#72255320

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12581 ⋊ C₂
Sylow	C₃ ⋊ #147787

Subgroups

	subgroup	G/H
center	C₂²	#70033
commutator	C₂²×C₃×Q₈	C₂²×C₄
Frattini	C₂²×#41	C₂²×D₃
Fitting	C₃×#12581	C₂
socle	C₂²×C₃	#1790
prime radical	#11017267	C₂
cosocle	C₂²×C₃×#41	C₂³
hypercenter	#12581	D₃
hypocenter	C₃	#147787
LCS	G ▷ C₂²×C₃×Q₈ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ #12581

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

Polycyclic presentation:

2^9,3|a^2=f,b^2=efg,d^2=i,e^2=i,b^a=be,c^a=cf,c^b=cd,d^a=dh,d^b=di,d^c=di,e^a=ei,e^b=ehi,e^c=ehi,e^d=ei,g^a=ghi,g^b=ghi,g^c=ghi,g^d=gi,g^e=gi,h^b=hi,j^c=j^2

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#147787, #72255519, #72270962
maximal normal subgroup of	D₃×#147787, #72255519, #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.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
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*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*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, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 4, 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);