#72270136

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#24288 ⋊ C₂
Sylow	C₃ ⋊ #6469085

Subgroups

	subgroup	G/H
center	C₄	D₃×#535
commutator, cosocle	C₂²×C₃×C₄	C₂⁵
Frattini	C₂²×C₄	C₂⁴×D₃
Fitting	C₃×#24288	C₂
socle	C₂×C₃	#34963
hypercenter	#24288	D₃
hypocenter	C₃	#6469085
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #24288

3444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 5 generators on 35 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270183, #72271460
maximal normal subgroup of	#72270183, #72271460

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