#72270141

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12743 ⋊ C₂
Sylow	C₃ ⋊ #500164

Subgroups

	subgroup	G/H
center	C₂	#10828139
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#12743	C₂
socle	C₂×C₃	#24033
prime radical	#11670494	C₂
hypercenter	#12743	D₃
hypocenter	C₃	#500164
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #12743

2080 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)(7,21)(8,22)(10,20)(11,23)(12,28)(13,29)(15,27)(17,19)(18,32)(24,26)(25,35)(30,31)(33,34);(2,3)(4,7,20,22,9,8,10,21)(5,12,27,29,14,13,15,28)(6,16,30,32,19,17,18,31)(11,23,33,35,24,26,25,34);(4,6,8,17,20,30,21,31,9,19,7,16,10,18,22,32)(5,11,13,23,27,33,28,35,14,24,12,26,15,25,29,34);(4,5)(6,11)(7,13)(8,12)(9,14)(10,15)(16,23)(17,26)(18,25)(19,24)(20,27)(21,29)(22,28)(30,33)(31,34)(32,35)

Polycyclic presentation:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270183, #72270453
maximal normal subgroup of	#72270183, #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, 2, F.5);;
gap> SetPower(coll, 3, F.8*F.9);;
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, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);