#72253847

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12766 ⋊ C₂
Sylow	C₃ ⋊ #511825

Subgroups

	subgroup	G/H
center	C₂	#10943959
commutator, cosocle	C₂×C₃×#41	C₂⁴
Frattini	C₂×#41	C₂³×D₃
Fitting	C₃×#12766	C₂
socle	C₂×C₃	#32898
hypercenter	#12766	D₃
hypocenter	C₃	#511825
LCS	G ▷ C₂×C₃×#41 ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #12766

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

Polycyclic presentation:

2^9,3|b^2=e,e^2=i,b^a=bg,c^a=ci,c^b=cf,d^b=dei,e^d=ei,f^a=fh,g^c=ghi,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	#72270941

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