#72270422

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#20788 ⋊ C₂
Sylow	C₃ ⋊ #495573

Subgroups

	subgroup	G/H
center	C₂	#10800422
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#20788	C₂
socle	C₂×C₃	#21408
prime radical	#11665265	C₂
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#20788	D₃
hypocenter	C₃	#495573
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2875 ◁ #20788

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

123 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270453, D₃×#495573
maximal normal subgroup of	#72270453, 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, 4, F.7*F.9);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*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, 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> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);