#72254142

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#20618 ⋊ C₂
Sylow	C₃ ⋊ #494622

Subgroups

	subgroup	G/H
center	C₂²	C₂×#5032
commutator	C₂×C₃×Q₈	C₂³×C₄
Frattini	C₂²×Q₈	C₂³×D₃
Fitting	C₃×#20618	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
prime radical	C₂×#84122	C₂
cosocle	C₂²×C₃×Q₈	C₂⁴
hypercenter	#20618	D₃
hypocenter	C₃	#494622
LCS	G ▷ C₂×C₃×Q₈ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ #20618

4056 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:

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

Polycyclic presentation:

2^9,3|a^2=hi,b^2=g,c^2=egi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=ci,c^b=ce,d^a=dhi,d^b=di,d^c=di,e^a=eh,e^b=ei,f^b=fhi,f^c=fhi,f^d=fi,f^e=fi,g^a=gh,g^f=gi,h^b=hi,j^a=j^2

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72254153, D₃×#494622, #72269931
maximal normal subgroup of	#72254153, D₃×#494622, #72269931

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