#72248355

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12574 ⋊ C₂
Sylow	C₃ ⋊ #147811

Subgroups

	subgroup	G/H
center	C₂²	#70033
commutator	C₂²×C₃×D₄	C₂²×C₄
Frattini	C₂³×D₄	C₂²×D₃
Fitting	C₃×#12574	C₂
socle	C₂²×C₃	#1790
prime radical	#11001922	C₂
cosocle	C₂³×C₃×D₄	C₂³
hypercenter	#12574	D₃
hypocenter	C₃	#147811
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #12574

3254 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 23 points:

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

Polycyclic presentation:

2^9,3|a^2=d,b^2=fgi,g^2=i,b^a=bdf,c^a=cdh,c^b=ce,d^b=di,e^a=eh,f^a=fi,f^b=fh,f^c=fhi,f^e=fi,g^a=gh,g^b=gh,g^c=ghi,g^e=gi,g^f=gi,h^b=hi,j^c=j^2

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72248361, #72248464, D₃×#147811
maximal normal subgroup of	#72248361, #72248464, D₃×#147811
maximal quotient of	#72253679

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