#72253803

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#23874 ⋊ C₂
Sylow	C₃ ⋊ #511825

Subgroups

	subgroup	G/H
center	C₂	#10943957
commutator, cosocle	C₂×C₃×#41	C₂⁴
Frattini	C₂×#41	C₂³×D₃
Fitting	C₃×#23874	C₂
socle	C₂×C₃	#32898
hypercenter	#23874	D₃
hypocenter	C₃	#511825
LCS	G ▷ C₂×C₃×#41 ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ #23874

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:

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

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=cfg,d^b=dei,e^d=ei,f^a=fi,f^b=fi,f^c=fh,g^a=gh,g^b=gi,g^c=gh,g^f=gi,h^b=hi,j^c=j^2

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#511825
maximal normal subgroup of	D₃×#511825
maximal quotient of	#72255313

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