#72228271

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #497868

Subgroups

	subgroup	G/H
center	C₂	#10813467
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#33463	C₂
socle	C₂×C₃	#22670
prime radical	C₃×#3414	C₂²
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#33463	D₃
hypocenter	C₃	#497868
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #33463

1176 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)(5,12,13,15)(10,14,24,25)(11,27,29,30)(16,28,32,31);(4,6,7,17,21,34,20,35,9,18,23,33,8,19,22,26)(5,11,14,28,12,27,24,32,13,29,25,31,15,30,10,16);(2,3)(5,10,13,24)(6,17,18,33)(7,20)(8,21)(12,25,15,14)(16,31)(19,35,34,26)(22,23)(27,30)(28,32);(4,5)(6,16)(7,10)(8,12)(9,13)(11,26)(14,22)(15,21)(17,30)(18,32)(19,28)(20,25)(23,24)(27,33)(29,35)(31,34)

Polycyclic presentation:

2^9,3|b^2=e,c^2=f,d^2=g,f^2=hi,h^2=i,b^a=bf,c^a=cfh,c^b=cf,d^a=dh,d^b=dg,e^a=ei,e^c=ei,f^a=fh,f^b=fhi,g^a=gi,h^a=hi,h^b=hi,j^b=j^2

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228279
maximal normal subgroup of	#72228279

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