#72253973

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#3025 ⋊ C₂
Sylow	C₃ ⋊ #498929

Subgroups

	subgroup	G/H
center	C₂²	C₂×#5032
commutator, cosocle	C₂²×C₃×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₂×C₃×#3025	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
hypercenter	C₂×#3025	D₃
hypocenter	C₃	#498929
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#3025

4612 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:

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

Polycyclic presentation:

2^9,3|b^2=i,e^2=i,b^a=bg,c^a=cf,c^b=ce,d^a=dh,e^a=eh,e^b=ei,e^c=ei,f^b=fi,g^c=gh,g^d=gi,g^e=gi,h^b=hi,j^c=j^2

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#498929, #72270922, #72270962
maximal normal subgroup of	D₃×#498929, #72270922, #72270962

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