#72248498

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13050 ⋊ C₂
Sylow	C₃ ⋊ #494600

Subgroups

	subgroup	G/H
center	C₂²	D₃×B(4,𝔽₂)
commutator	C₂×C₃×D₄	C₂³×C₄
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#13050	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
prime radical	C₂×#80292	C₂
cosocle	C₂²×C₃×D₄	C₂⁴
hypercenter	#13050	D₃
hypocenter	C₃	#494600
LCS	G ▷ C₂×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ #13050

4346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

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

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

Polycyclic presentation:

2^9,3|a^2=hi,b^2=f,c^2=ef,g^2=i,b^a=bg,c^a=ci,c^b=ce,d^a=di,e^a=eh,f^a=fh,g^b=ghi,g^c=ghi,g^e=gi,g^f=gi,h^b=hi,j^d=j^2

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#494600
maximal normal subgroup of	D₃×#494600
maximal quotient of	#72270231

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