#72269953

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #494058

Subgroups

	subgroup	G/H
center	C₂	#10795238
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#60176	C₂
socle	C₂×C₃	#20739
prime radical	C₂×C₃×#576	C₂²
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#60176	D₃
hypocenter	C₃	#494058
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2875 ◁ #60176

1890 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)(6,16,18,19)(7,22)(10,25,14,27)(11,20)(15,32,33,17)(21,34)(24,26,30,31)(28,35);(6,15)(7,21)(8,23)(10,24)(11,28)(13,29)(14,30)(16,32)(17,19)(18,33)(20,35)(22,34)(25,26)(27,31);(2,3)(4,6,7,17,23,16,34,33,9,18,22,32,8,19,21,15)(5,10,11,26,29,25,35,24,12,14,20,31,13,27,28,30);(4,5)(6,14)(7,20)(8,13)(9,12)(10,18)(11,22)(15,30)(16,27)(17,26)(19,25)(21,35)(23,29)(24,33)(28,34)(31,32)

Polycyclic presentation:

2^9,3|b^2=e,d^2=g,e^2=hi,h^2=i,b^a=bg,c^b=cef,d^b=dh,e^a=ei,e^c=eh,e^d=ei,f^a=fi,g^b=gi,h^c=hi,j^b=j^2

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#494058
maximal normal subgroup of	D₃×#494058

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