#72255128

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #383145

Subgroups

	subgroup	G/H
center	C₂²	#82608
commutator	C₂²×C₃×C₄	C₂³×C₄
Frattini	C₂³×C₄	C₂³×D₃
Fitting, prime radical	C₂×C₃×#3003	C₂
socle	C₂²×C₃	#2668
cosocle	C₂³×C₃×C₄	C₂⁴
hypercenter	C₂×#3003	D₃
hypocenter	C₃	#383145
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#3003

4984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 23 points:

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

Polycyclic presentation:

2^9,3|a^2=f,b^2=e,e^2=i,f^2=i,h^2=i,b^a=bfi,c^a=cfhi,c^b=cg,d^a=dh,d^b=dfghi,e^d=ei,f^b=fi,f^d=fi,g^a=gi,h^d=hi,j^b=j^2

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#383145, #72269869, #72271534
maximal normal subgroup of	D₃×#383145, #72269869, #72271534

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