#72247980

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #146656

Subgroups

	subgroup	G/H
center	C₂²	#72142
commutator	C₂×C₃×Q₈	C₂×C₄²
Frattini	C₂²×#41	C₂²×D₃
Fitting	C₃×#34415	C₂
socle	C₂²×C₃	#1790
prime radical	C₂×C₃×#535	C₂²
cosocle	C₂²×C₃×#41	C₂³
hypercenter	#34415	D₃
hypocenter	C₃	#146656
LCS	G ▷ C₂×C₃×Q₈ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ #34415

2256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9,3|a^2=h,b^2=e,c^2=d,f^2=i,g^2=i,b^a=bf,c^a=ch,c^b=cg,d^b=di,e^a=ei,e^c=eh,f^c=fhi,g^a=ghi,g^b=ghi,g^e=gi,g^f=gi,h^b=hi,j^b=j^2

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#146656, #72248059, #72254539
maximal normal subgroup of	D₃×#146656, #72248059, #72254539
maximal quotient of	#72270433

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