#72226005

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₈

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#21690 ⋊ C₂
Sylow	C₃ ⋊ #212918

Subgroups

	subgroup	G/H
center	C₂³	C₂×#940
commutator	C₃×C₄²	C₂³×C₄
Frattini	C₂×C₄²	C₂³×D₃
Fitting	C₃×#21690	C₂
socle	C₂³×C₃	C₂×#127
prime radical	C₂×#71276	C₂
cosocle	C₂×C₃×C₄²	C₂⁴
hypercenter	#21690	D₃
hypocenter	C₃	#212918
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #21690

2452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 768

A minimal faithful permutation representation — 4 generators on 23 points:

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

Polycyclic presentation:

2^9,3|a^2=e,c^2=eh,d^2=g,f^2=h,g^2=i,b^a=bf,c^a=cfh,d^a=dg,d^b=dgi,d^c=dg,f^a=fh,f^b=fh,f^c=fh,g^a=gi,g^b=gi,g^c=gi,j^b=j^2,j^c=j^2

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#212918
maximal normal subgroup of	D₃×#212918
maximal quotient of	#72258843

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