS	G ▷ C₂⁴×C₃ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ #8101

5464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 96

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

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

Polycyclic presentation:

2^9,3|a^2=d,b^2=de,c^2=i,b^a=bf,c^a=cg,d^c=di,e^a=eh,e^c=ei,f^b=fh,f^c=fi,g^a=gi,g^b=gi,j^a=j^2,j^c=j^2

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72248844, D₃×#106766, #72253433
maximal normal subgroup of	#72248844, D₃×#106766, #72253433
maximal quotient of	#72253527, #72271380

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