#72270119

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13076 ⋊ C₂
Sylow	C₃ ⋊ #495573

Subgroups

	subgroup	G/H
center	C₂	#10800419
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13076	C₂
socle	C₂×C₃	#21408
prime radical	#11665266	C₂
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#13076	D₃
hypocenter	C₃	#495573
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #13076

1564 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)(6,16,17,32)(7,19)(10,22,25,24)(11,27)(15,29,31,30)(18,33)(21,34,23,35)(26,28);(1,2,3)(6,15,17,31)(7,18)(8,20)(10,21,25,23)(11,26)(12,13)(16,29,32,30)(19,33)(22,34,24,35)(27,28);(4,6,7,15,8,16,18,30,9,17,19,31,20,32,33,29)(5,10,11,23,13,22,26,34,14,25,27,21,12,24,28,35);(4,5)(6,10)(7,11)(8,13)(9,14)(12,20)(15,23)(16,22)(17,25)(18,26)(19,27)(21,31)(24,32)(28,33)(29,35)(30,34)

Polycyclic presentation:

2^9,3|b^2=e,c^2=g,d^2=g,e^2=h,f^2=i,h^2=i,c^a=cg,c^b=cefi,d^b=dhi,e^c=eh,e^d=ei,f^a=fi,f^c=fi,g^b=gi,h^c=hi,j^d=j^2

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270183, D₃×#495573
maximal normal subgroup of	#72270183, D₃×#495573

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