#72255335

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12581 ⋊ C₂
Sylow	C₃ ⋊ #146488

Subgroups

	subgroup	G/H
center	C₂²	#70035
commutator	C₂²×C₃×C₄	C₂×C₄²
Frattini	C₂²×#41	C₂²×D₃
Fitting	C₃×#12581	C₂
socle	C₂²×C₃	#1781
prime radical	C₂²×#1009	C₂²
cosocle	C₂²×C₃×#41	C₂³
hypercenter	#12581	D₃
hypocenter	C₃	#146488
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ #12581

3614 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:

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

Polycyclic presentation:

2^9,3|a^2=f,b^2=efg,d^2=i,e^2=i,b^a=be,c^a=chi,c^b=cd,d^b=di,d^c=di,e^a=ei,e^b=eh,e^c=ei,g^a=gh,g^b=gh,g^c=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×#146488, #72255524, #72270962
maximal normal subgroup of	D₃×#146488, #72255524, #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, 1, F.6);;
gap> SetPower(coll, 2, F.5*F.6*F.7);;
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.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
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.8);;
gap> SetConjugate(coll, 5, 3, F.5*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.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);