#72255318

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #147787

Subgroups

	subgroup	G/H
center	C₂²	#72142
commutator	C₂²×C₃×Q₈	C₂²×C₄
Frattini	C₂²×#41	C₂²×D₃
Fitting, prime radical	C₃×#37297	C₂
socle	C₂²×C₃	#1790
cosocle	C₂²×C₃×#41	C₂³
hypercenter	#37297	D₃
hypocenter	C₃	#147787
LCS	G ▷ C₂²×C₃×Q₈ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ #37297

2538 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:

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

Polycyclic presentation:

2^9,3|a^2=f,b^2=d,d^2=i,e^2=i,b^a=be,c^a=cf,c^b=cdeg,d^c=di,e^a=ei,e^b=ei,e^c=eh,g^a=gh,g^b=gi,g^c=gh,g^e=gi,h^b=hi,j^b=j^2

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#147787, #72255448, #72270949
maximal normal subgroup of	D₃×#147787, #72255448, #72270949

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.6);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, 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);