#72270445

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12828 ⋊ C₂
Sylow	C₃ ⋊ #500650

Subgroups

	subgroup	G/H
center	C₂	#10830073
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#12828	C₂
socle	C₂×C₃	#24196
hypercenter	#12828	D₃
hypocenter	C₃	#500650
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #12828

1558 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)(6,15,16,33)(7,20)(8,21)(10,24,26,25)(11,29)(13,30)(14,17,32,34)(18,28)(19,22)(23,35,31,27);(2,3)(6,14)(7,19)(8,21)(10,23)(11,28)(13,30)(15,17)(16,32)(18,29)(20,22)(24,35)(25,27)(26,31)(33,34);(4,6,7,15,21,34,20,32,9,16,22,33,8,17,19,14)(5,10,11,25,30,35,29,31,12,26,18,24,13,27,28,23);(2,3)(4,5)(6,10,16,26)(7,18)(8,13)(9,12)(11,22)(14,31,32,23)(15,24,33,25)(17,27,34,35)(19,29)(20,28)(21,30)

Polycyclic presentation:

2^9,3|a^2=g,b^2=e,d^2=g,e^2=hi,h^2=i,b^a=bf,c^a=cg,c^b=ce,d^b=defgh,e^a=ei,e^c=eh,e^d=ehi,f^a=fi,f^b=fi,g^b=gi,h^c=hi,h^d=hi,j^a=j^2,j^c=j^2

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270453
maximal normal subgroup of	#72270453

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