#72253818

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#33570 ⋊ C₂
Sylow	C₃ ⋊ #511825

Subgroups

	subgroup	G/H
center	C₂	#10943956
commutator, cosocle	C₂×C₃×#41	C₂⁴
Frattini	C₂×#41	C₂³×D₃
Fitting	C₃×#33570	C₂
socle	C₂×C₃	#32898
hypercenter	#33570	D₃
hypocenter	C₃	#511825
LCS	G ▷ C₂×C₃×#41 ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #33570

4852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 35 points:

(2,3)(6,16)(7,20)(10,27)(12,19)(14,28)(15,17)(18,34)(21,24)(23,35)(26,31)(29,33)(30,32);(1,2,3)(5,11)(6,15)(7,20)(10,26)(13,22)(14,33)(16,17)(18,35)(21,24)(23,34)(27,31)(28,29);(4,6,7,15,9,18,20,35)(5,10,12,28,11,29,19,26)(8,23,24,34,25,17,21,16)(13,31,32,33,22,14,30,27);(4,5,9,11)(6,14,18,31)(7,19,20,12)(8,22,25,13)(10,23,29,17)(15,27,35,33)(16,28,34,26)(21,30,24,32)

Polycyclic presentation:

2^9,3|a^2=i,b^2=e,e^2=i,f^2=i,b^a=bf,c^a=ci,c^b=cei,d^b=defg,e^a=eh,e^c=ei,e^d=ei,f^a=fi,f^b=fhi,f^c=fhi,f^e=fi,g^a=ghi,g^b=ghi,g^c=ghi,g^e=gi,g^f=gi,h^b=hi,j^d=j^2

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#511825
maximal normal subgroup of	D₃×#511825
maximal quotient of	#72255448

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